Modeling Financial Market Volatility:
A Component Model Perspective
2018-1
Johan Stax Jakobsen
PhD Thesis
DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS
AARHUS BSS � AARHUS UNIVERSITY � DENMARK
MODELING FINANCIAL MARKET VOLATILITY: A
COMPONENT MODEL PERSPECTIVE
By Johan Stax Jakobsen
A PhD thesis submitted to
School of Business and Social Sciences, Aarhus University,
in partial fulfilment of the requirements of
the PhD degree in
Economics and Business Economics
January 2018
CREATESCenter for Research in Econometric Analysis of Time Series
This version: January 17, 2018 © Johan Stax Jakobsen
PREFACE
This dissertation is the final product of four years of PhD studies at the Department of
Economics and Business Economics, Aarhus School of Business and Social Sciences,
Aarhus University. I am thankful for the financial support and excellent research
facilities provided by the department that have made everything possible. I am also
grateful and proud for being a member of the renowned Center for Research in
Econometric Analysis of Time Series (CREATES) funded by the National Research
Foundation (DNRF78) that has allowed me to attend inspiring courses, taught by
some of the world’s leading scholars, and interesting seminars. Also, the rich group of
researchers has given me the option of valuable feedback for almost any problem.
I will furthermore acknowledge the external support from Augustinus Fonden, Etly
og Jørgen Stjerngrens Fond, Knud Højgaards Fond, Oticon Fonden and Rudolph
Als Fondet that made my two stays at Queensland University of Technology (QUT)
possible.
I would like to extend my gratitude to a number of people that have helped me
and made my life the last four years endlessly more pleasant than it otherwise would
have been without them. First and foremost, I wish to thank my main supervisor Prof.
Timo Teräsvirta who has shown great patience with me and always provided valuable
feedback to paper drafts and research ideas. His kind and insightful help will never
be forgotten.
During the winter months of the last two years, I had the pleasure of visiting
Prof. Stan Hurn and Dr. Annastiina Silvennoinen at the School of Economics and
Finance, QUT in Brisbane, Australia. I appreciate the hospitality and generosity of
the faculty and hope that I someday will return. In addition to Prof. Stan Hurn and Dr.
Annastiina Silvennoinen, I would, in particular, like to thank Prof. Adam Clements
and Prof. Russell Davidson for enlightening discussions. Also, I am grateful to John
Polichronis for allowing me to spend Christmas with him and his family while away
from my own, and to the PhD students at QUT for a lot of memorable weekend trips,
discussions about French politics and much more.
I appreciate the abundance of great colleagues at the Department of Economics
and Business Economics and CREATES. I am grateful to all of them for creating
an outstanding academic and social environment. Special thanks go to Prof. Niels
Haldrup for establishing and directing CREATES and to Solveig Nygaard Sørensen
i
ii
for always being helpful, proof-reading papers and making CREATES run efficiently.
I would also like to thank my fellow PhD students and in particular Alexander, Bo,
Boris, Carsten, Christian, Daniel, Jakob, Jonas, Jorge, Kasper, Mikkel, Simon, Strange,
and Søren for many interesting conversations, social activities and so much more.
It would not have been the same without you. Many of you will be close and dear
friends for the rest of my life.
I would like to put special thanks forward to a group of people that I hold close
to my heart. To Boris, my Bulgarian friend; I promise that I will continue to be a
great friend of you and your country. To Nicolai, my study mate from the first year
at the university and business partner; I am sure that the future will bring plenty of
new projects. To Daniel, my co-author with whom I share many interests; I hope
that the future will bring scientific contributions and plenty of fishing trips. Finally,
I would express my gratitude towards friends and family. Your unconditional love
and support mean everything to me. Special thanks to my twin brother Thomas for
sharing so many great experiences with me. In particular, countless fishing trips have
been peaceful escapes from the hazels and problems encountered during the last
four years.
Johan Stax Jakobsen
Aarhus, November 2017
UPDATED PREFACE
The predefence took place on December 21, 2017. I am grateful to the members
of the assessment committee consisting of Asger Lunde (Aarhus University and
CREATES), Peter Reinhardt Hansen (University of North Carolina and CREATES), and
Esther Ruiz Ortega (Charles III University of Madrid) for their valuable comments
and suggestions. Some of the suggestions have been incorporated into the present
version of the dissertation while other remain for future research.
Johan Stax Jakobsen
Aarhus, January 2018
iii
CONTENTS
Summary vii
Danish summary xi
1 Volatility persistence in the Realized Exponential GARCH model 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Persistence in a multiplicative REGARCH . . . . . . . . . . . . . . . . 4
1.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Empirical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2 Realized EGARCH models with time-varying unconditional variance 652.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.2 Four new Realized EGARCH models . . . . . . . . . . . . . . . . . . . 68
2.3 Estimation and inference . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.4 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.5 A VaR framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.6 Empirical application to stock market volatility . . . . . . . . . . . . . 78
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3 Introducing macro-finance variables into the Realized EGARCH frame-work 1093.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.2 Realized measures of volatility . . . . . . . . . . . . . . . . . . . . . . 111
3.3 Modeling framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.5 Forecasting methodology . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.6 Forecast evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.7 Empirical application . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
v
vi CONTENTS
3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
SUMMARY
This dissertation presents three self-contained papers on the modeling of latent
volatility of financial return series that is one of the most scrutinized research areas
in financial econometrics. This is with good reason since proper understanding
and modeling of volatility are of practical importance in the financial industry in
relation to for example risk management, portfolio allocation and pricing of financial
instruments. The AutoRegressive Conditional Heteroskedasticity (ARCH) model of
Engle (1982) and the Generalized ARCH (GARCH) model of Bollerslev (1986) have
fostered a huge and successful literature able to explain numerous stylized facts
of financial return series such as volatility clustering (Mandelbrot, 1963) and the
leverage effect (Black, 1976). Research has extended the early models in numerous
ways to deal with issues such as structural breaks, long-range dependence, time-
varying parameters, etc.
A successful part of the literature has focused on the development of multi-
component models to adequately capture the complex dynamics of financial volatil-
ity in a parsimonious way. Additive models with a quickly mean-reverting short-run
(high-frequency) and a persistent long-run (low-frequency) component were intro-
duced by Ding and Granger (1996) and Engle and Lee (1999). This notion of a short-
run and a long-run component is a common theme in the component literature and
rationalized using e.g. economic arguments (see e.g. Andersen and Bollerselv (1997)
(heterogeneity in news arrival) or Müller, Dacorogna, Davé, Olsen, Pictet, and von
Weizsäcker (1997) (investor time-horizon)) or structural breaks (see e.g. Lamoureux
and Lastrapes (1990)). More recently, there has been growing focus on multiplicative
component models (Engle and Rangle (2008) and Engle, Ghysels, and Sohn (2013),
among others). This dissertation in particular extends the literature on multiplicative
component models.
All chapters in this dissertation concern themselves with component structures
for modeling latent volatility. The papers extend the Realized Exponential GARCH
model (REGARCH) of Hansen and Huang (2016) in different directions. The RE-
GARCH is a state-of-the-art volatility model that uses realized measures of volatility
for predicting daily latent volatility. The first paper utilizes the idea of the GARCH-
MIDAS model of Engle et al. (2013) to increase the flexibility of the REGARCH in order
to accommodate evident long-range dependence in financial market volatility. The
vii
viii SUMMARY
second paper investigates whether insights from the literature on changing level of
unconditional variance in the GARCH framework are transferable to the Realized
EGARCH framework. The third paper examines the information content in different
macro-finance indicators for predicting latent volatility.
Chapter 1 - Volatility persistence in the Realized Exponential GARCH model (joint
work with Daniel Borup) - introduces parsimonious extensions of the Realized Expo-
nential GARCH model (REGARCH) of Hansen and Huang (2016) to capture evident
high-persistence in the conditional variance process. The extensions decompose the
conditional variance into a short-term and a long-term component. The latter utilizes
mixed-data sampling or a heterogeneous autoregressive structure, avoiding parame-
ter proliferation otherwise incurred by using the classical ARMA structures embedded
in the REGARCH. The proposed models are dynamically complete, which facilitates
multi-period forecasting. A thorough empirical investigation with the exchange-
traded index fund SPY that tracks the S&P 500 Index and 20 individual stocks shows
that our models better capture the autocorrelation structure in volatility. This leads
to substantial improvements in empirical fit and predictive ability (particular beyond
short horizons) relative to the original REGARCH.
Chapter 2 - Realized EGARCH models with time-varying unconditional variance (joint
work with Bo Laursen) - extends the Realized Exponential GARCH model (REGARCH)
of Hansen and Huang (2016) such that the unconditional variance is allowed to
change smoothly as a function of time. The model specification allows the conditional
variance to be multiplicatively decomposed into a stationary and non-stationary part.
The stationary part is specified as a zero mean REGARCH and the non-stationary
part represents the unconditional variance as a flexible function of time. We propose
four parametric alternatives inspired by the existing GARCH-literature: a smooth
transition time-varying structure, a flexible Fourier form, a quadratic spline, and a
cubic spline.
An application using data on the exchange-traded index fund SPY tests the models
empirically with both a forecasting and a Value-at-Risk exercise. The analysis shows
that the introduction of a non-stationary component modeled as a function of time
improves the in-sample fit of the model, but generally fails to provide out-of-sample
improvements.
Chapter 3 - Introducing macro-finance variables into the Realized EGARCH frame-
work - proposes two ways of including macro-finance indicators into the Realized
EGARCH model (REGARCH) of Hansen and Huang (2016). First, an additive speci-
fication, where the exogenous variables are added directly to the GARCH equation.
Secondly, a multiplicative component structure that separates the latent volatility
into a part modeled as a zero mean REGARCH and a part modeling the baseline
ix
volatility as a functions of the exogenous variables.
An empirical investigation with the exchange-traded index fund SPY that tracks
the S&P 500 Index and 20 individual stocks involving three macro-finance variables:
the policy uncertainty index of Baker, Bloom, and Davis (2016), the Arouba-Diebold-
Scotti business condition index of Aruoba, Diebold, and Scotti (2009) and VIX leads
to several interesting results. For the multiplicative decomposition, we realize large
in-sample and modest short horizon out-of-sample gains from including VIX as a
covariate, while the gains are smaller for the additive specification. This stipulates
that a multiplicative specification may be the preferred avenue when incorporating
implied volatility in GARCH type models. For ADS and EPU, we also find more mod-
est evidence of superior in-sample performance, but close to none out-of-sample
gains. Furthermore, our results corroborate that the additional information content
from including exogenous covariates is much smaller when working in a framework
utilizing realized measures of volatility.
x SUMMARY
References
Andersen, T. G., Bollerselv, T., 1997. Heterogeneous information arrivals and return
volatility dynamics: Uncovering the long-run in high frequency returns. The Journal
of Finance 52 (3), 975–1005.
Aruoba, S. B., Diebold, F. X., Scotti, C., 2009. Real-time measurement of business
conditions. Journal of Business & Economic Statistics 27 (4), 417–427.
Baker, S. R., Bloom, N., Davis, S. J., 2016. Measuring economic policy uncertainty. The
Quarterly Journal of Economics.
Black, F., 1976. Studies of stock market volatility changes. Proceedings of the American
Statistical Association, Business and Economics Statistics Section, 177–181.
Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal
of Econometrics 31 (3), 307–327.
Ding, Z., Granger, C. W., 1996. Modeling volatility persistence of speculative returns:
A new approach. Journal of Econometrics 73 (1), 185 – 215.
Engle, R. F., 1982. Autoregressive conditional heteroskedasticity with estimates of the
variance of united kingdom inflation. Econometrica 50 (4), 987–1008.
Engle, R. F., Ghysels, E., Sohn, B., 2013. Stock market volatility and macroeconomic
fundamentals. The Review of Economics and Statistics 95 (3), 776–797.
Engle, R. F., Lee, G., 1999. A long-run and short-run component model of stock
return volatility. In R. F. Engle and H. White (eds.), Cointegration, Causality, and
Forecasting: A Festschrift in Honour of Clive WJ Granger, 475–497.
Engle, R. F., Rangle, J. G., 2008. The Spline-GARCH model for low-frequency volatility
and its global macroeconomic causes. The Review of Financial Studies 21 (3),
1187–1222.
Hansen, P. R., Huang, Z., 2016. Exponential GARCH modeling with realized measures
of volatility. Journal of Business and Economic Statistics 34 (2), 269–287.
Lamoureux, C. G., Lastrapes, W. D., 1990. Persistence in variance, structural change,
and the GARCH model. Journal of Business & Economic Statistics 8, 225–234.
Mandelbrot, B., 1963. The variation of certain speculative prices. The Journal of
Business 36, 394–394.
Müller, U. A., Dacorogna, M. M., Davé, R. D., Olsen, R. B., Pictet, O. V., von Weizsäcker,
J. E., 1997. Volatilities of different time resolutions - analyzing the dynamics of
market components. Journal of Empirical Finance 4 (2), 213 – 239.
DANISH SUMMARY
Denne afhandling præsenterer tre selvstændige artikler omhandlende modellering af
latent volatilitet af finansielle afkastserier, der er et af de vigtigste forskningsområder
inden for finansiel økonometri. Dette er med god grund, da forståelse og modellering
af volatilitet er af praktisk relevans for den finansielle sektor inden for områder såsom
risikostyring, porteføljeallokering og prisfastsættelse af finansielle instrumenter. Den
AutoRegressive Conditional Heteroskedasticity (ARCH) model af Engle (1982) og
den Generalized ARCH (GARCH) model af Bollerslev (1986) har inspireret en stor og
succesfuld litteratur, der er i stand til at forklare en række karakteristika for finansielle
afkastserier såsom volatilitetsklynger (Mandelbrot, 1963) og gearingseffekten (Black,
1976). Forskningen har videreudviklet de tidlige modeller i et utal af retninger for
at tage højde for en række problemer som f.eks. strukturelle brud, long-memory og
tidsvarierende parametre, mv.
En succesfuld del af litteraturen har fokuseret på udviklingen af komponentmo-
deller, der på en simpel måde er i stand til at beskrive de komplekse dynamikker
for finansiel volatilitet. Additive modeller med en kortsigtet (højfrekvent) og persi-
stent langsigtet (lavfrekvent) komponent blev introduceret af Ding og Granger (1996)
og Engle og Lee (1999). Idéen om en kortsigtet og langsigtet komponent er et gen-
nemgående tema i komponentlitteraturen og rationaliseret ved bl.a. økonomiske
argumenter (se f.eks. Andersen og Bollerselv (1997) (heterogenitet i ankomsten af
nyheder) eller Müller et al. (1997) (forskelle i investorers tidshorisont)) eller struk-
turelle brud (se f.eks. Lamoureux og Lastrapes (1990)). Mere nyligt har der været
et voksende fokus på multiplikative komponentmodeller (Engle og Rangle (2008)
og Engle et al. (2013), med flere). Denne afhandling udvider primært litteraturen
vedrørende multiplikative komponentmodeller.
Alle kapitler i denne afhandling omhandler komponentstrukturer for modellering
af latent volatilitet. Artiklerne udvider den Realized Exponential GARCH model (RE-
GARCH) af Hansen og Huang (2016) i forskellige retninger. Denne model er en ’state
of the art’ volatilitetsmodel som bruger realiserede mål for volatilitet til at prediktere
daglig volatilitet. Den første artikel udnytter ideen bag GARCH-MIDAS modellen af
Engle et al. (2013) således, at den foreslåede model er i stand til at approksimere den
evidente long-range afhængighed i finansiel volatilitet. Den anden artikel undersøger,
om resultaterne fra litteraturen omkring niveauskift i den ubetingede forventning
xi
xii DANISH SUMMARY
af varians kan overføres til REGARCH-litteraturen. Den tredje artikel undersøger
informationsindholdet i forskellige makroøkonomiske og finansielle indikatorer for
prediktering af latent volatilitet.
Kapitel 1 - Volatility persistence in the Realized Exponential GARCH model (fælles
med Daniel Borup) - introducerer en simpel udvidelse af den Realized Exponential
GARCH model (REGARCH) af Hansen og Huang (2016), der er i stand til at beskrive
den evidente long-range afhængighed i den betingede varians-proces. Udvidelsen
dekomponerer den betingede varians i en kortsigtet og langsigtet del. Den langsig-
tede komponent udnytter et MIDAS-filter eller en heterogen autoregressiv struktur
således, at det store antal parametre krævet ved den klassiske ARMA struktur inde-
holdt i REGARCH undgås. De foreslåede modeller er dynamisk komplette, hvilket
muliggør forecasting flere perioder frem. En gennemarbejdet empirisk undersøgelse,
der anvender data fra den børshandlede indeksfond SPY, som tracker S&P 500 indek-
set, og 20 individuelle aktier, viser, at de nye modeller bedre er i stand til at matche
autokorrelationsstrukturen for volatilitet. Dette medfører substantiel forbedring af
empirisk fit og prediktiv evne.
Kapitel 2 - Realized EGARCH models with time-varying unconditional variance (fæl-
les med Bo Laursen) - udvider den Realized Exponential GARCH model (REGARCH)
af Hansen og Huang (2016) således, at den ubetingede varians er en funktion af
tid. Modelspecifikationen tillader en multiplikativ dekomponering af den betingede
varians i en stationær og ikke-stationær del. Den stationære del er specificeret som
en REGARCH med forventet værdi lig med nul og den ikke-stationære repræsenterer
den ubetingede varians som en fleksibel funktion af tid. Vi foreslår fire forskellige
parametriske alternativer inspireret af den eksisterende GARCH-litteratur: En smooth
transiton struktur, fleksibel fourier form, en kvadratisk spline og en kubisk spline. En
applikation med data fra den børshandlede fond SPY tester modellerne empirisk ved
hjælp af både en forecasting og en Value-at-Risk øvelse. Analysen viser, at introduk-
tionen af en ikke-stationær komponent specificeret som en funktion af tid forbedrer
in-sample fit, men generelt ikke medfører bedre out-of-sample fit.
Kapitel 3 - Introducing macro-finance variables into the Realized EGARCH framework -
foreslår to måder at inkludere makroøkonomiske og finansielle variable i den Realized
Exponential GARCH model (REGARCH) af Hansen og Huang (2016) og den robuste
udvidelse af Banulescu, Hansen, Huang, og Matei (2014). Den første specifikation er
en additiv version, hvor de eksogene variable er tilføjet GARCH-ligningen. Den anden
specifikation er en multiplikativ komponent struktur, som separerer den latente
volatilitet i en del beskrevet ved en REGARCH med forventet værdi lig med nul og en
del, der specificere baseline volatiliteten som en funktion af de eksogene variable.
En empirisk undersøgelse på SPY, som tracker S&P 500 indekset, og 20 individuel-
xiii
le aktier med tre forskellige makroøkonomiske og finansielle indikatorer: Economic
Policy Uncertainty index (EPU) af Baker et al. (2016), Arouba-Diebold-Scotti business
condition index (ADS) af Aruoba et al. (2009) og VIX fører til flere interessante resulta-
ter. For den multiplikative dekomponering, så finder vi store in-sample og moderat
out-of-sample forbedringer ved at inkludere VIX som en kovariat, mens forbedringer-
ne er mindre for den additive specification. Dette understreger, at den multiplikative
specifikation kan være at foretrække, når man anvender implied volatilitet i GARCH
modeller. For ADS og EPU, så finder vi også moderate evidens for bedre in-sample
performance, men tæt på ingen forbedringer out-of-sample. Derudover, så underbyg-
ger vores resultater, at den yderligere information fra inklusion af eksogene kovariater
er betydelig mindre, når man arbejder i et framework som udnytter realiserede mål
for volatilitet.
xiv DANISH SUMMARY
Litteratur
Andersen, T. G., Bollerselv, T., 1997. Heterogeneous information arrivals and return
volatility dynamics: Uncovering the long-run in high frequency returns. The Journal
of Finance 52 (3), 975–1005.
Aruoba, S. B., Diebold, F. X., Scotti, C., 2009. Real-time measurement of business
conditions. Journal of Business & Economic Statistics 27 (4), 417–427.
Baker, S. R., Bloom, N., Davis, S. J., 2016. Measuring economic policy uncertainty. The
Quarterly Journal of Economics.
Banulescu, G. D., Hansen, P. R., Huang, Z., Matei, M., 2014. Volatility during the finan-
cial crisis through the lens of high frequency data: A Realized EGARCH approach.
Black, F., 1976. Studies of stock market volatility changes. Proceedings of the American
Statistical Association, Business and Economics Statistics Section, 177–181.
Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal
of Econometrics 31 (3), 307–327.
Ding, Z., Granger, C. W., 1996. Modeling volatility persistence of speculative returns:
A new approach. Journal of Econometrics 73 (1), 185 – 215.
Engle, R. F., 1982. Autoregressive conditional heteroskedasticity with estimates of the
variance of united kingdom inflation. Econometrica 50 (4), 987–1008.
Engle, R. F., Ghysels, E., Sohn, B., 2013. Stock market volatility and macroeconomic
fundamentals. The Review of Economics and Statistics 95 (3), 776–797.
Engle, R. F., Lee, G., 1999. A long-run and short-run component model of stock
return volatility. In R. F. Engle and H. White (eds.), Cointegration, Causality, and
Forecasting: A Festschrift in Honour of Clive WJ Granger, 475–497.
Engle, R. F., Rangle, J. G., 2008. The Spline-GARCH model for low-frequency volatility
and its global macroeconomic causes. The Review of Financial Studies 21 (3),
1187–1222.
Hansen, P. R., Huang, Z., 2016. Exponential GARCH modeling with realized measures
of volatility. Journal of Business and Economic Statistics 34 (2), 269–287.
Lamoureux, C. G., Lastrapes, W. D., 1990. Persistence in variance, structural change,
and the GARCH model. Journal of Business & Economic Statistics 8, 225–234.
Mandelbrot, B., 1963. The variation of certain speculative prices. The Journal of
Business 36, 394–394.
xv
Müller, U. A., Dacorogna, M. M., Davé, R. D., Olsen, R. B., Pictet, O. V., von Weizsäcker,
J. E., 1997. Volatilities of different time resolutions - analyzing the dynamics of
market components. Journal of Empirical Finance 4 (2), 213 – 239.
C H A P T E R 1VOLATILITY PERSISTENCE IN THE REALIZED
EXPONENTIAL GARCH MODEL
Daniel BorupAarhus University and CREATES
Johan Stax JakobsenAarhus University and CREATES
Abstract
We introduce parsimonious extensions of the Realized Exponential GARCH model
(REGARCH) to capture evident long-range dependence in the conditional variance
process. The extensions decompose conditional variance into a short-term and a
long-term component. The latter utilizes mixed-data sampling or a heterogeneous
autoregressive structure, avoiding parameter proliferation otherwise incurred by us-
ing the classical ARMA structures embedded in the REGARCH. The proposed models
are dynamically complete, facilitating multi-period forecasting. A thorough empirical
investigation with an exchange-traded fund that tracks the S&P 500 Index and 20
individual stocks shows that our models better capture the autocorrelation structure
of volatility. This leads to substantial improvements in empirical fit and predictive
ability (particularly beyond short horizons) relative to the original REGARCH.
1
2 CHAPTER 1.
1.1 Introduction
The Realized GARCH model (RGARCH) and Realized Exponential GARCH model1
(REGARCH) (Hansen, Huang, and Shek, 2012; Hansen and Huang, 2016) provide an
advantageous structure for the joint modeling of stock returns and realized measures
of their volatility. The models facilitate exploitation of granular information in high-
frequency data by including realized measures, which constitute a much stronger
signal of latent volatility than squared returns (Andersen, Bollerslev, Diebold, and
Labys, 2001, 2003). Various models have been proposed to utilize similar information
with notable innovations including the GARCH-X model (Engle, 2002), the multi-
plicative error model (Engle and Gallo, 2006), and the HEAVY model (Shephard and
Sheppard, 2010).
It is, however, generally recognized that volatility is highly persistent. This persistence
is typically documented via a positive and slowly decaying autocorrelation function
(long-range dependence) or a persistence parameter close to unity, known as the
"integrated GARCH effect". Despite the empirical success of the R(E)GARCH mod-
els, these models do not adequately capture this dependency structure in volatility
(both latent and realized) without proliferation in parameters. Indeed, Hansen and
Huang (2016) point out that the REGARCH does a good job modeling the returns,
but falls short in terms of describing the dynamic properties of the realized mea-
sure. In the class of GARCH models without realized measures, several contributions
have been made to account for these two stylized features. A few notable references
include the Integrated GARCH (Engle and Bollerselv, 1986), the Fractionally Inte-
grated (E)GARCH (Baillie, Bollerslev, and Mikkelsen, 1996; Bollerslev and Mikkelsen,
1996), FIAPARCH (Tse, 1998), regime-switching GARCH (Diebold and Inoue, 2001),
HYGARCH (Davidson, 2004), the Spline-GARCH (Engle and Rangel, 2008), and the
time-varying component GJR-GARCH (Amado and Teräsvirta, 2013). In the class
of R(E)GARCH models, Vander Elst (2015) proposed a fractionally integrated RE-
GARCH, whereas Huang, Liu, and Wang (2016) suggested the addition of a weekly
and a monthly averaged realized measure in the GARCH equation of the RGARCH.
In this paper, we introduce parsimonious extensions of the REGARCH to capture this
evident high persistence by means of a decomposition of the conditional variance. We
utilize a multiplicative decomposition into a short-term and long-term component.
This structure is particularly useful since it enables explicit modelling of a "baseline
volatility", whose level arguably shifts over time, and is the basis around which short-
term movements occur. Such as structure is motivated by Mikosch and Starica (2004),
who show that long-range dependence and the integrated GARCH effect may be
1The REGARCH is a generalization of the RGARCH model with a more flexible specification of theleverage function supposed to better capture the asymmetric relationship between stock returns andvolatility.
1.1. INTRODUCTION 3
explained by level shifts in the unconditional variance, and by Amado and Teräsvirta
(2013), who support this finding empirically in a multiplicative component version of
the GJR-GARCH model.2
The idea of decomposing volatility originates from Engle and Lee (1999) and has pri-
marily been used to empirically support countercyclicality in stock market volatility
(see e.g. Engle et al. (2013) and Dominicy and Vander Elst (2015)). The multiplica-
tive component structure (see e.g. Feng (2004), Engle and Rangel (2008), Engle et al.
(2013) and Laursen and Jakobsen (2017)) is appealing since it is intuitive and facili-
tates parsimonious specifications of a slow-moving component in volatility. Moreover,
it allows for great flexibility as opposed to formal long-memory models employing,
e.g., fractional integration. Whether the high persistence arises due to structural
breaks, fractional integration or another source (see e.g. Lamoureux and Lastrapes
(1990), Diebold and Inoue (2001), Hillebrand (2005), McCloskey and Perron (2013),
and Varneskov and Perron (2017)) our proposed models are able to reproduce the
high persistence of volatility observed in stock return data and alleviate the integrated
GARCH effect, without formally belonging to the class of long-memory models. This
plays an important role in stationarity of the short-term component and existence
of the unconditional variance (which requires the persistence parameter |β| < 1),
but also provides a means to obtain improved multi-step forecasts by reducing the
long-lasting impact of the short-term component and its innovations (via faster con-
vergence to the baseline volatility).
When specifying our models, we retain the dynamics of the short-term component
like those from a first-order REGARCH, but model the long-term component either
via mixed-data sampling (MIDAS) or a heterogeneous autoregressive (HAR) structure.
The former specifies the slow-moving component as a weighted average of weekly
or monthly aggregates of the realized measure with the backward-looking window
and weights estimated from the data. The MIDAS concept was originally introduced
in a regression framework (Ghysels, Santa-Clara, and Valkanov, 2004, 2005; Ghysels,
Sinko, and Valkanov, 2007), allowing for the left-hand and right-hand variables to
be sampled at different frequencies. It has recently been incorporated successfully
into the GARCH framework with the GARCH-MIDAS proposal of Engle et al. (2013).
The latter is motivated by the simple, yet empirically successful HAR model by Corsi
(2009), which approximates the dependencies in volatility by a simple additive cas-
cade structure of a daily, weekly and monthly component of realized measures. Both
our extensions introduce only two or three additional parameters, hence avoid pa-
rameter proliferation otherwise incurred by means of the classical ARMA structures
embedded in the original REGARCH. Moreover, they remain dynamically complete.
2Conrad and Kleen (2016) also show formally that the autocorrelation function of squared returns isbetter captured by a multiplicative GARCH specification rather than its nested GARCH(1,1) model, arisingfrom the persistence in the long-term component.
4 CHAPTER 1.
That is, the models fully characterize the dynamic properties of all variables included
in the model. This property is especially relevant for forecasting purposes, since
it allows for multi-period forecasting. This contrasts GARCH-X models, which only
provide forecasts one period into the future, and related extensions including macroe-
conomic factors who typically rely on questionable assumptions about the included
variables’ dynamics.3
We apply our REGARCH-MIDAS and REGARCH-HAR to the exchange-traded in-
dex fund, SPY, which tracks the S&P 500 Index, and 20 individual stocks and compare
their performances to a quadratic REGARCH-Spline and a fractionally integrated
REGARCH, the FloEGARCH, (Vander Elst, 2015). We find that both our proposed
models better capture the autocorrelation structure of latent and realized volatility
relative to the original REGARCH, which is only able to capture the dependency
over the very short term. This leads to substantial improvements in empirical fit
(log-likelihood and information criteria) and predictive ability, particularly beyond
shorter horizons, when benchmarked to the original REGARCH. We document, ad-
ditionally, that the backward-looking horizon of the HAR specification is too short
to sufficiently capture autocorrelation beyond approximately one month. While the
REGARCH-Spline comes short relative to our proposals (with four-five extra param-
eters), the FloEGARCH performs well. It does, however, not perform better than
our best-performing REGARCH-MIDAS specifications in-sample and lack predictive
accuracy in the short-term. This leaves the REGARCH-MIDAS as a very attractive
model for capturing volatility persistence in the REGARCH framework and improving
forecasting performance.
The remainder of the paper is laid out as follows. Section 1.2 introduces our ex-
tensions to the original REGARCH: the REGARCH-MIDAS and the REGARCH-HAR.
Section 3.4 outlines the associated estimation procedure. Section 1.4 summarizes our
data set, examines the empirical fit and predictive ability of our proposed models, and
introduces a procedure for generating multi-period forecasts. Section 1.5 concludes.
Technical details concerning Proposition 1 are presented in the Appendix.
1.2 Persistence in a multiplicative REGARCH
Let {rt } denote a time series of returns, {xt } a (vector) time series of realized measures,
and {Ft } a filtration so that {rt , xt } is adapted to Ft . We define the conditional mean
by µt = E[rt |Ft−1] and the conditional variance by σ2t = Var[rt |Ft−1]. Our aim is to
allow for more flexible dependence structures in the state-of-the-art specification of
3For instance, the assumption of a random walk (Dominicy and Vander Elst, 2015)), use of outside-generated forecasts (usually from a standard autoregressive specification) of the exogenous variables inthe model (Conrad and Loch, 2015) or the assumption that the long-term component is constant for theforecasting horizon (Engle et al., 2013).
1.2. PERSISTENCE IN A MULTIPLICATIVE REGARCH 5
conditional variance provided by the REGARCH of Hansen and Huang (2016). To that
end, we define
rt =µt +σt zt , (1.1)
where {zt } is an i.i.d. innovation process with zero mean and unit variance, and
assume that the conditional variance can be multiplicatively decomposed into two
components
σ2t = ht g t . (1.2)
We refer to ht as the short-term component, supposed to capture day-to-day (high-
frequency) fluctuations in the conditional variance (see e.g. Engle et al. (2013), and
Wang and Ghysels (2015)). On the contrary, g t is supposed to capture secular (low-
frequency) movements in the conditional variance, henceforth referred to as the
long-term component or baseline volatility. With the multiplicative decomposition
in (1.2), we extend a daily REGARCH(1,1) (with a single realized measure) to
rt =µt +σt zt , (1.3)
loght =β loght−1 +τ(zt−1)+αut−1, (1.4)
log xt = ξ+φ logσ2t +δ(zt )+ut , (1.5)
log g t =ω+ f (xt−2, xt−3, . . . ;η), (1.6)
where f (·;η) is a Ft−1-measurable function, which can be linear or non-linear. The
equations are labelled as the "return equation", the "GARCH equation", the "mea-
surement equation", and the "long-term equation", respectively. For identification
purposes, we have omitted an intercept in (1.4). The leverage functions, τ(·) and
δ(·), facilitate modeling of the dependence between return innovations and volatility
innovations shown to be empirically important (see e.g. Christensen, Nielsen, and
Zhu (2010)). In addition, they play an important role in making the assumption of
independence between zt and ut empirically realistic (Hansen and Huang, 2016).
We adopt the quadratic form of the leverage functions based on the second-order
Hermite polynomial,
τ(z) = τ1z +τ2(z2 −1), (1.7)
δ(z) = δ1z +δ2(z2 −1). (1.8)
The leverage functions have a flexible form and imply E[τ(z)
] = E[δ(z)
] = 0 when
E [z] = 0 and Var[z] = 1. Thus, if |β| < 1, our identification restriction implies that
E[loght
] = 0 such that E[
logσ2t
]= E
[log g t
].4 In the (Quasi-)Maximum Likelihood
analysis below, we employ a Gaussian specification like Hansen and Huang (2016)
4The GARCH equation implies that loght = β j loght− j +∑ j−1
i=0 βi [τ(zt−1−i )+αut−1−i
]such that
loght has a stationary representation if |β| < 1.
6 CHAPTER 1.
with zt ∼ N (0,1) and ut ∼ N (0,σ2u), and zt ,ut mutually and serially independent.5
We check the validity of this approach via a parametric bootstrap in Section 3.4 below.
The return and GARCH equation are canonical in the GARCH literature. In the re-
turn equation, the conditional mean, µt , may be modeled in various ways including
a GARCH-in-Mean specification or simply as a constant.6 Following the latter ap-
proach, we estimate the constant µt = µ. In our multiplicative specification, the
GARCH equation drives the dynamics of the high-frequency part of latent volatility.
The dynamics are specified as a slightly modified version of the EGARCH model
of Nelson (1991) (different leverage function) with the addition of the term αut−1
that relates the latent volatility with the innovation to the realized measure. Hence,
α represents how informative the realized measure is about future volatility. The
persistence parameter β can be interpreted as the AR-coefficient in an AR(1) model
for loght with innovations τ(zt−1)+αut−1.
The measurement equation is the true innovation in the R(E)GARCH, which makes
the model dynamically complete. The equation links the ex-post realized measure
with the ex-ante conditional variance. Discrepancies between the two measures are
expected, since the conditional variance (and returns) refers to a close-to-close mar-
ket interval, whereas the realized measure is computed from a shorter, open-to-close
market interval. Hence, the realized measure is expected to be smaller than the con-
ditional variance on average. Additionally, the realized measure may be an imperfect
measure of volatility. Therefore, the equation includes both a proportional, ξ, and an
exponential, φ, correction parameter. The innovation term, ut , can be seen as the
true difference between ex-ante and ex-post volatility.
Given the high persistence of the conditional variance (documented in the empirical
section below), simply including additional lags in the ARMA structure embedded
in the original REGARCH is not a viable solution, keeping parameter proliferation
in mind (cf. Section 1.4). Instead, we utilize the multiplicative component structure,
which is both intuitively appealing and maintain parsimony. This is motivated by
Mikosch and Starica (2004) who showed that the high persistence can be explained
by level shifts in the unconditional variance (see also Diebold (1986) and Lamoureux
and Lastrapes (1990)). On this basis, Amado and Teräsvirta (2013) proposed a multi-
plicative decomposition of the GJR-GARCH model, where the "baseline volatility"
changes deterministically according to the passage of time. We may, therefore, enable
5Watanabe (2012), Louzis, Xanthopoulos-Sisinis, and Refenes (2013) and Louzis, Xanthopoulos-Sisinis,and Refenes (2014) assumed a skewed t-distribution in their Value-at-Risk applications.
6The mean is typically modeled as a constant since stock market returns generally are found to be closeto serially uncorrelated, see e.g. Ding, Granger, and Engle (1993) and references therein. Sometimes theassumption of zero mean, µ= 0, is imposed for simplicity and may in fact generate better out-of-sampleperformance, see e.g. Hansen and Huang (2016). However, in option-pricing applications a GARCH-in-Mean specification is usually employed, see e.g. Huang, Wang, and Hansen (2017).
1.2. PERSISTENCE IN A MULTIPLICATIVE REGARCH 7
capturing high persistence via the structure proposed above, when the long-term
component in (1.6) is specified as a slow-moving baseline volatility around which
stationary short-term fluctuations occur via the standard GARCH equation. Natu-
rally, this interpretation (and the existence of the unconditional variance) depends
on whether |β| < 1 holds in practice, which may be questionable on the basis on
former evidence for the original REGARCH (confirmed in Section 1.4). However, this
integrated GARCH effect is alleviated in our proposed models, where β is notably
below unity.
Whether high persistence of the conditional variance process arises due to structural
breaks, fractional integration or any other source, the long-term component, if mod-
eled accurately, facilitates high persistence in the REGARCH framework. That is, we
do not explicitly take a stance on the reason for the presence of high persistence. We
resort to this approach rather than developing a formal long-memory model (see e.g.
Bollerslev and Mikkelsen (1996) and Vander Elst (2015)), since prevailing ambiguity
about the origination of long memory somewhat distorts the judgement on the cor-
rect formal modeling. There exists a long list of explanations for long memory in a
time series of which a few are; (i) cross-sectional aggregation of short-memory time
series (Granger, 1980; Abadir and Talmain, 2002; Zaffaroni, 2004; Haldrup and Valdés,
2017), (ii) temporal aggregation across mixed-frequency series (Chambers, 1998),
(iii) aggregation through networks (Schennach, 2013), (iv) hidden cross-section de-
pendence in large-dimensional vector autoregressive systems (Chevillon, Hecq, and
Laurent, 2015), (v) structural breaks (Granger and Ding, 1996; Parke, 1999; Diebold
and Inoue, 2001; Perron and Qu, 2007), (vi) certain types of nonlinearity (Davidson
and Sibbertsen, 2005; Miller and Park, 2010), and (vii) economic agents’ learning
(Chevillon and Mavroeidis, 2017). The various explanations do, however, not nec-
essarily imply the same type of long memory (see e.g. Haldrup and Valdés (2017)
for several definitions). For instance, Parke (1999) formalizes the relation between
structural changes and fractional integration, whereas the expectation formation of
economic agents in Chevillon and Mavroeidis (2017) do not yield fractional integra-
tion, but rather apparent or spurious long memory (see e.g. Davidson and Sibbertsen
(2005) and Haldrup and Kruse (2014)).
For the remainder of this paper, we assume for clarity of exposition that xt is one-
dimensional, containing a single (potentially robust) realized measure consistently
estimating integrated variance (see e.g. (Andersen et al., 2001, 2003)), such as the
realized variance or the realized kernel (Barndorff-Nielsen, Hansen, Lunde, and Shep-
hard, 2008).7 We facilitate level shifts in the baseline volatility via the function f (·;η),
7This assumption is without loss of generality in the sense that additional realized measures (and theirassociated measurement equations) can be added, though we still approximate the long-range dependenceusing only past information of the realized variance, realized kernel or another related consistent estimatorfor integrated variance.
8 CHAPTER 1.
which takes as input past values of the realized measure. We make the dependence
on η explicit in the function f (·;η), and prefer that it is low-dimensional. If f (·;η) is
constant, we obtain the REGARCH as a special case. If f (·;η) is time-varying, past
information may assist in capturing the dependency structure of conditional variance
better, potentially leading to improved in-sample and out-of-sample properties of the
models. We propose in the following sections two ways to parsimoniously formulate
f (·;η) using non-overlapping weekly and monthly averages of the realized measure
to be consistent with the idea of a slow-moving, low-frequency component.8 We
model low-frequency movements in conditional variance using (aggregates of) past
information of the realized measure rather than tying it to macroeconomic state vari-
ables as in Engle et al. (2013) and Dominicy and Vander Elst (2015). Besides proving
empirically preferable (see e.g. Andersen and Varneskov (2014)), such a procedure
renders the model in (1.3)-(1.6) complete with dynamic specifications of all variables
included in the model. Consequently, forecasting can be conducted on the basis of
the (jointly estimated) empirical dynamics, which stands in contrast to incomplete
specifications using exogenous information (from e.g. macroeconomic variables).
The latter usually relies on unrealistic assumptions on the dynamics of the exogenous
variables (e.g. random walks (Dominicy and Vander Elst, 2015)), outside-generated
forecasts (usually from a standard autoregressive specification) of the exogenous
variables in the model (Conrad and Loch, 2015) or the assumption that the long-term
component is constant for the forecasting horizon (Engle et al., 2013). We do, however,
emphasize that our proposed model accommodates well the inclusion of exogenous
information if deemed appropriate.
In the following, we introduce two ways of modeling the low-frequency component,
g t , via formulations of f (·;η) that parsimoniously enable high persistence in the RE-
GARCH formulation, leading to the REGARCH-MIDAS model and the REGARCH-HAR
model.
1.2.1 The Realized EGARCH-MIDAS model
Inspired by the GARCH-MIDAS model of Engle et al. (2013), we consider the following
MIDAS specification of the long-term component
log g t =ω+λK∑
k=1Γk
(γ)
y (N )t−1,k , (1.9)
where Γk(γ)
is a parametrized (by the vector γ) non-negative weighting function sat-
isfying the restriction∑K
k=1Γk(γ)= 1, and y (N )
t ,k = 1N
∑Ni=1 log xt−N (k−1)−i is an N -day
8Excluding information in the realized measure on day t −1 from the function f (·;η) is consistent withthe formulations in the GARCH-MIDAS framework of Engle et al. (2013). The idea is to separate the effectsof the realized measure into two, such that the day-to-day effects is (mainly) contained in the short-termcomponent ht via ut−1 and the long-term component captures the information contained in the realizedmeasure further back in time.
1.2. PERSISTENCE IN A MULTIPLICATIVE REGARCH 9
average of the logarithm of the realized measure. Hence, the value of N determines
the frequency of the data feeding into the low-frequency component. We consider in
the following N ∈ {5,22}, corresponding to weekly and monthly averages.
By estimatingγ, for a given weighting function and choice of K , the term∑K
k=1Γk(γ)
yt−1,k
acts as a filter, which extracts the empirically relevant information from past values
of the realized measure with assigned importance given by the estimated λ. That is,
the lag selection process is allowed to be data driven. In practice, we need to choose
a value for K and a weighting scheme. Conventional weighting schemes are based on
the exponential, exponential Almon lag, or the beta-weight specification. A detailed
discussion can be found in Ghysels et al. (2007), who studied the choice of weighting
function in the context of MIDAS regression models. We employ in the following the
two-parameter beta-weight specification defined by
Γk(γ1,γ2
)= (k/K
)γ1−1 (1−k/K
)γ2−1∑Kj=1
(j /K
)γ1−1 (1− j /K
)γ2−1 (1.10)
due to its flexible form. We restrict γ2 > 1, which ensures a monotonically decreas-
ing weighting scheme and avoid counterintuitive schemes with, e.g., most weight
assigned to the most distant observation (see Engle et al. (2013) and Asgharian,
Christiansen, and Hou (2016) for a similar restriction).9 We then examine a single-
parameter case in which we impose γ1 = 1 and a case where γ1 is a free parameter.
More rich structures for the weighting scheme can obviously be considered by intro-
ducing additional parameters, but we will not explore that route, since one important
aim of the MIDAS models is parsimony.
As long as the weighting function is reasonably flexible, the choice of lag length
of the MIDAS component, K , is of limited importance if chosen reasonably large. The
reason is that the estimated γ assigns the relevant weights to each lag simultaneously
while estimating the entire model. Should one want to determine an ‘optimal’ K , we
simply suggest to estimate the model for a range of values of K and choose that for
which higher values lead to no sizeable gain in the maximized log-likelihood value
(see also the empirical section below).
The REGARCH-MIDAS framework proposed here is easily extendable in several
ways. For instance, a multivariate extension is simply obtained by adding additional
MIDAS components to (1.9). Hence, we may add additional high-frequency based
measures such as the daily range, the realized quarticity (see e.g. Bollerslev, Patton,
and Quaedvlieg (2016)) or additional, different estimators of integrated variance. If
the relationship between macroeconomic variables and volatility is of interest, one
may also include indicators such as GDP and production growth rates, or inflation
9We found in our empirical section below that this restriction was only binding in a few cases.
10 CHAPTER 1.
rates (see e.g. Engle et al. (2013)), despite them being of different frequencies. An-
other direction of interest is the understanding of different aggregation schemes of
higher-frequency variables. For example, by considering a rolling window of non-
overlapping averages, our approach differs slightly from that initially proposed in
Engle et al. (2013) who used overlapping averages in the GARCH-MIDAS context.
1.2.2 The Realized EGARCH-HAR model
Inspired by Corsi (2009), we suggest the following HAR-specification of the long-term
component
log g t =ω+γ11
5
5∑i=1
log xt−i−1 +γ21
22
22∑i=i
log xt−i−1. (1.11)
The argument for this particular lag structure is motived by the heterogeneous market
hypothesis (Müller et al., 1993), which suggests an account of the heterogeneity
in information arrival due to e.g. different trading frequencies of financial market
participants. See Corsi (2009) for a more detailed discussion. This particular choice
of lag structure including the lagged weekly and monthly average of the logarithm
of the realized measure is intuitive and has been empirically successful, but is not
data driven as opposed to the MIDAS lag structure. The lag structure can be seen as a
special case of the step-function MIDAS specification in Forsberg and Ghysels (2007),
which was, indeed, inspired by Corsi (2009).
1.3 Estimation
We estimate the models using (Quasi-)Maximum Likelihood (QML) consistent with
the procedures in Hansen et al. (2012) and Hansen and Huang (2016). The log-
likelihood function can be factorized as
L(r, x;θ) =T∑
t=1`t (rt , xt ;θ) =
T∑t=1
[`t (rt ;θ)+`t (xt |rt ;θ)], (1.12)
where θ = (µ,β,τ1,τ2,α,ξ,φ,δ1,δ2,ω,η,σ2u)′ is the vector of parameters in (1.3)-(1.6),
and `t (rt ;θ) is the partial log-likelihood, measuring the goodness of fit of the return
distribution. Given the distributional assumptions, zt ∼ N (0,1) and ut ∼ N (0,σ2u),
and zt ,ut mutually and serially independent, we have
`t (rt ;θ) =−1
2
[log2π+ logσ2
t + z2t
], (1.13)
`t (xt |rt ;θ) =−1
2
[log2π+ logσ2
u + u2t
σ2u
], (1.14)
where zt = zt (θ) = (rt −µ)/σt . We initialize the conditional variance process to be
equal to its unconditional mean, i.e. logh0 = 0. Alternatively, one can treat logh0 as an
1.3. ESTIMATION 11
unknown parameter and estimate it as in Hansen and Huang (2016), who show that
the initial value is asymptotically negligible. To initialize the long-term component,
log g t , at the beginning of the sample, we simply set past values of log xt equal to
log x1 for the length of the backward-looking horizon in the MIDAS-filter. This is
done to avoid giving our proposed models an unfair advantage by utilizing more
data than the benchmark REGARCH. To avoid inferior local optima in the numerical
optimization, we perturb starting values and re-estimate the parameters for each
perturbation.
1.3.1 Score function
Since the scores define the first order conditions for the maximum-likelihood esti-
mator and facilitate direct computation of standard errors for the coefficients, we
present closed-form expressions for the scores in the following. To simplify nota-
tion, we write τ(z) = τ′a(z) and δ(z) = δ′b(z) with a(z) = b(z) =(z, z2 −1
)′, and
let azt = ∂a(zt )/∂zt and bzt = ∂b(zt )/∂zt . In addition, we define θ1 = (β,τ1,τ2,α)′,θ2 = (ξ,φ,δ1,δ2)′, mt = (loght , a(zt )′,ut )′, and nt = (1, logσ2
t ,b(zt )′)′.
Proposition 1 (Scores). The scores, ∂`∂θ =∑Tt=1
∂`t∂θ , are given from
∂`t
∂θ=
B(zt ,ut )hµ,t −[
zt −δ′ ut
σ2u
bzt
]1σt
B(zt ,ut )hθ1,t
B(zt ,ut )hθ2,t + ut
σ2u
nt
B(zt ,ut )hω,t +D(zt ,ut )gω,t
B(zt ,ut )hη,t +D(zt ,ut )gη,t
12
u2t −σ2
u
σ4u
, (1.15)
where
A(zt ) = ∂ loght+1
∂ loght= (
β−αφ)+ 1
2
(αδ′bzt −τ′azt
)zt , (1.16)
B(zt ,ut ) = ∂`t
∂ loght=−1
2
[(1− z2
t )+ ut
σ2u
(δ′bzt zt −2φ
)], (1.17)
C (zt ) = ∂ loght+1
∂ log g t=−αφ+ 1
2
(αδ′bzt −τ′azt
)zt , (1.18)
D(zt ,ut ) = ∂`t
∂ log g t=−1
2
[(1− z2
t )+ ut
σ2u
(δ′bzt zt −2φ
)]. (1.19)
12 CHAPTER 1.
Furthermore, we have
hµ,t+1 = ∂ loght+1
∂µ= A(zt )hµ,t +
(αδ′bzt −τ′azt
) 1
σt, (1.20)
hθ1,t+1 = ∂ loght+1
∂θ1= A(zt )hθ1,t +mt , (1.21)
hθ2,t+1 = ∂ loght+1
∂θ2= A(zt )hθ2,t +αnt , (1.22)
hω,t+1 = ∂ loght+1
∂ω= A(zt )hω,t +C (zt ), (1.23)
hη,t+1 = ∂ loght+1
∂η= A(zt )hη,t +C (zt )gη,t , (1.24)
where gη,t depends on the specification of f (·;η) and is therefore presented in Appendix
A.1.
By corollary, the score function is a Martingale Difference Sequence (MDS), pro-
vided that E[zt |Ft−1
]= 0, E[
z2t |Ft−1
]= 1, E
[ut |zt ,Ft−1
]= 0, and E[
u2t |zt ,Ft−1
]=σ2
u ,
which is useful for future analysis of the asymptotic properties of the QML estima-
tor.10
1.3.2 Asymptotic Properties
It is commonly acknowledged that the asymptotic analysis of even conventional
GARCH models is challenging (see e.g. Francq and Zakoïan (2010)), causing most
models to be introduced without accompanying asymptotic properties of their esti-
mators. Most recently, the asymptotic theory of the EGARCH(1,1) model was devel-
oped by Wintenberger (2013). Han and Kristensen (2014) and Han (2015) conclude
that inference for the QML estimator is quite robust to the level of persistence in
covariates included in GARCH-X models, irrespective of them being stationary or not.
However, no such analysis has, to our knowledge, been developed for the original RE-
GARCH. The MDS properties following Proposition 1 apply to the original REGARCH
as well, leading Hansen and Huang (2016) to conjecture that the limiting distribution
of the estimators is normal. We follow the same route and leave the development of
the asymptotic theory for estimators of the REGARCH-MIDAS and REGARCH-HAR
for future research. Hence, we conjecture that
pT (θ−θ)
d−→ N (0,T J−1I J−1), (1.25)
where I is the limit of the outer-product of the scores and J is the negative limit of
the Hessian matrix for the log-likelihood function. In practice, we rely on estimates
of these two components in the sandwich formula for computing robust standard
10These are the same conditions as in Hansen and Huang (2016) and we refer the reader hereto forfurther details.
1.4. EMPIRICAL RESULTS 13
errors of the coefficients.
To check the validity of this approach, we employ a parametric bootstrapping tech-
nique (Paparoditis and Politis, 2009) with 999 replications and a sample size of 2,500
observations (approximately 10 years, similar to the size of the rolling in-sample
window used in the forecasting exercise below). Figure A.1 depicts the empirical
standardized distribution of a subset of the estimated parameters.
¿ Insert Figure A.1 about here À
It stands out that the in-sample distribution of the estimated parameters for both the
REGARCH, REGARCH-MIDAS and REGRACH-HAR is generally in agreement with a
standard normal distribution. We also compared the bootstrapped standard errors
with the robust QML standard errors computed from the sandwich-formula in (1.25),
which are reported in the empirical section below. The standard errors were quite
similar, which suggests in conjunction with Figure A.1 that the QML approach and
associated inferences are valid. We do, however, note that the QML standard errors
are slightly smaller on average relative to the bootstrapped standard errors, causing
us to be careful in not putting too much weight on the role of standard errors in the
interpretation of the results below.
1.4 Empirical results
In this section, we examine the empirical fit as well as the forecasting performance of
the REGARCH-MIDAS and REGARCH-HAR, including an outline of the forecasting
procedures involved with the proposed models. We mainly comment on the weekly
REGARCH-MIDAS, since its empirical results are qualitatively similar to those from
the monthly version.
1.4.1 Data
The full sample data set consists of daily close-to-close returns and the daily realized
kernels (RK) of the SPY exchange-traded fund that tracks the S&P 500 Index and 20
individual stocks for the 2002/01-2013/12 period. In the computation of the realized
kernel, we use tick-by-tick data, restrict attention to the official trading hours 9:30:00
and 16:00:00 New York time, and employ the Parzen kernel as in Barndorff-Nielsen,
Hansen, Lunde, and Shephard (2011). See also Barndorff-Nielsen et al. (2008) and
Barndorff-Nielsen, Hansen, Lunde, and Shephard (2009) for additional details.11 For
each stock, we remove short trading days where trading occurred in a span of less
than 20,000 seconds (compared to typically 23,400 for a full trading day). We also
remove data on February 27, 2007, which contains an extreme outlier associated
11The data was kindly provided to us by Asger Lunde.
14 CHAPTER 1.
with a computer glitch on the New York Exchange that day. This leaves a sample
size for each stock of about 3,000 observations. Table A.1 reports summary statistics
of the daily returns and the logarithm of daily realized kernels. Figure A.2 depicts
the evolution of returns, squared returns, realized kernel and the autocorrelation
function (ACF) of the logarithm of the realized kernel for SPY.
¿ Insert Table A.1 about here À
¿ Insert Figure A.2 about here À
We compute outlier-robust estimates of return skewness and kurtosis (Kim and White,
2004; Teräsvirta and Zhao, 2011) along with their conventional estimates. The robust
measures point to negligible skewness and quite mild kurtosis in the return series.
This stands in contrast to the moderately skewed, severely fat-tailed distributions sug-
gested by the conventional measures, corroborating the findings in Kim and White
(2004) that stylized facts of returns series change when using robust estimators.
We estimate the fractional integrated parameter d in the logarithm of the realized
kernel with the two-step exact local Whittle estimator of Shimotsu (2010). Over the
full sample all series have d > 0.5, suggesting that volatility is highly persistent.12
This finding is supported by the slowly decaying ACF of the logarithm of the realized
kernel for SPY. Since the conventional ACF may be biased for the unobserved ACF
of the logarithm conditional variance due to the presence of measurement errors,13
we also compute the instrumented ACF proposed by Hansen and Lunde (2014). We
use the authors’ preferred specification with multiple instruments (four through
ten) and optimal combination. The instrumented ACF show a similar pattern as
the conventional ACF, but points toward an even higher degree of persistence. We
also conducted a (Dickey-Fuller) unit root test across all asset considered using the
instrumented persistence parameter (cf. Table A.2).
¿ Insert Table A.2 about here À
The (biased) conventional least square estimates point to moderate persistence and
strong rejection of a unit root. The persistence parameter is, as expected, notably
higher when using the instrumented variables estimator of Hansen and Lunde (2014),
however the null hypothesis of a unit root remains rejected for all assets. Collectively,
these findings motivate a modeling framework that is capable of capturing a high
12We estimated the parameters with m = bT q c for q =∈ {0.5,0.55, . . . ,0.8}, leading to no alterations of theconclusions obtained for q = 0.65. See also Wenger, Leschinski, and Sibbertsen (2017) for a comprehensiveempirical study on long memory in volatility and the choice of estimator of d .
13The element of microstructure noise is, arguably, low, given the construction of the realized kernel,however sampling error may still be present, causing the differences in the conventional and instrumentedACF.
1.4. EMPIRICAL RESULTS 15
degree of persistence. Given the requirement that |β| < 1, this also motivates a frame-
work that pulls β away from unity. This is where the proposed REGARCH-MIDAS and
REGARCH-HAR prove useful.
1.4.2 In-sample results
In this section, we examine the empirical fit of the proposed REGARCH-HAR and
REGARCH-MIDAS using the full sample of observations for SPY and the 20 individual
stocks. We start out by discussing the choice of lag length for the MIDAS component,
K , in the following subsection.
Choice of lag length, K
As noted above, the REGARCH-HAR utilizes by construction lagged information
equal to four weeks (approximately one month) to describe the dynamics of the
realized measure, whereas the REGARCH-MIDAS allows the researcher to explore
and subsequently choose a suitable lag length, possibly beyond four weeks. For the
original two-parameter setting as well as the single-parameter setting, Figure A.3
depicts the estimated lag weights and associated maximized log-likelihood values of
the weekly REGRACH-MIDAS on SPY for a range of K starting with four lags up to
104 lags (approximately two years).
¿ Insert Figure A.3 about here À
The figure yields a number of interesting insights. First, the maximized log-likelihood
values and associated patterns are very similar across the single-parameter and
two-parameter case. The maximized log-likelihood values initially increase until lag
25-50, after which the values reach a ceiling. This observation is corroborated by the
estimated lag functions in the lower panel of the figure. Their patterns show that
recent information matters the most with the information content decaying to zero
for lags approximately equal to 20 in the two-parameter setting and 25 in the single-
parameter setting. Hence, based on the figure we may conclude that information
up to half a year in the past is most important for explaining the dynamics of the
conditional variance. This is generally supported by a similar analysis using monthly
averages rather than weekly in the MIDAS component, but the monthly specification
seems to indicate that additional past information is relevant (cf. Figure A.4).
Secondly, a REGARCH-MIDAS with information only up to the past four weeks pro-
vides only a slightly greater log-likelihood value than the REGARCH-HAR (cf. Table
A.3 below). This indicates that the step-function approximation in the REGARCH-
HAR does a reasonable job at capturing the information content up to four weeks
in the past. Collectively, however, these findings also suggest that the information
lag in the REGARCH-HAR is too short. Based on these findings, we proceed in the
16 CHAPTER 1.
following with a value of K = 52 for the weekly MIDAS and K = 12 for the monthly
MIDAS uniformly in all subsequent analyses, including the individual stock results.
Note that we choose K larger than what the initial analysis suggests for the weekly
specification, since we want consistency between the weekly and monthly specifica-
tions and greater flexibility when applying the choice to the individual stocks. We do,
however, emphasize that it is free for the researcher to optimize over the choice of K
for each individual asset to achieve an even better fit.
Benchmark models
For comparative purposes, we estimate (using QML) two direct antecedents of
the REGARCH-MIDAS and REGARCH-HAR proposed in this paper. The first is a
REGARCH-Spline (REGARCH-S), with the only difference stemming from the spec-
ification of the long-term component. That is, we consider the quadratic spline
formulation
log g t =ω+ c0t
T+
K∑k=1
ck
(max
{ t
T− tk−1
T,0
})2
, (1.26)
where {t0 = 0, t1, t2, . . . , tK = T } denotes a partition of the time horizon T in K + 1
equidistant intervals. Consequently, the smooth fluctuations in the long-term com-
ponent arises from the (deterministic) passage of time instead of (stochastic) move-
ments in the realized kernel as prescribed by the REGARCH-HAR and REGARCH-
MIDAS.14 The formulation of the long-term component originates from Engle and
Rangel (2008) and is also examined in Engle et al. (2013) and Laursen and Jakobsen
(2017), to which we refer for further details. The number of knots, K , is selected using
the BIC information criterion.15
The second benchmark is the FloEGRACH of Vander Elst (2015), which incorpo-
rates fractional integration in the GARCH equation of the REGARCH in a similar vein
to the development of the FI(E)GARCH model of Baillie et al. (1996) and Bollerslev
and Mikkelsen (1996). The model, thus, explicitly incorporates long-memory via frac-
tionally integrated polynomials in the ARMA structure defined via the parameter d .
In contrast to our proposals and the REGARCH-S, the FloEGARCH do not formulate a
multiplicative component structure. Following Vander Elst (2015), we implement a
FloEGARCH(1,d ,1), which is defined as
rt =µ+σt zt , (1.27)
logσ2t =ω+ (1−β)L−1(1−L)−d (
τ(zt−1)+αut−1)
, (1.28)
log xt = ξ+φ logσ2t +δ(zt )+ut , (1.29)
14When the long-term component is specified as a deterministic component it follows that E[logσ2t ] =
log gt .15In a similar spirit to the choice of K for the REGARCH-MIDAS, we apply the number of knots
determined in the estimation on SPY uniformly in all subsequent analyses.
1.4. EMPIRICAL RESULTS 17
where (1−L)d is the fractional differencing operator. The infinite polynomial can be
written as
(1−β)L−1(1−L)−d =∞∑
n=0
(n∑
m=0βmψ−d ,n−m
)Ln , (1.30)
where ψ−d ,k =ψ−d ,k−1k−1+d
k and ψ−d ,0 = 1. In the implementation, we truncate the
infinite sum at 1,000, similar to Bollerslev and Mikkelsen (1996) and Vander Elst
(2015), and initialize the process similarly to Vander Elst (2015).
For completeness, we also estimate a multiplicative version of the EGARCH(1,1)
model (Nelson, 1991) defined by
rt =µ+σt zt , (1.31)
loght =β loght−1 +τ1zt−1 +α(|zt−1|−
p2/π
), (1.32)
log g t =ω. (1.33)
Results for the S&P 500 Index
In Table A.3, we report estimated parameters, their standard errors, and the associated
maximized log-likelihood values for the models under consideration.
¿ Insert Table A.3 about here ÀWe derive a number of notable findings. First, the multiplicative component struc-
tures lead to substantial increases in the maximized log-likelihood value relative
to the original REGARCH. It is worth noting that the null hypothesis of no MIDAS
component, λ= 0 such that f (·;η) = 0, renders γ1 and γ2 unidentified nuisance pa-
rameters. Hence, assessing the statistical significance of the differences in maximized
log-likelihood values via a standard LR test and a limiting χ2 distribution is infeasible.
We follow conventional approaches (see e.g. Hansen et al. (2012); Engle et al. (2013);
Hansen and Huang (2016)) and comment only on log-likelihood differences relative
to the original REGARCH, but note that comparing twice this difference with the
critical value of the χ2 distribution with appropriate degrees of freedom can be indica-
tive of significance.16 For instance, the LR statistic associated with the log-likelihood
gain of the weekly REGARCH-MIDAS is 92.06, compared to a 5% critical value of
5.99, which strongly indicates significance of the log-likelihood improvement. On a
similar data set, Huang et al. (2016) find a log-likelihood gain of approximately 16.5
points (LR statistic of 32.91), when introducing a HAR modification of the RGARCH
of Hansen et al. (2012).17 Addressing this issue, we nuance our interpretation of the
16Most recently, Conrad and Kleen (2016) have developed a misspecification test for comparison of theGARCH-MIDAS model of Engle et al. (2013) and its nested GARCH model.
17The RGARCH by Hansen et al. (2012) is obtained as a special case of the REGARCH (with similarrealized measures) by a proportionality restriction on the leverage function in the GARCH equation, (1.4),via τ(zt ) =αδ(zt ).
18 CHAPTER 1.
log-likelihood gains by information criteria, which hold the number of parameters
up against the maximized log-likelihood.
The substantial increases in log-likelihood value by only a small increase in the
number of parameters in the REGACRH-MIDAS and REGARCH-HAR lead to system-
atic improvements in information criteria. Despite the noticeably greater number
of parameters in the REGARCH-S, the increase in the log-likelihood value is only
comparable to that of the REGARCH-HAR, leading to a modest improvement in the
AIC, only a slight improvement in the BIC, and even a worsening of the HQIC. The
FloEGARCH comes closest to the REGARCH-MIDAS specifications, but is still short
about seven likelihood points. Since it only introduces one additional parameter, the
information criteria are comparable to those of the REGARCH-MIDAS.18 We have also
considered higher-order versions of the original REGARCH(p,q), with p, q ∈ {1, . . . ,5}.
The best fitting version, the REGARCH(5,5), provides a likelihood gain close to, but
still less than the REGARCH-MIDAS models. This gain is, however, obtained with
the inclusion of an additional eight parameters, causing the information criteria to
deteriorate.19
Secondly, we confirm the finding in the former section that the single-parameter
REGARCH-MIDAS performs comparable to the two-parameter version. Additionally,
for the same number of parameters, the single-parameter REGARCH-MIDAS provides
a considerable 16-point likelihood gain relative to the REGARCH-HAR. This suggests
that the HAR formulation is too short-sighted to fully capture the conditional variance
dynamics (despite providing a substantial gain relative to the original REGARCH)
by using only the most recent month’s realized kernels. The differences of the lag
functions, as depicted in Figure A.5, corroborate this point, by attaching a positive
weight on observations further than a month in the past.
¿ Insert Figure A.5 about here À
The cascade structure as evidenced in Corsi (2009) and Huang et al. (2016) of the
HAR formulation is clear from the figure as well, leading to the conclusion that it con-
stitutes a rather successful, yet suboptimal, approximation of the beta-lag function
used in the MIDAS formulation.
18It is also noteworthy that the FloEGARCH attaches a positive weight to information four years inthe past (1,000 daily lags), whereas the REGARCH-MIDAS only carries information from the last year.This suggests that the outperformance of the REGARCH-MIDAS relative to the FloEGARCH is somewhatconservative.
19It also stands out from Table A.3 that the improvements in maximised value from all models underconsideration arises from a better modeling of the realized measure and not returns, which comes asno surprise given the motivation behind their development and that the original REGARCH is already avery successful model in fitting returns while lacking adequate modelling of the realized measure, as putforward in Hansen and Huang (2016).
1.4. EMPIRICAL RESULTS 19
In Figure A.6, we depict the fitted conditional variance along with the long-term
components of each multiplicative component model under consideration.
¿ Insert Figure A.6 about here À
The long-term component of the REGARCH-MIDAS models appear smooth and do,
indeed, resemble a time-varying baseline volatility. The long-term component in the
REGARCH-HAR is less smooth in contrast to that from the REGACRH-Spline, which
is excessively smooth. To elaborate on the pertinence of the long-term component,
we compute for each model the variance ratio given by
VR = Var[log g t ]
Var[loght g t ], (1.34)
which reveals how much of the variation in the fitted conditional variance can be
attributed to the long-term component. The last row in Table A.3 suggests that the
long-term component contribution is important with more than two-thirds of the
variation for the REGARCH-HAR and REGARCH-MIDAS formulations - noticeably
larger than that for the REGARCH-S. Moreover, the monthly aggregation scheme for
the realized kernel leads to a smoother slow-moving component and, by implication,
a smaller VR ratio.
In terms of parameter estimates and associated standard errors, the values are very
similar across the various REGARCH extensions for most of the intersection of pa-
rameters. The leverage effect appears to be supported in all model formulations,
and estimated values of φ are less than unity with relatively small standard errors,
consistent with the realized measure being computed from open-to-close data and
conditional variance referring to the close-to-close period. Moreover, estimated λ is
close to 0.9 and precisely estimated, suggesting that past information in the realized
kernels are highly informative on conditional variance. The fractional integration pa-
rameter, d , is estimated to 0.65 in the FloEGARCH, confirming the high persistence in
the conditional variance process also suggested by the summary statistics presented
above. Note also that the parameters of the beta-weight function are imprecisely
estimated when γ1 = 1 is not imposed. The reason is that two almost identical weight
structures may be obtained for two (possibly very) different combinations of γ1 and
γ2, leaving the pair imprecisely estimated. Importantly, the estimated values of β are
considerably smaller in our proposed models relative to the original REGARCH. A
similar, but less pronounced result, is obtained for the REGARCH-S. This reduction
in estimated β plays an important role in satisfying the condition that |β| < 1 and
alleviating the integrated GARCH effect. This occurs intuitively since we enable a
flexible level of the baseline volatility which the short-term movements fluctuates
around. Lastly, the measurement equations in the REGARCH-MIDAS and REGARCH-
HAR have smaller estimated residual variances, σ2u , than the original REGARCH. This
20 CHAPTER 1.
may indicate that the new models also provide a better empirical fit of the realized
measure via the multiplicative component specifications proposed here.
Autocorrelation function of conditional variance and realized kernel
In this section, we consider the implications of the REGARCH-HAR and REGARCH-
MIDAS on the ACF of the conditional variance and the realized kernel relative to
the original formulation in REGARCH. We depict in Figure A.7 the simulated and
sample ACF of the logarithm of the conditional variance, logσ2t , for the REGARCH,
REGARCH(5,5), REGARCH-HAR, single-parameter and two-parameter REGARCH-
MIDAS, and FloEGARCH on SPY. The simulated ACF is obtained using the estimated
parameters in Table A.3 with a sample size of 3,750 (approximately 15 years) and
10,000 Monte Carlo replications, whereas the sample ACF is based on the fitted
conditional variance.
¿ Insert Figure A.7 about here À
In general and for a given model, the closer the simulated and sample ACF are to
each other, the larger is the degree of internal consistency in modeling the depen-
dency structure of conditional variance. We note that the original REGARCH is only
able to capture the autocorrelation structure over the very short term. Moreover,
the REGARCH(5,5) does not substantially improve upon the REGARCH. The simu-
lated ACF of the REGARCH-HAR is closer to the sample ACF, but starts diverging at
about lag 30. Only the REGARCH-MIDAS models and the FloEGARCH are capable of
capturing the pattern of the autocorrelation structure over a long horizon. It should
also be noted that the results for the REGARCH-MIDAS is for a particular choice
of K = 52 and K = 12 for the weekly and monthly versions, respectively. Larger val-
ues of K , for a given model, may provide an even greater degree of fit. Indeed, the
monthly REGARCH-MIDAS trades off some fit in the short term for improved ac-
curacy in the long term by using a cruder aggregation scheme of the realized measure.
In Figure A.8, we depict simulated and sample ACFs of the logarithm of the real-
ized kernel for each model to provide an insight into whether the models are able to
capture the autocorrelation structure of the market realized variance.
¿ Insert Figure A.8 about here À
The picture is, expectedly, similar to the one in Figure A.7. With only two or three
additional parameters, the REGARCH-HAR and especially the REGARCH-MIDAS
specifications provide a noticeable increase in the ability to capture the dynamics of
the realized measure relative to the REGARCH. This suggests that the multiplicative
component structure used in the REGARCH-HAR and REGARCH-MIDAS consti-
tutes a very appealing and parsimonious way of capturing high persistence in the
REGARCH framework.
1.4. EMPIRICAL RESULTS 21
Results for individual stocks
The conclusions for the SPY above also apply to individual stocks, for which detailed
results are presented in Appendix A.4. In summary, Table A.4 reports the differences
in log-likelihood values for our proposed models and their benchmarks relative to
the original REGARCH.
¿ Insert Table A.4 about here À
First, the REGARCH-HAR and REGARCH-MIDAS provide systematically large gains
relative to the original REGARCH for all stocks. The two competing benchmarks,
REGARCH-S and FloEGARCH, also provide sizeable gains. Despite this, the REGARCH-
MIDAS specification is the preferred choice for all but two stocks. It also stands out
that the weekly REGARCH-MIDAS consistently outperforms the REGARCH-HAR.
This is generally the case for the monthly REGARCH-MIDAS as well, albeit with a few
exceptions. These exceptions may relate to its crude aggregation scheme, which sac-
rifices too much fit of the autocorrelation structure in the short term for better fit in
the long-term compared to the relatively short-sighted formulation in the REGARCH-
HAR. On this basis, we may conjecture that a framework which incorporates both
daily, weekly and monthly aggregates (sort of hybrid between a HAR and MIDAS
specification) would fit particularly well. The information criteria in the Appendix
corroborate these findings.
In Table A.5 we report the estimated β for all stocks.
¿ Insert Table A.5 about here À
They are all very similar and close to unity in the original REGARCH, but are substan-
tially reduced in the REGARCH-MIDAS and REGARCH-HAR - even more so than for
the S&P 500 Index.
1.4.3 Forecasting with the REGARCH-MIDAS and REGARCH-HAR
In this section, we detail how to generate one- and multi-step forecasts using the
REGARCH-MIDAS and REGARCH-HAR. We note that our models are dynamically
complete. By implication, they are capable of generating multi-period forecasts with-
out imposing (unrealistic) assumptions on the dynamics of the realized measure
(such as the random walk), as usually done in the GARCH-X model that otherwise
are only suitable for one-step ahead forecasting. This feature turns out to be valu-
able below, when we evaluate the predictive ability of the REGARCH-MIDAS and
REGARCH-HAR relative to that of the original REGARCH and the benchmark models.
22 CHAPTER 1.
One-step and multi-step forecasting
Denote by k, k ≥ 1, the forecast horizon measured in days. Our aim is to forecast
the conditional variance k days into the future. To that end, we note that for k = 1
one-step ahead forecasting can be easily achieved directly via the GARCH equation
in (1.4). For multi-period forecasting (k > 1), we note that recursive substitution of
the GARCH equation implies
loght+k =βk loght +k∑
j=1β j−1
(τ(zt+k− j )+αut+k− j
), (1.35)
such that
logσ2t+k = loght+k g t+k =βk loght +
k∑j=1
β j−1(τ(zt+k− j )+αut+k− j
)+ log g t+k .
(1.36)
Multi-period forecasts of logσ2t+k may then be obtained via
logσ2t+k|t ≡ E[logσ2
t+k |Ft ] =βk loght +βk−1 (τ(zt )+αut
)+ log g t+k . (1.37)
Consequently, the contribution of the short-term component to the forecast is easily
computed with known quantities at time t , namely ht ,ut , zt . To obtain g t+k , we gen-
erate recursively, using estimated parameters, the future path of the realized measure
using the measurement equation in (1.5). It is worth noting that for multi-step fore-
cast horizons a lower magnitude of β causes the forecast to converge more rapidly
towards the baseline volatility, determined by (the forecast of) the long-term compo-
nent. Because this baseline volatility is allowed to be time-varying, a lower magnitude
of β is preferable since it generates more flexibility and reduces a long-lasting impact
on the forecast from the most recent ht and its innovation. By implication, the abil-
ity to generate reasonable forecasts of the long-term component is valuable, which
strongly motivates the dynamic completeness of the models.20
Jensen’s inequality stipulates that exp{E[logσ2t+k |Ft ]} 6= E[exp{logσ2
t+k }|Ft ] such that
we need to consider the distributional aspects of logσ2t+k|t to obtain an unbiased
forecast of σ2t+k|t . As a solution, we utilize a simulation procedure with empirical dis-
tributions of zt and ut . Using M simulations and re-sampling the estimated residuals,
the resulting forecast of the conditional variance given by
σ2t+k|t =
1
M
M∑m=1
exp{logσ2t+k|t ,m} (1.38)
20We found, indeed, that setting gt+k = gt leads to notably inferior forecasting performance relative tothe case that exploits the estimated dynamics of the realized kernel.
1.4. EMPIRICAL RESULTS 23
is unbiased. In the implementation, we estimate model parameters on a rolling
basis with 10 years of data (2,500 observations) and leave the remaining (about 500)
observations for (pseudo) out-of-sample evaluation. The empirical distribution of
zt and ut is similarly obtained using the same historical window of observations.
Forecasting with the REGARCH follows directly from the above with log g t+h =ω.
Forecast evaluation
Given the latent nature of the conditional variance, we require a proxy, σ2t , of σ2
t for
forecast evaluation. To that end, we employ the adjusted realized kernel in line with
e.g. Huang et al. (2016) and Sharma and Vipul (2016) given by σ2t = κRKt , where
κ=∑T
t=1 r 2t∑T
t=1 RKt. (1.39)
The adjustment is needed since the realized measure is a measure of open-to-close
variance, whereas the forecast generated by the REGARCH framework measures close-
to-close variance. We compute κ on the basis of the out-of-sample period. A second
implication of using the realized kernel as proxy is that we implicitly restrict ourselves
to the choice of robust loss functions (Hansen and Lunde, 2006; Patton, 2011) when
quantifying the forecast precisions in order to obtain consistent ranking of forecasts.
Let Li ,t+k (σ2t+k ,σ2
t+k|t ) denote the loss function for the i ’th k-step ahead forecast. Two
such robust functions are the Squared Prediction Error (SPE) and Quasi-Likelihood
(QLIKE) loss function given as
L(SPE)i ,t+k (σ2
t+k ,σ2t+k|t ) = (σ2
t+k −σ2t+k|t )2, (1.40)
L(QLIKE)i ,t+k (σ2
t+k ,σ2t+k|t ) = σ2
t+k
σ2t+k|t
− log
σ2t+k
σ2t+k|t
−1. (1.41)
In both cases, a value of zero is obtained for a perfect forecast. The SPE (QLIKE) loss
function penalizes forecast error symmetrically (asymmetrically), and the QLIKE
often gives rise to more power in statistical forecast evaluation procedures, espe-
cially when comparing losses across different regimes (see e.g. Borup and Thyrsgaard
(2017)).
Given the objective of evaluating whether the REGARCH-MIDAS and REGARCH-HAR
provide an improvement in forecasts relative to the REGARCH, we use the Diebold-
Mariano test (Diebold and Mariano, 1995).21 Let the loss differentials from the i ’th
21We acknowledge that the Diebold-Mariano test is technically not appropriate for comparing forecastsof nested models since the limiting distribution is non-standard under the null hypothesis (see e.g. Clarkand McCracken (2001) and Clark and West (2007)). The adjusted mean squared errors of Clark andWest (2007) or the bootstrapping procedure of Clark and McCracken (2015) are appropriate alterationsto standard inferences. However, since we estimate our models on a rolling basis with a finite, fixed
24 CHAPTER 1.
model relative to the REGARCH (abbreviated REG) be given by di ,t = Li ,t+k (σ2t ,σ2
t+k|t )−LREG,t+k (σ2
t ,σ2t+k|t ). The Diebold-Mariano test of equal predictive ability can be con-
ducted using the conventional t-statistic
S = T 1/2 d√V
, (1.42)
where d = T −1 ∑Tt=1 di ,t and V is an estimate of the long-run variance of the loss
differentials. We employ in the following a HAC estimator and follow state-of-the art
good practice by using the data-dependent bandwidth selection by Andrews (1991)
based on an AR(1) approximation and a Bartlett kernel.22 We perform the test against
the alternative that the i ’th forecast losses are smaller than the ones arising from the
original REGARCH and evaluate S in the standard normal distribution.
We also do a Model Confidence Set (MCS) procedure (Hansen, Lunde, and Nason,
2011) to compare the predictive accuracy of all our proposed models to that of the
REGACRH-Spline and the FloEGARCH. For a fixed significance level, α, the proce-
dure identifies the MCS, M∗α, from the set of competing models, M0, which contains
the best models with 1−α probability (asymptotically as the length of the out-of-
sample window approaches infinity). The procedure is conducted recursively based
on an equivalence test for any M ⊆ M0 and an elimination rule, which identifies
and removes a given model from M in case of rejection of the equivalence test. The
equivalence test is based on pairwise comparisons using the statistic Si j in (1.42)
for all i , j ∈ M and the range statistic TM = maxi , j∈M {|Si j |}, where the eliminated
model is identified by argmaxi∈M sup j∈M {Si j }. Following Hansen et al. (2011), we
implement the procedure using a block bootstrap and 105 replications.
Forecasting results
Figure A.9 depicts Theil’s U statistic in terms of the ratio of forecast losses on the
SPY arising from forecasts generated by the original REGARCH to those from the
REGARCH-HAR and the weekly REGARCH-MIDAS (single-parameter) on horizons
k = 1, . . . ,22. It depicts their associated statistical significance, too. Quantitatively and
qualitatively similar results for the remaining MIDAS specifications are left out, but
are available upon request.
window size, the asymptotic framework of Giacomini and White (2006) provides a rigorous justification forproceeding with the Diebold-Mariano test statistic evaluated in a standard normal distribution. See alsoDiebold (2015) for a discussion.
22Admittedly, the high persistence in both the realized kernels and the forecasts generated by themodels under consideration may transmit to the loss differentials, leading to a potential need for a long-memory robust variance estimator in (1.42). In fact, Kruse, Leschinski, and Will (2016) show that thestandard Diebold-Mariano test statistic is most likely oversized in these cases. However, this transmissioncritically depends on the unbiasedness and (loading on) a common long memory between the forecasts(see their Propositions 2-4), leaving a further examination out of the scope of this paper.
1.4. EMPIRICAL RESULTS 25
¿ Insert Figure A.9 about here À
The figure convincingly concludes that both the REGARCH-HAR and REGARCH-
MIDAS improve upon the forecasting performance of the original REGARCH for all
forecast horizons. These improvements tend to grow as the forecast horizon increases
from a few percentages to roughly 30-40% depending on the loss function. This
indicates the usefulness of modeling a slow-moving component, particularly for
forecasting beyond short horizons. In general, the improvements are statistically
significant for all horizons, except for the shorter horizons in the REGARCH-MIDAS
case.23 Table A.6 reports results from a similar analysis on the 20 individual stocks.
¿ Insert Table A.6 about here À
Also on the individual stock basis, both the REGARCH-HAR and REGARCH-MIDAS
provide substantial improvements on the original REGARCH, in particular at longer
horizons. The REGARCH-MIDAS outperforms the REGARCH-HAR with a system-
atically larger improvement for all horizons and based on statistical significance.
Moreover, only a few stocks are not significantly favoring the REGARCH-MIDAS over
the original REGARCH.
Having established the improvement upon the original REGARCH, we turn to a
complete comparison of all our proposed models, the REGARCH-Spline and the
FloEGARCH. Table A.7 reports the percentage of stocks (including SPY) for which a
given model is included in the MCS at an α= 10% significance level.
¿ Insert Table A.7 about here À
The inclusion frequency of our proposed REGARCH-MIDAS models are high and
indicate superiority over all competing models in both the short-term and beyond.
Interestingly, the cruder, monthly aggregation scheme dominates for longer horizons,
whereas the finer, weekly scheme is preferred for short horizons. The REGARCH-
Spline shows moderate improvement over the original REGARCH, but is less fre-
quently included in the MCS compared to our proposed REGARCH-MIDAS and
REGARCH-HAR. The FloEGARCH performs relatively bad for horizons 2,3,4 and 5,
but is increasingly included in the MCS as the forecast horizon increases, reaching
similar performance as the REGARCH-MIDAS models at monthly predictions. These
findings indicate the usefulness of the flexibility obtained via the multiplicative com-
ponent structure as opposed to, e.g., incorporating fractional integration as in the
FloEGARCH.
23We have also examined the models’ predictive ability of cumulative forecasts for a 5,10, and 22horizon. Consistent with the findings for the point forecasts, both the REGARCH-HAR and REGARCH-MIDAS provide substantial and statistically significant improvements relative to the original REGARCH.
26 CHAPTER 1.
1.5 Conclusion
We introduce two extensions of the otherwise successful REGARCH model to capture
the evident high persistence observed in stock return volatility series. Both exten-
sions exploit a multiplicative decomposition of the conditional variance process
into a short-term and a long-term component. The latter is modeled either using
mixed-data sampling or a heterogeneous autoregressive structure, giving rise to the
REGARCH-MIDAS and REGARCH-HAR models, respectively. Both models lead to
substantial in-sample improvements of the REGARCH with the REGARCH-MIDAS
dominating the REGARCH-HAR. Evidently, the backward-looking horizon of the
HAR specification is too short to adequately capture the autocorrelation structure of
volatility for horizons longer than a month.
Our suggested models are dynamically complete, facilitating multi-period forecasting
in contrast to e.g. the GARCH-X or models tying the slow-moving behavior of volatil-
ity to e.g. macroeconomic state variables. Coupled with a lower estimated β and
time-varying baseline volatility, we show in a forecasting exercise that the REGARCH-
MIDAS and REGARCH-HAR leads to significant improvements in predictive ability of
the REGARCH, particularly beyond short horizons.
Similarly to the original REGARCH, our proposed models involve an easy multi-
variate extension, enabling the inclusion of for instance additional realized measures,
macroeconomic variables or event-related dummies (e.g. from policy announce-
ments). Some additional questions remain for future research. On the empirical side,
applications to other asset classes exhibiting high persistence such as commodities,24
bonds or exchange rates, or the use of our proposed models in estimating the (term
structure of) variance risk premia, or investigating the risk-return relationship via
the return equation (see e.g. Christensen et al. (2010)) are of potential interest. On
the theoretical side, development of a misspecification tests for comparison of our
models with the nested REGARCH and asymptotic properties of the QML estimator
would prove very useful.
Acknowledgement
We thank Timo Teräsvirta, Asger Lunde, Peter Reinhard Hansen, Esther Ruiz Ortega,
Bent Jesper Christensen, Jorge Wolfgang Hansen and participants at research semi-
nars at Aarhus University for useful comments and suggestions. We also thank Asger
Lunde for providing cleaned high-frequency tick data. The authors acknowledge
support from CREATES - Center for Research in Econometric Analysis of Time Se-
ries (DNRF78), funded by the Danish National Research Foundation. Some of this
24See e.g. Lunde and Olesen (2013) for an application of the REGARCH to commodities.
1.5. CONCLUSION 27
research was carried out while D. Borup was visiting the Department of Economics,
University of Pennsylvania, and the generosity and hospitality of the department is
gratefully acknowledged. An earlier version of this paper was circulated under the
title "Long-range dependence in the Realized (Exponential) GARCH framework".
28 CHAPTER 1.
1.6 References
Abadir, K., Talmain, G., 2002. Aggregation, persistence and volatility in a macro model.
The Review of Economic Studies 69, 749–779.
Amado, C., Teräsvirta, T., 2013. Modelling volatility by variance decomposition. Jour-
nal of Econometrics 175, 142–153.
Andersen, T. G., Bollerslev, T., Diebold, F. X., Labys, P., 2001. The distribution of realized
exchange rate volatility. Journal of the American Statistical Association 96 (453),
42–55.
Andersen, T. G., Bollerslev, T., Diebold, F. X., Labys, P., 2003. Modeling and forecasting
realized volatility. Econometrica 71 (2), 579–625.
Andersen, T. G., Varneskov, R., 2014. On the informational efficiency of option-implied
and time series forecasts of realized volatility, CREATES Research Paper.
Andrews, D. W. K., 1991. Heteroskedasticity and autocorrelation consistent covariance
matrix estimation. Econometrica 59 (3), 817–858.
Asgharian, H., Christiansen, C., Hou, A. J., 2016. Macro-finance determinants of
the long-run stock-bond correlation: The DCC-MIDAS specification. Journal of
Financial Econometrics 14 (3), 617.
Baillie, R. T., Bollerslev, T., Mikkelsen, H. O., 1996. Fractionally integrated generalized
autoregressive conditional heteroskedasticity. Journal of Econometrics 74 (1), 3–30.
Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N., 2008. Designing
realized kernels to measure the ex post variation of equity prices in the presence of
noise. Econometrica 76 (6), 1481–1536.
Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N., 2009. Realised kernels
in practice: Trades and quotes. Econometrics Journal 12 (3), 1–32.
Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N., 2011. Multivariate
realised kernels: Consistent positive semi-definite estimators of the covariation of
equity prices with noise and non-synchronous trading. Journal of Econometrics
162 (2), 149–169.
Bollerslev, T., Mikkelsen, H. O., 1996. Modeling and pricing long memory in stock
market volatility. Journal of Econometrics 73 (1), 542–547.
Bollerslev, T., Patton, A. J., Quaedvlieg, R., 2016. Exploiting the errors: A simple ap-
proach for improved volatility forecasting. Journal of Econometrics 192 (1), 1–18.
Borup, D., Thyrsgaard, M., 2017. Statistical tests for equal predictive ability across
multiple forecasting methods, CREATES Research Paper.
1.6. REFERENCES 29
Chambers, M. J., 1998. Long memory and aggregation in macroeconomic time series.
International Economic Review 39, 1053–1072.
Chevillon, G., Hecq, A., Laurent, S., 2015. Long memory through marginalization of
large systems and hidden cross-section dependence, working paper.
Chevillon, G., Mavroeidis, S., 2017. Learning can generate long memory. Journal of
Econometrics 198, 1–9.
Christensen, B. J., Nielsen, M. O., Zhu, J., 2010. Long memory in stock market volatility
and the volatility-in-mean effect: The FIEGARCH-M model. Journal of Empirical
Finance 17, 460–470.
Clark, T. E., McCracken, M., 2001. Tests of equal forecast accuracy and encompassing
for nested models. Journal of Econometrics 105 (1), 85–110.
Clark, T. E., McCracken, M., 2015. Nested forecast model comparisons: A new ap-
proach to testing equal accuracy. Journal of Econometrics 186, 160–177.
Clark, T. E., West, K. D., 2007. Approximately normal tests for equal predictive accuracy
in nested models. Journal of Econometrics 138 (1), 291–311.
Conrad, C., Kleen, O., 2016. On the statistical properties of multiplicative GARCH
models, working paper.
Conrad, C., Loch, K., 2015. Anticipating long-term stock market volatility. Journal of
Applied Econometrics 30, 1090–1114.
Corsi, F., 2009. A simple approximate long-memory model of realised volatility. Jour-
nal of Financial Econometrics 7 (2), 174–196.
Davidson, J., 2004. Moment and memory properties of linear conditional het-
eroscedasticity models, and a new model. Journal of Business and Economic
Statistics 22 (1), 16–29.
Davidson, J., Sibbertsen, P., 2005. Generating schemes for long memory processes:
Regimes, aggregation and linearity. Journal of Econometrics 128, 253–282.
Diebold, F. X., 1986. Modeling the persistence of conditional variance: A comment.
Econometric Reviews 5 (1), 51–56.
Diebold, F. X., 2015. Comparing predictive accuracy, twenty years later: A personal
perspective on the use and abuse of the diebold-mariano tests. Journal of Business
and Economic Statistics 33 (1), 1–9.
Diebold, F. X., Inoue, A., 2001. Long memory and regime switching. Journal of Econo-
metrics 105, 131–159.
30 CHAPTER 1.
Diebold, F. X., Mariano, R. S., 1995. Comparing predictive accuracy. Journal of Busi-
ness and Economic Statistics 13 (3), 253–263.
Ding, Z., Granger, C. W., Engle, R. F., 1993. A long memory property of stock market
returns and a new model. Journal of Empirical Finance 1 (1), 83 – 106.
Dominicy, Y., Vander Elst, H., 2015. Macro-driven VaR forecasts: From very high to
very low-frequency data, working paper.
Engle, R., 2002. New frontiers for ARCH models. Journal of Applied Econometrics 17,
425–446.
Engle, R., Bollerselv, T., 1986. Modeling the persistence in conditional variances.
Econometric Reviews 5 (1), 1–50.
Engle, R. F., Gallo, G. M., 2006. A multiple indicators model for volatility using intra-
daily data. Journal of Econometrics 131, 3–27.
Engle, R. F., Ghysels, E., Sohn, B., 2013. Stock market volatility and macroeconomic
fundamentals. The Review of Economics and Statistics 95 (3), 776–797.
Engle, R. F., Lee, G., 1999. A long-run and short-run component model of stock
return volatility. In R. F. Engle and H. White (eds.), Cointegration, Causality, and
Forecasting: A Festschrift in Honour of Clive WJ Granger, 475–497.
Engle, R. F., Rangel, J. G., 2008. The Spline-GARCH model for low-frequency volatility
and its global macroeconomic causes. The Review of Financial Studies 21 (3),
1187–1222.
Feng, Y., 2004. Simultaneously modeling conditional heteroskedasticity and scale
change. Econometric Theory 20 (3), 563–596.
Forsberg, L., Ghysels, E., 2007. Why do absolute returns predict volatility so well?
Journal of Financial Econometrics 5 (1), 31.
Francq, C., Zakoïan, J.-M., 2010. GARCH Models: Structure, statistical inference and
financial applications,. New York: Wiley.
Fuller, W. A., 1996. Introduction to statistical time series, 2nd Edition. Wiley.
Ghysels, E., Santa-Clara, P., Valkanov, R., 2004. The MIDAS touch: Mixed data sampling
regression models, working paper.
Ghysels, E., Santa-Clara, P., Valkanov, R., 2005. There is a risk-return trade-off after all.
Journal of Financial Economics 76 (3), 509 – 548.
Ghysels, E., Sinko, A., Valkanov, R., 2007. MIDAS regressions: Further results and new
directions. Econometric Reviews 26 (1), 53–90.
1.6. REFERENCES 31
Giacomini, R., White, H., 2006. Tests of conditional predictive ability. Econometrica
74 (6), 1545–1578.
Granger, C. W., 1980. Long memory relationships and the aggregation of dynamic
models. Journal of Econometrics 14, 227–238.
Granger, C. W., Ding, Z., 1996. Varieties of long memory models. Journal of Econo-
metrics 73, 61–77.
Haldrup, N., Kruse, R., 2014. Discriminating between fractional integration and spuri-
ous long memory, working Paper.
Haldrup, N., Valdés, J. E. V., 2017. Long memory, fractional integration, and cross-
sectional aggregation. Journal of Econometrics 199, 1–11.
Han, H., 2015. Asymptotic properties of GARCH-X processes. Journal of Financial
Econometrics 13 (1), 188–221.
Han, H., Kristensen, D., 2014. Asymptotic theory for the QMLE in GARCH-X models
with stationary and nonstationary covariates. Journal of Business and Economic
Statistics 32, 416–429.
Hansen, P. R., Huang, Z., 2016. Exponential GARCH modeling with realized measures
of volatility. Journal of Business and Economic Statistics 34 (2), 269–287.
Hansen, P. R., Huang, Z., Shek, H. H., 2012. Realized GARCH: A joint model for return
and realized mesures of volatility. Journal of Applied Econometrics 27, 877–906.
Hansen, P. R., Lunde, A., 2006. Consistent ranking of volatility models. Journal of
Econometrics 131 (1-2), 97–121.
Hansen, P. R., Lunde, A., 2014. Estimating the persistence and the autocorrelation
function of a time series that is measured with error. Econometric Theory 30, 60–93.
Hansen, P. R., Lunde, A., Nason, J. M., 2011. The model confidence set. Econometrica
79 (2), 453–497.
Hillebrand, E., 2005. Neglecting parameter changes in GARCH models. Journal of
Econometrics 129, 121 – 138.
Huang, Z., Liu, H., Wang, T., 2016. Modeling long memory volatility using realized
measures of volatility: A realized HAR GARCH model. Economic Modelling 52,
812–821.
Huang, Z., Wang, T., Hansen, P. R., 2017. Option pricing with the Realized GARCH
model: An analytical approximation approach. Journal of Futures Markets 37 (4),
328–358.
32 CHAPTER 1.
Kim, T.-H., White, H., 2004. On more robust estimation of skewness and kurtosis.
Finance Research Letters 1, 56–73.
Kruse, R., Leschinski, C., Will, C., 2016. Comparing predictive ability under long
memory - with an application to volatility forecasting, CREATES Research Paper,
2016-17.
Lamoureux, C. G., Lastrapes, W. D., 1990. Persistence in variance, structural change,
and the GARCH model. Journal of Business and Economic Statistics 8, 225–234.
Laursen, B., Jakobsen, J. S., 2017. Realized EGARCH model with time-varying uncon-
ditional variance, working Paper.
Louzis, D. P., Xanthopoulos-Sisinis, S., Refenes, A. P., 9 2013. The role of high-frequency
intra-daily data, daily range and implied volatility in multi-period Value-at-Risk
forecasting. Journal of Forecasting 32 (6), 561–576.
Louzis, D. P., Xanthopoulos-Sisinis, S., Refenes, A. P., 2014. Realized volatility models
and alternative Value-at-Risk prediction strategies. Economic Modelling 40, 101 –
116.
Lunde, A., Olesen, K. V., 2013. Modeling and forecasting the distribution of energy
forward returns, CREATES Research Paper, 2013-19.
McCloskey, A., Perron, P., 2013. Memory parameter estimation in the presence of level
shifts and deterministic trends. Econometric Theory 29 (6), 1196–1237.
Mikosch, T., Starica, C., 2004. Nonstationarities in financial time series, the long-range
dependence, and IGARCH effects. The Review of Economics and Statistics 86 (1),
378–390.
Miller, J. I., Park, J. Y., 2010. Nonlinearity, nonstationarity, and thick tails: How they
interact to generate persistence in memory. Journal of Econometrics 155, 83–89.
Müller, U. A., et al., 1993. Fractals and intrinsic time - a challenge to econometricians.
In: 39th International AEA Conference on Real Time Econometrics, 14-15 October
1993, Luxembourg.
Nelson, D. B., 1991. Conditional heteroskedasticity in asset returns: A new approach.
Econometrica 59 (2), 347–370.
Paparoditis, E., Politis, D. N., 2009. Resampling and subsampling for financial time
series. In "Handbook of financial time series", Springer, 983–999.
Parke, W. R., 1999. What is fractional integration? The Review of Economics and
Statistics 81, 632–638.
1.6. REFERENCES 33
Patton, A. J., 2011. Volatility forecast comparison using imperfect volatility proxies.
Journal of Econometrics 160 (1), 246–256.
Perron, P., Qu, Z., 2007. An analytical evaluation of the log-periodogram: Estimate in
the presence of level shifts, working Paper.
Schennach, S. M., 2013. Long memory via networking, working paper.
Sharma, P., Vipul, 2016. Forecasting stock market volatility using Realized GARCH
model: International evidence. The Quarterly Review of Economics and Finance
59, 222–230.
Shephard, N., Sheppard, K., 2010. Realising the future: Forecasting with high-
frequency-based volatility (HEAVY) models. Journal of Applied Econometrics 25,
197–231.
Shimotsu, K., 2010. Exact local whittle estimation of fractional integration with un-
known mean and time trend. Econometric Theory 26 (2), 501–540.
Teräsvirta, T., Zhao, Z., 2011. Stylized facts of return series, robust estimates and three
popular models of volatility. Applied Financal Economics 21 (1-2), 67–94.
Tse, Y., 1998. The conditional heteroskedasticity of the yen-dollar exchange rate.
Journal of Applied Econometrics 13 (1), 49–55.
Vander Elst, H., 2015. FloGARCH: Realizing long memory and asymmetries in returns
volatility, working paper.
Varneskov, R., Perron, P., 2017. Combining long memory and level shifts in modeling
and forecasting the volatility of asset returns. Quantitative Finance, 1–23.
Wang, F., Ghysels, E., 2015. Econometric analysis of volatility component models.
Econometric Theory 32 (2), 362–393.
Watanabe, T., 3 2012. Quantile forecasts of financial returns using Realized GARCH
models. Japanese Economic Review 63 (1), 68–80.
Wenger, K., Leschinski, C., Sibbertsen, P., 2017. Long memory of volatility, working
paper.
Wintenberger, O., 2013. Continuous invertibility and stable qml estimation of the
EGARCH(1,1) model. Scandinavian Journal of Statistics 40, 846–867.
Zaffaroni, P., 2004. Contemporaneous aggregation of linear dynamic models in large
economies. Journal of Econometrics 120, 75–102.
34 CHAPTER 1.
Appendix
A.1 Derivation of score function
First, consider A(zt ) = ∂ loght+1/∂ loght and C (zt ) = ∂ loght+1/∂ log g t . From zt =rt−µσt
, it can easily be shown that
zt
loght= zt
log g t=−1
2zt . (A.1)
From ut = log xt −φ logσ2t −δ(zt ), we find
∂ut
∂ loght= −δ′ ∂b(zt )
∂zt
∂zt
loght−φ=−δ′bzt
∂zt
loght−φ, (A.2)
∂ut
∂ log g t= −δ′ ∂b(zt )
∂zt
∂zt
log g t−φ=−δ′bzt
∂zt
log g t−φ. (A.3)
Similarly, we have
∂τ(zt )
∂ loght= τ′
∂a(zt )
∂zt
∂zt
loght= τ′azt
∂zt
loght, (A.4)
∂τ(zt )
∂ log g t= τ′
∂a(zt )
∂zt
∂zt
log g t= τ′azt
∂zt
log g t. (A.5)
Inserting the above components in the following expressions for A(zt ) and C (zt )
A(zt ) = ∂ loght+1
∂ loght= β+ ∂τ(zt )
∂ loght+α ∂ut
∂ loght, (A.6)
C (zt ) = ∂ loght+1
∂ log g t= ∂τ(zt )
∂ log g t+α ∂ut
∂ log g t, (A.7)
yields
A(zt ) = (β−αφ)+ 1
2
(αδ′bzt −τ′azt
)zt , (A.8)
C (zt ) = −αφ+ 1
2
(αδ′bzt −τ′azt
)zt . (A.9)
Next, we turn to B(zt ,ut ) = ∂`t /∂ loght and D(zt ,ut ) = ∂`t /∂ log g t . The terms loght
and log g t enter the log-likelihood contribution at time t directly due to logσ2t =
loght + log g t and indirectly through z2t and u2
t . Thus, we have
B(zt ,ut ) = −1
2
[1+ ∂z2
t
∂ loght+ 1
σ2u
2ut∂ut
∂ loght
], (A.10)
D(zt ,ut ) = −1
2
[1+ ∂z2
t
∂ log g t+ 1
σ2u
2ut∂ut
∂ log g t
]. (A.11)
A.1. DERIVATION OF SCORE FUNCTION 35
We note that∂`t
∂ log g t= ∂`t
∂ loght=−z2
t . (A.12)
Combining the different expressions yields
B(zt ,ut ) = −1
2
[(1− z2
t )+ ut
σ2u
(δ′bzt zt −2φ
)], (A.13)
D(zt ,ut ) = −1
2
[(1− z2
t )+ ut
σ2u
(δ′bzt zt −2φ
)]. (A.14)
Now, we turn to the derivatives of loght+1 with respect to the different parameters.
For hµ,t+1 = ∂ht+1/∂µ, we have
hµ,t+1 =β∂ loght
∂µ+ ∂τ(zt )
∂µ+α∂ut
∂µ, (A.15)
where
∂τ(zt )
∂µ= ∂τ(zt )
∂zt
∂zt
∂µ= τ′azt
[−1
2zt∂ loght
∂µ− 1
σt
], (A.16)
∂ut
∂µ= −φ∂ loght
∂µ−δ′bzt
∂zt
∂µ
= −φ∂ loght
∂µ−δ′bzt
[−1
2zt∂ loght
∂µ− 1
σt
]. (A.17)
Inserting (A.16) and (A.17) in (A.15) and rearranging yields
hµ,t+1 =[(β−αφ)+ 1
2
[αδ′bzt −τ′azt
]zt
]∂ loght
∂µ+
[αδ′bzt −τ′azt
] 1
σt
= A(zt )hµ,t +[αδ′bzt −τ′azt
] 1
σt. (A.18)
For hθ1,t+1 = ∂ht+1/∂θ1, we have
hθ1,t+1 =β∂ loght
∂θ1+ ∂τ(zt )
∂θ1+α∂ut
∂θ1+ (loght , zt , z2
t −1,ut )′. (A.19)
However, we remember that τ(zt ) and ut only depend on θ1 through loght such that
we can reduce the first three terms into one
hθ1,t+1 = ∂ loght+1
∂ loght
∂ loght
∂θ1+ (loght , zt , z2
t −1,ut )′
= A(zt )hθ1,t +mt . (A.20)
36 CHAPTER 1.
For hθ2,t+1 = ∂ht+1/∂θ2, hω,t+1 = ∂ht+1/∂ω and hη,t+1 = ∂ht+1/∂η, we obtain
hθ2,t+1 = ∂ loght+1
∂ loght
∂ loght
∂θ2+α(1, logσ2
t , zt , z2t −1)′
= A(zt )hθ2,t +nt , (A.21)
hω,t+1 = ∂ loght+1
∂ loght
∂ loght
∂ω+ ∂ loght+1
∂ log g t
∂ log g t
∂ω
= A(zt )hω,t +C (zt ), (A.22)
hη,t+1 = ∂ loght+1
∂ loght
∂ loght
∂η+ ∂ loght+1
∂ log g t
∂ log g t
∂η
= A(zt )hη,t +C (zt )gη,t , (A.23)
respectively. Finally, we turn to the scores. The parameter µ enters the log-likelihood
contribution at time t through loght , zt , and u2t such that
∂`t
∂µ= −1
2hµ,t −
z2t
∂µ− 1
2
1
σ2u
∂u2t
∂µ
= ∂`t
∂ loght
∂ loght
∂µ−
[zt −δ′ ut
σ2u
bzt
]1
σt
= B(zt ,ut )hµ,t −[
zt −δ′ ut
σ2u
bzt
]1
σt. (A.24)
Since θ1 only enters the log-likelihood contribution at time t indirectly through loght ,
an application of the chain-rule yields
∂`t
∂θ1= B(zt ,ut )hθ1,t . (A.25)
The parameter vector θ2 also enters through u2t ,
∂`t
∂θ2= B(zt ,ut )hθ2,t +
ut
σ2u
nt . (A.26)
The parameters ω and η enter through loght and log g t ,
∂`t
∂ω= B(zt ,ut )hω,t +D(zt ,ut )gω,t ,
∂`t
∂η= B(zt ,ut )hη,t +D(zt ,ut )gη,t . (A.27)
The parameter σ2u only enters directly in the log-likelihood contribution such that
∂`t
∂σ2u
= 1
2
u2t −σ2
u
σ2u
. (A.28)
Stacking the above scores,
∂`t
∂θ=
(∂`t
∂µ,∂`t
∂θ′1,∂`t
∂θ′2,∂`t
∂ω,∂`t
∂η′,∂`t
∂σ2u
)′, (A.29)
yields the result in Proposition 1.
A.1. DERIVATION OF SCORE FUNCTION 37
A.1.1 Derivatives specific to the long-run component
In the REGARCH-HAR with f (·;η) given by (1.11), we have η= (γ1,γ2)′ such that
gη,t =(
15
∑5i=1 log xt−i−1
122
∑22i=1 log xt−i−1
). (A.30)
In the two-parameter REGARCH-MIDAS with f (·;η) given by (1.9), we have η =(λ,γ1,γ2)′ such that
gη,t =
∑Kk=1πk (γ1,γ2)yt−1,k∑K
k=1
(γ1−1)(1− k
K
)γ2−1(kK
)γ1−1 ∑Kj=1
(1− j
K
)γ2−1(
kK −
(j
K
)−1)(
jK
)γ1,i −1
[∑Kj=1
(j
K
)γ1−1(1− j
K
)γ2−1]2 yt−1,k
∑Kk=1
(γ2−1)(1− k
K
)γ2−1(kK
)γ1−1 ∑Kj=1
(1− j
K
)γ2−1(1− k
K −(1− j
K
)−1)(
jK
)γ1−1
[∑Kj=1
(j
K
)γ1−1(1− j
K
)γ2−1]2 yt−1,k
.
(A.31)
In the single-parameter REGARCH-MIDAS with f (·;η) given by (1.9), we have η =(λ,γ2)′ such that
gη,t =
∑K
k=1πk (γ1,γ2)yt−1,k∑Kk=1
(γ2−1)(1− k
K
)γ2−1 ∑Kj=1
(1− j
K
)γ2−1(1− k
K −(1− j
K
)−1)
[∑Kj=1
(1− j
K
)γ2−1]2 yt−1,k
. (A.32)
38 CHAPTER 1.
A.2 Figures
This page is intentionally left blank
A.2. FIGURES 39
Figure A.1: Standardized empirical distribution of estimated parametersThis figure depicts the standardized empirical distribution of a subset of the QML parametersusing a parametric bootstrap with resampling of the empirical residuals from the estimationon SPY (Paparoditis and Politis, 2009). We use 999 bootstrap replications and a sample size of2500 observations in the estimation. The left column depicts results for the original REGARCH,the middle column for the weekly, single-parameter REGARCH-MIDAS, and the right columnfor the REGARCH-HAR.
40 CHAPTER 1.
2002 2005 2008 2010 2013-0.15
-0.1
-0.05
0
0.05
0.1
0.15
2002 2005 2008 2010 20130
1
2
3
4
5
6
2002 2005 2008 2010 20130
0.5
1
1.5
2
2.5
3
50 100 150 200 2500
0.2
0.4
0.6
0.8
1
Figure A.2: Summary statistics for SPY daily returns and realized kernelThis figure depicts the evolution of SPY daily returns (upper-left panel), annualized squaredreturns (upper-right panel), annualized realized kernel (lower-left panel), and autocorrelationfunction of the logarithm of the realized kernel (lower-right panel). The solid line indicates theconventional autocorrelation function, whereas the dashed line indicates the instrumentedvariable autocorrelation function of Hansen and Lunde (2014) using their preferred instru-ments (four through ten) and optimal combination.
A.2. FIGURES 41
1 5 10 15 20 25 30 35 40
Lags (weeks)
0
0.1
0.2
0.3
0.4
0.5
0.6
K = 4K = 10K = 15K = 20K = 25K = 30K = 40K = 50K = 75K = 100
1 5 10 15 20 25 30 35 40
Lags (weeks)
0
0.1
0.2
0.3
0.4
0.5
0.6
K = 4K = 10K = 15K = 20K = 25K = 30K = 40K = 50K = 75K = 100
10 20 30 40 50 60 70 80 90 100
K
-5595
-5590
-5585
-5580
-5575
10 20 30 40 50 60 70 80 90 100
K
-5595
-5590
-5585
-5580
-5575
Figure A.3: Backward-looking horizon, K , for weekly REGARCH-MIDASThis figure depicts in the first row panel the estimated lag functions for SPY in the weeklytwo-parameter setting (left panel) and weekly single-parameter setting (right panel) for arange of values of K . The second row panel depicts the maximized log-likelihood values forK = 4, . . . ,104 weeks.
42 CHAPTER 1.
1 5 10 15 20 25
Lags (months)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
K = 5K = 10K = 15K = 20K = 25
1 5 10 15 20 25
Lags (months)
0
0.1
0.2
0.3
0.4
0.5
0.6
K = 5K = 10K = 15K = 20K = 25
5 10 15 20 25
K
-5590
-5585
-5580
-5575
-5570
-5565
-5560
5 10 15 20 25
K
-5590
-5585
-5580
-5575
-5570
-5565
-5560
Figure A.4: Backward-looking horizon, K , for monthly REGARCH-MIDASThis figure depicts in the first row panel the estimated lag functions for SPY in the monthlytwo-parameter setting (left panel) and monthly single-parameter setting (right panel) for arange of values of K . The second row panel depicts the maximized log-likelihood values forK = 4, . . . ,26 months.
A.2. FIGURES 43
1 5 10 15 20 25
Lags (weeks)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Weekly REGARCH-MIDASWeekly REGARCH-MIDAS (single-parameter)Monthly REGARCH-MIDASMonthly REGARCH-MIDAS (single-parameter)REGARCH-HAR
Figure A.5: Estimated SPY weighting functionsThis figure depicts the estimated weighting functions in our proposed models for SPY withK = 52 and K = 12 in the weekly and monthly REGARCH-MIDAS, respectively. Blue lines relateto the weekly REGARCH-MIDAS, red lines relate to the monthly REGARCH-MIDAS, and thegreen line to the REGARCH-HAR. Solid lines refer to the two-parameter weighting function,whereas dashed lines refer to the restricted, single-parameter weighting function.
44 CHAPTER 1.
2002 2005 2008 2010 20130
0.2
0.4
0.6
0.8
1
1.2REGARCH-Spline
2002 2005 2008 2010 20130
0.2
0.4
0.6
0.8
1
1.2REGARCH-HAR
2002 2005 2008 2010 20130
0.2
0.4
0.6
0.8
1
1.2Weekly REGARCH-MIDAS (single-parameter)
2002 2005 2008 2010 20130
0.2
0.4
0.6
0.8
1
1.2Weekly REGARCH-MIDAS
2002 2005 2008 2010 20130
0.2
0.4
0.6
0.8
1
1.2Monthly REGARCH-MIDAS (single-parameter)
2002 2005 2008 2010 20130
0.2
0.4
0.6
0.8
1
1.2Monthly REGARCH-MIDAS
Figure A.6: Evolution of the conditional variance and the long-term componentThis figure depicts the evolution of the fitted conditional variance together with its long-term component from the multiplicative REGARCH modifications contained in Table A.3.The upper-left panel refers to the REGARCH-S, the upper-right panel to the REGARCH-HAR,the middle-left panel to the weekly single-parameter REGARCH-MIDAS, the middle-rightpanel to the weekly two-parameter REGARCH-MIDAS, the lower-left panel to the monthlysingle-parameter REGARCH-MIDAS, and lower-right panel to the monthly two-parameterREGARCH-MIDAS.
A.2. FIGURES 45
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SampleREGARCH
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SampleREGARCH(5,5)
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SampleWeekly REGARCH-MIDAS
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SampleWeekly REGARCH-MIDAS (single-parameter)
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SampleMonthly REGARCH-MIDAS
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SampleMonthly REGARCH-MIDAS (single-parameter)
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SampleREGARCH-HAR
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SampleFloEGARCH
Figure A.7: Simulated and sample autocorrelation function of logσ2t
This figure depicts the simulated (dashed line) and sample (solid line) autocorrelation functionof logσ2
t for the REGARCH, REGARCH(5,5), REGARCH-MIDAS, REGARCH-HAR and the FloE-GARCH. We use the estimated parameters for SPY reported in Table A.3 and K = 52 (K = 12)for the weekly (monthly) REGARCH-MIDAS. See Section 1.4.2 for additional details on theircomputation.
46 CHAPTER 1.
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SampleREGARCH
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SampleREGARCH(5,5)
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SampleWeekly REGARCH-MIDAS
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SampleWeekly REGARCH-MIDAS (single-parameter)
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SampleMonthly REGARCH-MIDAS
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SampleMonthly REGARCH-MIDAS (single-parameter)
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SampleREGARCH-HAR
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SampleFloEGARCH
Figure A.8: Simulated and sample autocorrelation function of logRKtThis figure depicts the simulated (dashed line) and sample (solid line) autocorrelation functionof logRKt for the REGARCH, REGARCH(5,5), REGARCH-MIDAS, REGARCH-HAR and theFloEGARCH. We use the estimated parameters for SPY reported in Table A.3 and K = 52(K = 12) for the weekly (monthly) REGARCH-MIDAS. See Section 1.4.2 for additional details ontheir computation.
A.2. FIGURES 47
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 220.9
1.0
1.1
1.2
1.3
1.4
1.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 220.9
1.0
1.1
1.2
1.3
1.4
1.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 220.9
1.0
1.1
1.2
1.3
1.4
1.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 220.9
1.0
1.1
1.2
1.3
1.4
1.5
Figure A.9: Forecast evaluation of REGARCH-MIDAS and REGARCH-HARThis figure depicts the ratio of forecast losses of the REGARCH-MIDAS and REGARCH-HARto the original REGARCH. Values exceeding unity indicate improvements in predictive abilityof our proposed models. Full circles indicate whether difference in forecast loss (for a givenforecast horizon) is significant on a 5% significance level using a Diebold-Mariano test forequal predictive ability. Empty circles indicate insignificance. See Section 1.4.3 for additionaldetails. The left panel uses the QLIKE loss function in (1.41), whereas the right panel uses theSPE loss function in (1.40). The upper panel reports results for the weekly single-parameterREGARCH-MIDAS and the lower panel for the REGARCH-HAR (results for the remainingREGARCH-MIDAS specifications are similar and are available upon request).
48 CHAPTER 1.
A.3 Tables
This page is intentionally left blank
A.3. TABLES 49
Tab
leA
.1:S
um
mar
yst
atis
tics
for
dai
lyre
turn
san
dre
aliz
edke
rnel
Th
ista
ble
rep
ort
ssu
mm
ary
stat
isti
csfo
rth
ed
aily
retu
rns
and
the
loga
rith
mo
fth
ere
aliz
edke
rnel
.D
aily
retu
rns
and
the
real
ized
kern
elar
ein
per
cen
tage
s.R
obu
stsk
ewn
ess
and
kurt
osis
are
from
Kim
and
Wh
ite
(200
4).T
he
frac
tion
alin
tegr
ated
par
amet
erd
ises
tim
ated
usi
ng
the
two-
step
exac
tlo
calW
hit
tle
esti
mat
or
ofS
him
ots
u(2
010)
and
ban
dw
idth
cho
ice
ofm
=bT
0.65c.
Ret
urn
Log(
RK
)
No.
of
ob
s.M
ean
Std
.D
ev.
Skew
Ro
bu
stSk
ewE
x.K
urt
.R
ob
ust
Ex.
Ku
rt.
Med
ian
Mea
nSt
d.
Dev
.M
edia
nd
SP50
03,
020
0.02
1.32
0.07
-0.0
811
.67
1.03
0.08
-0.3
51.
00-0
.50
0.66
AA
3,00
40.
002.
730.
230.
028.
920.
990.
001.
130.
860.
980.
64A
IG2,
999
-0.0
04.
551.
420.
0154
.58
1.17
-0.0
31.
071.
300.
880.
64A
XP
2,99
40.
072.
440.
550.
0411
.12
1.07
0.02
0.60
1.18
0.38
0.70
BA
2,99
60.
071.
890.
230.
014.
070.
840.
070.
600.
820.
470.
64C
AT2,
998
0.07
2.09
0.11
0.03
5.06
0.92
0.06
0.73
0.82
0.59
0.67
DD
2,99
50.
041.
78-0
.04
0.01
5.68
0.88
0.04
0.51
0.85
0.38
0.63
DIS
2,99
70.
071.
910.
51-0
.02
6.76
0.88
0.06
0.55
0.88
0.38
0.66
GE
3,00
50.
021.
990.
380.
0310
.30
1.06
0.00
0.40
1.05
0.22
0.68
IBM
2,99
60.
031.
530.
200.
016.
870.
870.
020.
100.
83-0
.05
0.65
INT
C3,
016
0.03
2.20
-0.2
2-0
.00
6.09
0.90
0.04
0.85
0.80
0.74
0.63
JNJ
2,99
70.
031.
16-0
.28
0.03
20.0
60.
950.
02-0
.28
0.86
-0.4
30.
68K
O2,
999
0.04
1.24
0.32
-0.0
211
.96
0.92
0.04
-0.1
00.
81-0
.22
0.63
MM
M2,
992
0.05
1.45
-0.0
60.
025.
540.
970.
060.
130.
800.
030.
64M
RK
2,99
40.
031.
80-1
.21
0.04
24.1
80.
870.
030.
380.
850.
260.
61M
SFT
3,01
60.
031.
810.
370.
028.
940.
960.
000.
460.
810.
320.
63P
G2,
998
0.04
1.14
-0.0
20.
016.
740.
920.
03-0
.18
0.77
-0.3
00.
61V
Z2,
995
0.03
1.57
0.34
-0.0
37.
370.
900.
050.
310.
890.
140.
67W
HR
2,99
20.
072.
520.
400.
035.
140.
96-0
.01
1.01
0.86
0.92
0.58
WM
T3,
001
0.03
1.31
0.30
-0.0
35.
570.
880.
020.
050.
80-0
.09
0.65
XO
M3,
001
0.05
1.60
0.34
-0.0
113
.37
0.81
0.07
0.24
0.86
0.14
0.68
50 CHAPTER 1.
Table A.2: Persistence parameters (π) and unit root tests (DF)This table reports estimated autoregressive persistence parameters, π, and unit roottests, DF. The first column contains the conventional least squares estimator, whereasthe following two columns contain the instrumented variables estimator from Hansenand Lunde (2014) using the first lag as instrument and their preferred specifica-tion (four through ten) with optimal combination, respectively. The following threecolumns contain the Dickey-Fuller unit root test using each estimate of the per-sistence parameter. The 1%, 5% and 10% critical values are -20.7, -14.1 and -11.3,respectively (see Fuller (1996), Table 10.A.1).
πOLS π1 π4:10 DFOLS DF1 DF4:10
SP500 0.883 0.959 0.985 -354.3 -124.8 -45.8
AA 0.865 0.961 0.985 -405.3 -116.6 -44.8AIG 0.919 0.966 0.990 -242.4 -103.1 -30.5AXP 0.926 0.980 0.992 -222.7 -59.0 -23.5BA 0.847 0.956 0.987 -458.6 -131.7 -37.4CAT 0.866 0.949 0.988 -400.6 -151.9 -35.6DD 0.856 0.952 0.983 -431.6 -143.8 -51.8DIS 0.866 0.956 0.986 -401.6 -132.3 -41.4GE 0.904 0.969 0.990 -287.6 -93.8 -30.6IBM 0.870 0.959 0.983 -389.5 -122.3 -52.0INTC 0.869 0.951 0.985 -395.4 -148.0 -45.5JNJ 0.852 0.955 0.988 -443.6 -134.3 -37.0KO 0.836 0.953 0.985 -492.7 -140.4 -45.7MMM 0.833 0.940 0.981 -499.5 -178.2 -57.3MRK 0.815 0.942 0.983 -552.7 -174.7 -49.6MSFT 0.857 0.951 0.981 -429.7 -146.7 -56.3PG 0.818 0.937 0.980 -546.2 -188.9 -58.5VZ 0.861 0.961 0.987 -414.7 -118.0 -38.5WHR 0.823 0.938 0.986 -528.4 -186.6 -41.5WMT 0.844 0.957 0.985 -467.4 -127.8 -43.8XOM 0.878 0.954 0.980 -366.7 -137.8 -59.4
A.3. TABLES 51
Tab
leA
.3:F
ull
sam
ple
resu
lts
for
SPY
Th
ista
ble
rep
orts
esti
mat
edp
aram
eter
s,ro
bu
stst
and
ard
erro
rs(u
sin
gth
esa
nd
wic
hfo
rmu
laan
dre
por
ted
inp
aren
thes
es),
nu
mb
erof
par
amet
ers
(p),
info
rmat
ion
crit
eria
,var
ian
cera
tio
fro
m(1
.34)
and
par
tial
,as
wel
las
full
max
imiz
edlo
g-lik
elih
oo
dva
lue
for
each
mo
del
un
der
con
sid
erat
ion
.Res
ult
sfo
rth
eR
EG
AR
CH
-MID
AS
are
for
K=
52(K
=12
)in
the
wee
kly
(mo
nth
ly)
case
.
EG
AR
CH
RE
GA
RC
HR
EG
AR
CH
-MID
AS
RE
GA
RC
H-M
IDA
SR
EG
AR
CH
-MID
AS
RE
GA
RC
H-M
IDA
SR
EG
AR
CH
-HA
RR
EG
AR
CH
-SFl
oE
GA
RC
H(w
eekl
y)(w
eekl
y)(m
on
thly
)(m
on
thly
)(s
ingl
e-p
aram
eter
)(s
ingl
e-p
aram
eter
)
µ0.
020(
0.01
35)
0.01
6(0.
0127
)0.
015(
0.01
44)
0.01
5(0.
0143
)0.
016(
0.01
43)
0.01
6(0.
0143
)0.
014(
0.01
44)
0.02
4(0.
0143
)0.
015(
0.01
01)
β0.
981(
0.00
25)
0.97
2(0.
0036
)0.
761(
0.01
66)
0.84
2(0.
0118
)0.
872(
0.00
98)
0.88
0(0.
0094
)0.
734(
0.01
80)
0.94
3(0.
0058
)0.
176(
0.02
74)
α0.
121(
0.01
44)
0.33
8(0.
0225
)0.
337(
0.02
74)
0.32
9(0.
0250
)0.
324(
0.02
39)
0.32
4(0.
0238
)0.
355(
0.02
70)
0.32
4(0.
0216
)0.
370(
0.02
26)
ξ−0
.265
(0.0
267)
−0.2
71(0
.026
9)−0
.270
(0.0
269)
−0.2
69(0
.026
9)−0
.269
(0.0
269)
−0.2
72(0
.026
8)−0
.264
(0.0
264)
−0.2
74(0
.026
7)σ
2 u0.
155(
0.00
58)
0.15
0(0.
0057
)0.
150(
0.00
57)
0.15
0(0.
0057
)0.
150(
0.00
57)
0.15
1(0.
0057
)0.
153(
0.00
57)
0.15
0(0.
0057
)τ
1−0
.138
(0.0
118)
−0.1
48(0
.007
4)−0
.170
(0.0
084)
−0.1
66(0
.008
1)−0
.164
(0.0
079)
−0.1
63(0
.007
9)−0
.171
(0.0
085)
−0.1
50(0
.007
5)−0
.170
(0.0
083)
τ2
0.04
0(0.
0049
)0.
047(
0.00
55)
0.04
5(0.
0053
)0.
044(
0.00
51)
0.04
4(0.
0051
)0.
047(
0.00
56)
0.04
1(0.
0051
)0.
051(
0.00
54)
δ1
−0.1
13(0
.008
3)−0
.115
(0.0
083)
−0.1
15(0
.008
3)−0
.115
(0.0
083)
−0.1
15(0
.008
3)−0
.114
(0.0
083)
−0.1
12(0
.008
4)−0
.115
(0.0
082)
δ2
0.04
9(0.
0059
)0.
051(
0.00
60)
0.05
0(0.
0059
)0.
050(
0.00
59)
0.05
0(0.
0059
)0.
051(
0.00
60)
0.05
0(0.
0062
)0.
051(
0.00
59)
φ0.
962(
0.02
53)
0.96
8(0.
0167
)0.
969(
0.01
87)
0.97
0(0.
0198
)0.
970(
0.02
01)
0.96
4(0.
0163
)0.
961(
0.02
32)
0.96
9(0.
0231
)ω
0.05
8(0.
1632
)−0
.089
(0.1
255)
0.24
3(0.
0397
)0.
225(
0.04
58)
0.22
2(0.
0499
)0.
213(
0.05
15)
0.23
5(0.
0386
)0.
175(
0.23
66)
−0.0
92(0
.167
0)λ
0.94
7(0.
0298
)0.
906(
0.03
27)
0.91
4(0.
0426
)0.
888(
0.03
97)
γ1
0.02
5(0.
4475
)−0
.518
(0.7
866)
0.29
4(0.
0429
)γ
26.
337(
6.33
25)
12.5
45(1
.834
9)2.
063(
2.84
31)
8.50
8(1.
4365
)0.
620(
0.04
61)
d0.
649(
0.01
42)
p5
1114
1314
1313
1812
logL
−5,6
23.5
5−5
,577
.52
−5,5
78.0
2−5
,577
.52
−5,5
78.2
3−5
,595
.10
−5,5
89.4
4−5
,584
.65
logL
p−4
,213
.84
−4,1
48.7
1−4
,159
.70
−4,1
57.4
3−4
,156
.60
−4,1
56.1
0−4
,160
.22
−4,1
48.6
2−4
,162
.54
AIC
11,2
69.1
011
,183
.04
11,1
82.0
311
,183
.04
11,1
82.4
611
,216
.20
11,2
14.8
711
,193
.29
BIC
11,3
35.2
411
,267
.23
11,2
60.2
011
,267
.22
11,2
60.6
211
,294
.37
11,3
23.1
111
,265
.45
HQ
IC11
,423
.39
11,3
79.4
111
,364
.37
11,3
79.4
011
,364
.79
11,3
98.5
311
,467
.34
11,3
61.6
0V
R0.
730.
650.
610.
590.
740.
40
52 CHAPTER 1.Tab
leA
.4:Differen
cein
maxim
izedlo
g-likeliho
od
relativeto
RE
GA
RC
HT
his
table
repo
rtsth
ed
ifferences
inth
em
aximized
log-likelih
oo
dvalu
esfo
ro
ur
pro
po
sedm
od
elsan
dth
eR
EG
AR
CH
-Splin
ean
dFlo
EG
AR
CH
relativeto
the
origin
alRE
GA
RC
H.P
ositive
values
ind
icateim
provem
ents
inem
piricalfi
t.We
repo
rtresu
ltsfo
rall20
ind
ividu
alstocks
and
inclu
de
SPY
for
com
parative
pu
rpo
ses.Gray
shad
edareas
ind
icateth
em
od
elwith
the
high
estlikeliho
od
gainrelative
toth
eR
EG
AR
CH
.
RE
GA
RC
H-M
IDA
SR
EG
AR
CH
-MID
AS
RE
GA
RC
H-M
IDA
SR
EG
AR
CH
-MID
AS
RE
GA
RC
H-H
AR
RE
GA
RC
H-S
FloE
GA
RC
H(w
eekly)(w
eekly)(m
on
thly)
(mo
nth
ly)(sin
gle-param
eter)(sin
gle-param
eter)
SP500
46.045.5
46.045.3
28.534.1
38.9
AA
46.545.0
41.340.9
35.928.4
40.1A
IG126.2
120.0116.3
112.5119.7
103.3123.2
AX
P41.6
40.941.6
41.427.4
35.840.5
BA
43.542.4
36.936.9
28.227.7
39.4C
AT50.6
47.334.8
34.441.3
23.542.6
DD
34.931.0
30.630.2
19.932.9
29.3D
IS53.0
50.737.6
36.838.4
34.943.7
GE
40.940.6
40.140.1
28.037.4
41.3IB
M31.1
27.322.5
22.519.5
19.916.9
INT
C63.4
60.450.8
50.845.6
43.352.7
JNJ
30.830.8
22.221.5
19.223.4
26.7K
O41.6
37.531.9
31.035.3
31.434.0
MM
M35.0
32.726.2
25.922.2
20.732.0
MR
K31.3
27.723.6
23.021.0
36.027.5
MSF
T47.6
43.641.4
41.135.9
36.436.5
PG
46.742.3
28.427.7
37.822.5
31.3V
Z29.0
28.923.7
23.714.7
17.320.4
WH
R71.8
69.169.2
68.967.0
48.058.4
WM
T40.6
36.129.4
29.328.5
24.128.4
XO
M44.0
40.326.0
25.335.8
22.816.2
SP500
46.045.5
46.045.3
28.534.1
38.9
Mean
47.444.8
39.138.5
35.733.5
39.0
A.3. TABLES 53
Tab
leA
.5:E
stim
ated
βac
ross
vari
ou
sR
EG
AR
CH
spec
ifica
tio
ns
Th
ista
ble
rep
orts
esti
mat
edβ
for
our
pro
pos
edm
odel
san
dth
eR
EG
AR
CH
-Sp
line
and
FloE
GA
RC
Hre
lati
veto
the
orig
inal
RE
GA
RC
H.W
ere
por
tres
ult
sfo
ral
l20
ind
ivid
ual
sto
cks
and
incl
ud
eSP
Yfo
rco
mp
arat
ive
pu
rpo
ses.
RE
GA
RC
HR
EG
AR
CH
-MID
AS
RE
GA
RC
H-M
IDA
SR
EG
AR
CH
-MID
AS
RE
GA
RC
H-M
IDA
SR
EG
AR
CH
-HA
RR
EG
AR
CH
-SFl
oE
GA
RC
H(w
eekl
y)(w
eekl
y)(m
on
thly
)(m
on
thly
)(s
ingl
e-p
aram
eter
)(s
ingl
e-p
aram
eter
)
SP50
00.
972
0.76
10.
842
0.87
20.
880
0.73
40.
943
0.17
6
AA
0.97
20.
649
0.71
10.
820
0.82
40.
638
0.93
30.
195
AIG
0.97
20.
577
0.62
30.
742
0.73
70.
578
0.86
40.
121
AX
P0.
987
0.71
00.
790
0.86
10.
865
0.81
10.
943
0.19
5B
A0.
977
0.60
60.
656
0.83
70.
836
0.58
30.
936
0.13
3C
AT0.
973
0.55
30.
584
0.82
50.
831
0.53
70.
941
0.10
4D
D0.
972
0.62
30.
710
0.86
40.
871
0.59
10.
932
0.17
9D
IS0.
981
0.52
00.
554
0.83
20.
841
0.50
80.
942
0.08
0G
E0.
982
0.81
70.
742
0.84
30.
845
0.80
10.
935
0.11
7IB
M0.
974
0.61
20.
638
0.89
10.
893
0.56
80.
948
0.16
1IN
TC
0.97
20.
602
0.65
40.
812
0.81
20.
565
0.91
30.
131
JNJ
0.97
60.
627
0.62
00.
850
0.85
60.
570
0.95
00.
093
KO
0.97
30.
562
0.61
70.
827
0.82
90.
558
0.93
20.
121
MM
M0.
967
0.58
70.
634
0.86
30.
856
0.55
40.
938
0.13
7M
RK
0.97
10.
552
0.65
70.
864
0.84
90.
494
0.92
60.
108
MSF
T0.
968
0.61
30.
689
0.82
60.
834
0.58
50.
917
0.15
8P
G0.
961
0.54
70.
567
0.83
10.
837
0.52
10.
930
0.13
8V
Z0.
979
0.64
90.
691
0.87
50.
871
0.59
90.
953
0.15
6W
HR
0.96
30.
554
0.63
10.
745
0.74
20.
531
0.90
50.
092
WM
T0.
979
0.53
60.
582
0.85
20.
850
0.50
00.
947
0.11
3X
OM
0.96
70.
598
0.61
00.
873
0.88
00.
561
0.94
70.
150
Mea
n0.
973
0.61
20.
657
0.83
80.
840
0.59
00.
932
0.13
6
54 CHAPTER 1.
Table
A.6:Fo
recastevaluatio
nfo
rin
divid
ualsto
cksT
his
table
repo
rtsa
sum
mary
ofth
ek
-steps
ahead
pred
ictiveab
ilityo
fthe
RE
GA
RC
H-H
AR
and
weekly
single-p
arameter
RE
GA
RC
H-M
IDA
Sb
ench
-m
arkedagain
stthe
originalR
EG
AR
CH
.Statisticalsignifi
cance
ofthe
differen
cesin
forecastlossesis
assessedby
mean
softh
eD
iebold
-Marian
otestfor
equ
alpred
ictiveab
ility.For
eachfo
recastho
rizon
and
stock
we
categorize
the
ou
tcom
eo
fthe
testaccord
ing
toth
esize
ofth
ep
-value
and
repo
rtthe
nu
mb
ero
fstocks
falling
into
eachcatego
ryin
the
table.Fo
rin
stance,th
elastrow
inth
eleftPan
elAin
dicates
thatfo
rth
e22-step
sah
eadfo
recast,the
weekly
RE
GA
RC
H-M
IDA
So
utp
erform
sR
EG
AR
CH
15/20tim
esatth
e1%
signifi
cance
leveland
1/20tim
esatth
e10%
signifi
cance
level(bu
tno
tatthe
5%an
d1%
level),wh
ereas2/20
times
the
differen
cein
forecastp
erform
ance
was
insign
ifican
tand
2/20tim
esth
eR
EG
AR
CH
was
perfo
rmin
gth
eb
est.A
san
oth
erm
easure
ofim
provem
ent,w
ealso
repo
rtth
em
edian
ratioo
fforecast
loss
ofo
ur
pro
po
sedm
od
elsrelative
toth
eo
riginalR
EG
AR
CH
.An
um
ber
above
un
ityin
dicates
sup
erior
perfo
rman
ceo
fou
rp
rop
osed
mo
del.See
Section
1.4.3fo
rad
ditio
nald
etails.
Pan
elA:Q
LIK
Elo
ssfu
nctio
n
Weekly
RE
GA
RC
H-M
IDA
SR
EG
AR
CH
-HA
R
Ho
rizon
1%5%
10%In
sign.
RE
GA
RC
HM
edian
loss
ratioH
orizo
n1%
5%10%
Insign
.R
EG
AR
CH
Med
ianlo
ssratio
k=
15
23
55
1.02k
=1
41
15
91.01
k=
26
32
54
1.03k
=2
32
17
71.02
k=
39
22
43
1.05k
=3
44
14
71.02
k=
49
11
63
1.05k
=4
70
32
81.02
k=
59
11
54
1.05k
=5
71
24
61.03
k=
1011
22
14
1.11k
=10
101
22
51.09
k=
1513
20
14
1.22k
=15
121
12
41.15
k=
2215
01
22
1.33k
=22
131
04
21.26
Pan
elB:Sq
uared
Pred
ition
Erro
rlo
ssfu
nctio
n
Weekly
RE
GA
RC
H-M
IDA
SR
EG
AR
CH
-HA
R
Ho
rizon
1%5%
10%In
sign.
RE
GA
RC
HM
edian
loss
ratioH
orizo
n1%
5%10%
Insign
.R
EG
AR
CH
Med
ianlo
ssratio
k=
16
21
47
1.01k
=1
34
11
111.00
k=
210
21
43
1.04k
=2
45
21
81.02
k=
39
41
42
1.08k
=3
65
02
71.03
k=
49
50
42
1.08k
=4
73
03
71.04
k=
59
41
33
1.09k
=5
82
12
71.04
k=
1013
30
22
1.17k
=10
121
00
71.11
k=
1514
12
12
1.30k
=15
121
01
61.19
k=
2215
20
12
1.48k
=22
111
11
61.32
A.3. TABLES 55
Tab
leA
.7:M
od
elC
on
fid
ence
Sete
valu
atio
nfo
rin
div
idu
alst
ock
san
dSP
YT
his
tab
lere
po
rts
the
per
cen
tage
of
sto
cks
and
SPY
for
wh
ich
agi
ven
mo
del
isin
clu
ded
inth
eM
CS
on
a10
%si
gnifi
can
cele
vela
nd
on
e-st
epan
dm
ult
i-st
epah
ead
fore
cast
ing.
For
inst
ance
,th
eR
EG
AR
CH
was
incl
ud
edin
the
MC
Sfo
r33
%o
fth
est
ock
sw
ith
afo
reca
sth
ori
zon
of
k=
22w
hen
eval
uat
edu
sin
gth
eQ
LIK
Elo
ssfu
nct
ion
,wh
erea
sth
em
on
thly
RE
GA
RC
H-M
IDA
Sw
asin
clu
ded
inth
eM
CS
for
90%
of
the
sto
cks.
We
use
ab
lock
bo
ots
trap
wit
h10
5in
the
imp
lem
enta
tio
n.S
eeSe
ctio
n1.
4.3
for
add
itio
nal
det
ails
.
Pan
elA
:QL
IKE
loss
fun
ctio
n
Ho
rizo
nR
EG
AR
CH
RE
GA
RC
H-M
IDA
SR
EG
AR
CH
-MID
AS
RE
GA
RC
H-M
IDA
SR
EG
AR
CH
-MID
AS
RE
GA
RC
H-H
AR
RE
GA
RC
H-S
Flo
EG
AR
CH
(wee
kly)
(wee
kly)
(mo
nth
ly)
(mo
nth
ly)
(sin
gle-
par
amet
er)
(sin
gle-
par
amet
er)
k=
10.
570.
670.
810.
520.
570.
710.
380.
76k
=2
0.29
0.67
0.71
0.52
0.52
0.67
0.38
0.05
k=
30.
330.
670.
710.
520.
570.
670.
430.
10k
=4
0.38
0.76
0.81
0.71
0.67
0.67
0.43
0.14
k=
50.
430.
710.
760.
810.
760.
670.
520.
38k
=10
0.29
0.67
0.76
0.81
0.76
0.67
0.52
0.76
k=
150.
290.
710.
860.
900.
760.
710.
520.
81k
=22
0.33
0.76
0.76
0.90
0.90
0.71
0.76
0.86
Pan
elB
:Sq
uar
edP
red
itio
nE
rro
rlo
ssfu
nct
ion
k=
10.
430.
570.
710.
570.
670.
520.
520.
86k
=2
0.29
0.76
0.81
0.67
0.71
0.62
0.48
0.19
k=
30.
330.
760.
810.
670.
670.
670.
430.
10k
=4
0.33
0.76
0.81
0.67
0.57
0.62
0.52
0.19
k=
50.
290.
760.
810.
760.
760.
710.
520.
33k
=10
0.10
0.71
0.81
0.81
0.86
0.76
0.62
0.76
k=
150.
240.
710.
860.
900.
860.
760.
760.
81k
=22
0.14
0.71
0.76
0.86
0.86
0.67
0.67
0.86
56 CHAPTER 1.
A.4 In-sample results for individual stocks
Table A.8: REGARCHThis table reports full-sample estimated parameters, information criteria as well as full maxi-mized log-likelihood value for the original REGARCH.
AA AIG AXP BA CAT DD DIS GE IBM INTC
µ 0.015 -0.017 0.051 0.074 0.074 0.041 0.053 0.021 0.029 0.026β 0.972 0.972 0.987 0.977 0.973 0.972 0.981 0.982 0.974 0.972α 0.355 0.606 0.394 0.322 0.376 0.413 0.356 0.418 0.438 0.478ξ -0.518 -0.296 -0.385 -0.446 -0.590 -0.207 -0.327 -0.342 -0.375 -0.285σ2
u 0.136 0.201 0.148 0.135 0.130 0.147 0.146 0.153 0.129 0.129τ1 -0.054 -0.086 -0.085 -0.062 -0.056 -0.076 -0.076 -0.063 -0.072 -0.051τ2 0.039 0.039 0.039 0.036 0.016 0.023 0.022 0.029 0.014 0.021δ1 -0.061 -0.049 -0.065 -0.054 -0.069 -0.071 -0.079 -0.045 -0.063 -0.038δ2 0.063 0.042 0.060 0.076 0.043 0.048 0.047 0.049 0.037 0.036φ 1.055 0.855 0.993 1.046 1.105 0.961 0.981 0.985 0.962 0.924ω 1.544 1.513 1.135 1.041 1.155 0.770 1.067 0.865 0.549 1.353
logL -7,883.37 -8,493.81 -7,122.76 -7,000.25 -7,237.72 -6,751.76 -6,965.93 -6,832.49 -6,176.02 -7,343.71AIC 15,788.74 17,009.63 14,267.52 14,022.51 14,497.45 13,525.52 13,953.85 13,686.98 12,374.05 14,709.42BIC 15,854.83 17,075.70 14,333.57 14,088.57 14,563.51 13,591.57 14,019.91 13,753.08 12,440.11 14,775.55
HQIC 15,942.92 17,163.77 14,421.62 14,176.62 14,651.58 13,679.63 14,107.98 13,841.17 12,528.17 14,863.68
JNJ KO MMM MRK MSFT PG VZ WHR WMT XOM
µ 0.030 0.029 0.044 0.023 0.035 0.026 0.028 0.074 0.017 0.042β 0.976 0.973 0.967 0.971 0.968 0.961 0.979 0.963 0.979 0.967α 0.359 0.399 0.357 0.257 0.447 0.373 0.334 0.287 0.307 0.357ξ -0.123 -0.150 -0.342 -1.017 -0.381 -0.159 -0.185 -1.112 -0.259 -0.291σ2
u 0.151 0.144 0.147 0.191 0.137 0.151 0.153 0.171 0.137 0.123τ1 -0.066 -0.065 -0.080 -0.034 -0.044 -0.059 -0.064 -0.040 -0.035 -0.087τ2 0.038 0.027 0.010 0.001 0.013 0.025 0.039 0.024 0.028 0.041δ1 -0.031 -0.053 -0.071 -0.057 -0.036 -0.053 -0.061 -0.043 -0.028 -0.105δ2 0.058 0.064 0.041 0.010 0.033 0.056 0.059 0.068 0.058 0.050φ 0.961 0.949 1.067 1.450 1.003 1.084 1.009 1.344 1.120 1.082ω -0.071 0.112 0.437 0.974 0.900 0.021 0.530 1.538 0.350 0.554
logL -5,426.50 -5,694.49 -6,291.30 -7,470.15 -6,832.84 -5,651.66 -6,419.04 -8,210.50 -5,951.18 -6,113.27AIC 10,875.01 11,410.98 12,604.59 14,962.29 13,687.67 11,325.33 12,860.09 16,443.00 11,924.36 12,248.54BIC 10,941.07 11,477.05 12,670.64 15,028.34 13,753.80 11,391.39 12,926.14 16,509.04 11,990.44 12,314.61
HQIC 11,029.13 11,565.12 12,758.68 15,116.40 13,841.94 11,479.46 13,014.20 16,597.08 12,078.52 12,402.69
A.4. IN-SAMPLE RESULTS FOR INDIVIDUAL STOCKS 57
Table A.9: Weekly REGARCH-MIDASThis table reports full-sample estimated parameters, information criteria, variance ratio from(1.34) as well as full maximized log-likelihood value for the weekly two-parameter REGARCH-MIDAS. Results are for K = 52.
AA AIG AXP BA CAT DD DIS GE IBM INTC
µ 0.011 -0.016 0.056 0.078 0.071 0.044 0.063 0.025 0.033 0.031β 0.649 0.577 0.710 0.606 0.553 0.623 0.520 0.817 0.612 0.602α 0.392 0.589 0.410 0.360 0.432 0.455 0.429 0.452 0.479 0.531ξ -0.487 -0.291 -0.392 -0.441 -0.572 -0.202 -0.322 -0.333 -0.378 -0.276σ2
u 0.133 0.191 0.144 0.132 0.126 0.144 0.142 0.149 0.127 0.125τ1 -0.062 -0.094 -0.096 -0.076 -0.062 -0.090 -0.089 -0.068 -0.086 -0.061τ2 0.044 0.050 0.048 0.047 0.022 0.025 0.029 0.035 0.015 0.018δ1 -0.063 -0.048 -0.065 -0.054 -0.069 -0.071 -0.080 -0.046 -0.064 -0.040δ2 0.061 0.047 0.059 0.075 0.042 0.047 0.045 0.050 0.037 0.032φ 1.039 0.872 0.999 1.044 1.093 0.959 0.976 0.978 0.963 0.920ω 0.538 0.361 0.402 0.448 0.547 0.224 0.340 0.362 0.392 0.331λ 0.900 1.117 0.973 0.908 0.873 0.999 0.993 0.945 0.981 1.038γ1 -0.156 -0.577 -0.041 -0.214 -0.822 -0.758 -0.866 2.003 -0.971 -0.296γ2 6.538 1.481 6.938 6.994 2.134 1.000 1.130 27.928 1.004 4.883
logL -7,836.88 -8,367.58 -7,081.18 -6,956.80 -7,187.14 -6,716.86 -6,912.97 -6,791.63 -6,144.97 -7,280.34AIC 15,701.75 16,763.15 14,190.36 13,941.60 14,402.28 13,461.72 13,853.93 13,611.26 12,317.94 14,588.67BIC 15,785.86 16,847.24 14,274.42 14,025.67 14,486.36 13,545.79 13,938.01 13,695.38 12,402.01 14,672.84
HQIC 15,897.98 16,959.33 14,386.49 14,137.75 14,598.44 13,657.86 14,050.09 13,807.49 12,514.09 14,785.01VR 0.82 0.87 0.89 0.85 0.85 0.82 0.89 0.80 0.83 0.84
JNJ KO MMM MRK MSFT PG VZ WHR WMT XOM
µ 0.034 0.032 0.045 0.025 0.038 0.028 0.031 0.067 0.021 0.051β 0.627 0.562 0.587 0.552 0.613 0.547 0.649 0.554 0.536 0.598α 0.411 0.453 0.415 0.297 0.492 0.416 0.384 0.315 0.342 0.382ξ -0.128 -0.149 -0.340 -1.002 -0.365 -0.158 -0.177 -1.055 -0.254 -0.288σ2
u 0.147 0.141 0.143 0.188 0.133 0.147 0.150 0.164 0.133 0.120τ1 -0.076 -0.071 -0.086 -0.039 -0.048 -0.068 -0.071 -0.048 -0.045 -0.102τ2 0.049 0.036 0.018 0.002 0.015 0.031 0.048 0.032 0.036 0.046δ1 -0.029 -0.053 -0.070 -0.057 -0.038 -0.053 -0.060 -0.045 -0.027 -0.105δ2 0.058 0.065 0.041 0.010 0.033 0.055 0.058 0.067 0.056 0.049φ 0.958 0.936 1.060 1.437 0.986 1.071 1.000 1.315 1.117 1.084ω 0.103 0.149 0.324 0.712 0.391 0.132 0.192 0.861 0.225 0.279λ 0.949 1.017 0.873 0.647 0.951 0.863 0.938 0.701 0.855 0.851γ1 1.471 -0.875 -0.638 -0.563 -0.477 -0.957 0.001 -0.034 -0.776 -1.116γ2 48.950 1.000 3.334 4.365 3.124 1.000 11.393 7.918 2.321 1.000
logL -5,395.67 -5,652.84 -6,256.25 -7,438.83 -6,785.22 -5,604.99 -6,390.01 -8,138.73 -5,910.60 -6,069.31AIC 10,819.34 11,333.68 12,540.50 14,905.66 13,598.45 11,237.98 12,808.02 16,305.46 11,849.19 12,166.62BIC 10,903.42 11,417.77 12,624.56 14,989.73 13,682.61 11,322.06 12,892.09 16,389.52 11,933.29 12,250.72
HQIC 11,015.50 11,529.86 12,736.61 15,101.79 13,794.78 11,434.14 13,004.16 16,501.58 12,045.39 12,362.82VR 0.83 0.84 0.80 0.83 0.82 0.80 0.84 0.81 0.87 0.80
58 CHAPTER 1.
Table A.10: Weekly REGARCH-MIDAS (single-parameter)This table reports full-sample estimated parameters, information criteria, variance ratio from(1.34) as well as full maximized log-likelihood value for the weekly single-parameter REGARCH-MIDAS. Results are for K = 52.
AA AIG AXP BA CAT DD DIS GE IBM INTC
µ 0.011 -0.017 0.057 0.078 0.071 0.044 0.064 0.025 0.033 0.032β 0.711 0.623 0.790 0.656 0.584 0.710 0.554 0.742 0.638 0.654α 0.390 0.592 0.408 0.359 0.433 0.451 0.431 0.459 0.482 0.532ξ -0.491 -0.301 -0.392 -0.442 -0.573 -0.204 -0.321 -0.333 -0.378 -0.276σ2
u 0.133 0.191 0.144 0.132 0.126 0.144 0.142 0.149 0.127 0.125τ1 -0.062 -0.095 -0.095 -0.076 -0.061 -0.089 -0.089 -0.068 -0.085 -0.060τ2 0.043 0.049 0.046 0.046 0.022 0.025 0.029 0.036 0.016 0.018δ1 -0.063 -0.048 -0.065 -0.054 -0.069 -0.071 -0.079 -0.046 -0.063 -0.040δ2 0.061 0.047 0.059 0.075 0.043 0.048 0.045 0.049 0.037 0.032φ 1.041 0.876 1.000 1.045 1.093 0.959 0.975 0.978 0.962 0.920ω 0.563 0.406 0.409 0.459 0.571 0.256 0.358 0.356 0.398 0.353λ 0.878 1.078 0.959 0.889 0.844 0.945 0.964 0.964 0.942 1.014γ2 22.519 26.382 17.205 27.678 40.966 27.351 39.739 19.960 44.999 25.944
logL -7,838.40 -8,373.86 -7,081.88 -6,957.85 -7,190.46 -6,720.73 -6,915.24 -6,791.92 -6,148.72 -7,283.34AIC 15,702.80 16,773.72 14,189.76 13,941.71 14,406.93 13,467.46 13,856.48 13,609.84 12,323.44 14,592.68BIC 15,780.90 16,851.80 14,267.82 14,019.78 14,485.01 13,545.53 13,934.55 13,687.95 12,401.51 14,670.83
HQIC 15,885.01 16,955.88 14,371.88 14,123.85 14,589.08 13,649.59 14,038.63 13,792.06 12,505.58 14,774.99VR 0.80 0.86 0.87 0.83 0.84 0.78 0.88 0.84 0.82 0.82
JNJ KO MMM MRK MSFT PG VZ WHR WMT XOM
µ 0.034 0.032 0.045 0.024 0.038 0.028 0.031 0.068 0.021 0.050β 0.620 0.617 0.634 0.657 0.689 0.567 0.691 0.631 0.582 0.610α 0.411 0.453 0.414 0.293 0.492 0.420 0.381 0.315 0.343 0.388ξ -0.129 -0.150 -0.343 -1.012 -0.364 -0.157 -0.177 -1.052 -0.256 -0.287σ2
u 0.147 0.141 0.144 0.188 0.134 0.148 0.150 0.164 0.134 0.120τ1 -0.076 -0.071 -0.086 -0.039 -0.048 -0.068 -0.071 -0.048 -0.045 -0.103τ2 0.049 0.036 0.017 0.002 0.015 0.031 0.048 0.032 0.036 0.046δ1 -0.029 -0.053 -0.070 -0.057 -0.038 -0.054 -0.060 -0.044 -0.027 -0.105δ2 0.058 0.064 0.041 0.010 0.033 0.055 0.058 0.067 0.056 0.048φ 0.958 0.939 1.064 1.446 0.986 1.073 1.001 1.313 1.119 1.082ω 0.104 0.148 0.332 0.723 0.407 0.125 0.197 0.871 0.229 0.288λ 0.953 0.973 0.838 0.625 0.920 0.820 0.923 0.691 0.828 0.820γ2 37.685 35.949 35.219 28.038 24.932 46.647 28.890 20.578 39.253 53.373
logL -5,395.73 -5,657.00 -6,258.62 -7,442.47 -6,789.21 -5,609.35 -6,390.15 -8,141.40 -5,915.05 -6,072.96AIC 10,817.46 11,340.01 12,543.24 14,910.94 13,604.43 11,244.70 12,806.31 16,308.79 11,856.10 12,171.92BIC 10,895.53 11,418.09 12,621.29 14,989.00 13,682.59 11,322.78 12,884.37 16,386.85 11,934.20 12,250.01
HQIC 10,999.61 11,522.17 12,725.34 15,093.06 13,786.74 11,426.86 12,988.44 16,490.90 12,038.29 12,354.10VR 0.83 0.83 0.78 0.79 0.79 0.78 0.83 0.79 0.86 0.79
A.4. IN-SAMPLE RESULTS FOR INDIVIDUAL STOCKS 59
Table A.11: Monthly REGARCH-MIDASThis table reports full-sample estimated parameters, information criteria, variance ratio from(1.34) as well as full maximized log-likelihood value for the monthly two-parameter REGARCH-MIDAS. Results are for K = 12.
AA AIG AXP BA CAT DD DIS GE IBM INTC
µ 0.011 -0.018 0.057 0.078 0.071 0.044 0.064 0.025 0.034 0.032β 0.820 0.742 0.861 0.837 0.825 0.864 0.832 0.843 0.891 0.812α 0.383 0.590 0.404 0.345 0.414 0.432 0.390 0.445 0.451 0.515ξ -0.492 -0.306 -0.391 -0.441 -0.567 -0.201 -0.335 -0.333 -0.379 -0.276σ2
u 0.133 0.193 0.144 0.132 0.127 0.144 0.143 0.149 0.127 0.126τ1 -0.062 -0.092 -0.094 -0.073 -0.063 -0.086 -0.087 -0.069 -0.078 -0.060τ2 0.041 0.048 0.044 0.042 0.020 0.023 0.026 0.034 0.014 0.018δ1 -0.063 -0.048 -0.065 -0.054 -0.070 -0.071 -0.080 -0.046 -0.063 -0.040δ2 0.061 0.049 0.059 0.075 0.044 0.048 0.044 0.050 0.036 0.033φ 1.042 0.881 0.999 1.044 1.089 0.957 0.987 0.978 0.966 0.920ω 0.577 0.405 0.404 0.467 0.565 0.241 0.361 0.355 0.390 0.349λ 0.864 1.080 0.959 0.869 0.842 0.951 0.951 0.952 0.897 1.009γ1 -0.583 -1.605 0.025 1.398 -0.865 -0.266 -0.878 0.424 0.800 1.028γ2 4.112 1.000 5.262 13.651 3.036 2.728 1.933 8.113 6.714 9.921
logL -7,842.06 -8,377.55 -7,081.13 -6,963.39 -7,202.89 -6,721.18 -6,928.36 -6,792.40 -6,153.55 -7,292.92AIC 15,712.13 16,783.10 14,190.26 13,954.77 14,433.77 13,470.37 13,884.72 13,612.79 12,335.11 14,613.85BIC 15,796.24 16,867.19 14,274.33 14,038.85 14,517.86 13,554.44 13,968.80 13,696.91 12,419.18 14,698.01
HQIC 15,908.35 16,979.28 14,386.39 14,150.92 14,629.94 13,666.51 14,080.88 13,809.03 12,531.26 14,810.18VR 0.72 0.82 0.83 0.73 0.70 0.65 0.76 0.78 0.60 0.74
JNJ KO MMM MRK MSFT PG VZ WHR WMT XOM
µ 0.034 0.032 0.045 0.024 0.038 0.028 0.031 0.067 0.021 0.049β 0.850 0.827 0.863 0.864 0.826 0.831 0.875 0.745 0.852 0.873α 0.388 0.434 0.380 0.277 0.476 0.400 0.359 0.314 0.325 0.361ξ -0.127 -0.149 -0.344 -1.006 -0.367 -0.159 -0.180 -1.032 -0.256 -0.294σ2
u 0.148 0.142 0.144 0.188 0.134 0.149 0.151 0.165 0.134 0.121τ1 -0.074 -0.069 -0.084 -0.038 -0.050 -0.065 -0.069 -0.048 -0.040 -0.095τ2 0.045 0.033 0.013 0.001 0.014 0.028 0.044 0.031 0.032 0.044δ1 -0.030 -0.053 -0.070 -0.057 -0.039 -0.053 -0.060 -0.045 -0.028 -0.105δ2 0.058 0.064 0.041 0.010 0.033 0.056 0.058 0.067 0.057 0.049φ 0.958 0.941 1.071 1.439 0.988 1.080 1.005 1.301 1.120 1.089ω 0.082 0.141 0.330 0.730 0.407 0.118 0.205 0.874 0.225 0.297λ 0.914 0.962 0.783 0.595 0.906 0.794 0.877 0.686 0.807 0.761γ1 -1.427 -1.388 2.524 3.325 0.013 -0.931 1.769 3.777 1.456 -0.651γ2 1.000 1.000 18.021 21.725 5.094 1.783 13.707 33.579 13.546 2.319
logL -5,404.34 -5,662.62 -6,265.13 -7,446.52 -6,791.44 -5,623.29 -6,395.34 -8,141.29 -5,921.83 -6,087.31AIC 10,836.68 11,353.24 12,558.27 14,921.04 13,610.88 11,274.58 12,818.68 16,310.58 11,871.65 12,202.61BIC 10,920.76 11,437.32 12,642.32 15,005.10 13,695.05 11,358.66 12,902.75 16,394.64 11,955.75 12,286.71
HQIC 11,032.84 11,549.41 12,754.38 15,117.17 13,807.22 11,470.75 13,014.82 16,506.70 12,067.85 12,398.81VR 0.67 0.70 0.58 0.63 0.69 0.59 0.68 0.74 0.72 0.54
60 CHAPTER 1.
Table A.12: Monthly REGARCH-MIDAS (single-parameter)This table reports full-sample estimated parameters, information criteria, variance ratiofrom (1.34) as well as full maximized log-likelihood value for the monthly single-parameterREGARCH-MIDAS. Results are for K = 12.
AA AIG AXP BA CAT DD DIS GE IBM INTC
µ 0.012 -0.017 0.057 0.078 0.071 0.045 0.065 0.025 0.034 0.032β 0.824 0.737 0.865 0.836 0.831 0.871 0.841 0.845 0.893 0.812α 0.383 0.591 0.404 0.345 0.413 0.431 0.387 0.445 0.450 0.515ξ -0.492 -0.311 -0.391 -0.441 -0.568 -0.201 -0.337 -0.333 -0.378 -0.276σ2
u 0.134 0.192 0.144 0.132 0.127 0.144 0.143 0.149 0.127 0.126τ1 -0.062 -0.093 -0.094 -0.073 -0.063 -0.085 -0.087 -0.069 -0.078 -0.060τ2 0.041 0.048 0.044 0.042 0.019 0.023 0.026 0.034 0.014 0.018δ1 -0.063 -0.047 -0.065 -0.054 -0.070 -0.071 -0.080 -0.046 -0.063 -0.040δ2 0.061 0.048 0.059 0.075 0.044 0.048 0.044 0.050 0.036 0.033φ 1.042 0.883 0.999 1.045 1.090 0.957 0.988 0.978 0.966 0.920ω 0.587 0.431 0.406 0.466 0.576 0.252 0.373 0.356 0.390 0.349λ 0.855 1.054 0.955 0.871 0.828 0.930 0.933 0.950 0.894 1.009γ2 12.772 21.575 9.843 11.378 13.169 8.224 10.921 11.252 7.447 9.787
logL -7,842.51 -8,381.29 -7,081.41 -6,963.39 -7,203.34 -6,721.53 -6,929.11 -6,792.42 -6,153.56 -7,292.92AIC 15,711.02 16,788.57 14,188.81 13,952.79 14,432.67 13,469.06 13,884.22 13,610.84 12,333.13 14,611.85BIC 15,789.13 16,866.65 14,266.87 14,030.86 14,510.75 13,547.13 13,962.29 13,688.95 12,411.20 14,690.00
HQIC 15,893.23 16,970.74 14,370.94 14,134.93 14,614.83 13,651.19 14,066.36 13,793.06 12,515.27 14,794.16VR 0.72 0.82 0.83 0.73 0.70 0.64 0.76 0.78 0.59 0.74
JNJ KO MMM MRK MSFT PG VZ WHR WMT XOM
µ 0.034 0.033 0.045 0.024 0.038 0.028 0.031 0.067 0.021 0.049β 0.856 0.829 0.856 0.849 0.834 0.837 0.871 0.742 0.850 0.880α 0.387 0.434 0.381 0.277 0.474 0.399 0.360 0.314 0.326 0.361ξ -0.127 -0.149 -0.345 -1.016 -0.367 -0.159 -0.180 -1.037 -0.256 -0.293σ2
u 0.148 0.142 0.144 0.188 0.134 0.149 0.151 0.165 0.134 0.121τ1 -0.073 -0.069 -0.085 -0.038 -0.050 -0.065 -0.069 -0.048 -0.041 -0.095τ2 0.045 0.033 0.013 0.001 0.014 0.028 0.044 0.030 0.032 0.044δ1 -0.030 -0.052 -0.070 -0.058 -0.039 -0.053 -0.060 -0.044 -0.028 -0.105δ2 0.058 0.064 0.041 0.010 0.033 0.056 0.058 0.067 0.057 0.049φ 0.958 0.940 1.072 1.448 0.988 1.080 1.005 1.304 1.120 1.089ω 0.077 0.140 0.329 0.728 0.409 0.116 0.203 0.874 0.224 0.304λ 0.889 0.938 0.792 0.602 0.901 0.772 0.883 0.686 0.809 0.739γ2 14.140 14.689 10.161 10.076 9.315 11.923 9.798 15.579 11.051 10.146
logL -5,405.05 -5,663.48 -6,265.40 -7,447.15 -6,791.78 -5,623.97 -6,395.39 -8,141.55 -5,921.85 -6,087.94AIC 10,836.10 11,352.96 12,556.81 14,920.31 13,609.55 11,273.94 12,816.77 16,309.10 11,869.70 12,201.87BIC 10,914.18 11,431.04 12,634.86 14,998.37 13,687.71 11,352.02 12,894.84 16,387.16 11,947.79 12,279.96
HQIC 11,018.25 11,535.13 12,738.91 15,102.43 13,791.87 11,456.10 12,998.90 16,491.21 12,051.89 12,384.05VR 0.65 0.70 0.59 0.65 0.69 0.58 0.68 0.74 0.72 0.52
A.4. IN-SAMPLE RESULTS FOR INDIVIDUAL STOCKS 61
Table A.13: REGARCH-HARThis table reports full-sample estimated parameters, information criteria, variance ratio from(1.34) as well as full maximized log-likelihood value for the REGARCH-HAR.
AA AIG AXP BA CAT DD DIS GE IBM INTC
µ 0.011 -0.017 0.056 0.077 0.071 0.043 0.063 0.024 0.032 0.030β 0.638 0.578 0.811 0.583 0.537 0.591 0.508 0.801 0.568 0.565α 0.396 0.585 0.420 0.367 0.434 0.465 0.438 0.459 0.486 0.538ξ -0.495 -0.310 -0.391 -0.445 -0.576 -0.203 -0.322 -0.337 -0.377 -0.280σ2
u 0.134 0.191 0.145 0.133 0.127 0.145 0.143 0.150 0.127 0.126τ1 -0.062 -0.094 -0.095 -0.076 -0.061 -0.089 -0.089 -0.069 -0.086 -0.060τ2 0.045 0.050 0.046 0.048 0.022 0.026 0.029 0.036 0.016 0.019δ1 -0.063 -0.047 -0.065 -0.055 -0.069 -0.070 -0.080 -0.046 -0.064 -0.040δ2 0.061 0.048 0.060 0.075 0.043 0.048 0.044 0.051 0.037 0.033φ 1.043 0.881 0.998 1.047 1.096 0.958 0.976 0.980 0.959 0.922ω 0.572 0.418 0.429 0.465 0.571 0.256 0.360 0.377 0.398 0.366γ1 0.321 0.391 0.033 0.373 0.465 0.497 0.506 0.001 0.597 0.453γ2 0.552 0.677 0.898 0.510 0.381 0.453 0.457 0.920 0.356 0.551
logL -7,847.46 -8,374.10 -7,095.31 -6,972.04 -7,196.44 -6,731.86 -6,927.53 -6,804.48 -6,156.54 -7,298.11AIC 15,720.92 16,774.20 14,216.63 13,970.07 14,418.88 13,489.73 13,881.07 13,634.95 12,339.08 14,622.23BIC 15,799.03 16,852.28 14,294.69 14,048.14 14,496.96 13,567.79 13,959.14 13,713.06 12,417.15 14,700.38
HQIC 15,903.13 16,956.37 14,398.75 14,152.21 14,601.04 13,671.86 14,063.21 13,817.17 12,521.22 14,804.54VR 0.82 0.86 0.84 0.85 0.85 0.82 0.88 0.80 0.84 0.84
JNJ KO MMM MRK MSFT PG VZ WHR WMT XOM
µ 0.034 0.032 0.044 0.025 0.039 0.028 0.030 0.068 0.020 0.049β 0.570 0.558 0.554 0.494 0.585 0.521 0.599 0.531 0.500 0.561α 0.419 0.458 0.423 0.303 0.502 0.421 0.394 0.317 0.349 0.389ξ -0.130 -0.149 -0.339 -0.998 -0.362 -0.157 -0.180 -1.066 -0.254 -0.288σ2
u 0.148 0.141 0.145 0.189 0.134 0.148 0.152 0.165 0.135 0.120τ1 -0.076 -0.072 -0.085 -0.039 -0.046 -0.068 -0.072 -0.047 -0.045 -0.103τ2 0.050 0.037 0.018 0.002 0.016 0.031 0.049 0.032 0.037 0.046δ1 -0.029 -0.053 -0.069 -0.056 -0.038 -0.053 -0.061 -0.044 -0.028 -0.105δ2 0.059 0.065 0.042 0.010 0.033 0.055 0.059 0.066 0.056 0.049φ 0.959 0.936 1.058 1.433 0.982 1.069 1.001 1.321 1.117 1.082ω 0.106 0.148 0.331 0.719 0.408 0.126 0.200 0.881 0.228 0.287γ1 0.480 0.499 0.457 0.356 0.423 0.500 0.437 0.246 0.472 0.556γ2 0.474 0.480 0.388 0.275 0.501 0.329 0.485 0.435 0.361 0.273
logL -5,407.30 -5,659.23 -6,269.06 -7,449.17 -6,796.98 -5,613.81 -6,404.37 -8,143.54 -5,922.70 -6,077.50AIC 10,840.60 11,344.46 12,564.13 14,924.33 13,619.96 11,253.63 12,834.74 16,313.08 11,871.40 12,181.01BIC 10,918.68 11,422.54 12,642.18 15,002.40 13,698.11 11,331.71 12,912.80 16,391.13 11,949.49 12,259.10
HQIC 11,022.75 11,526.62 12,746.23 15,106.46 13,802.27 11,435.78 13,016.87 16,495.18 12,053.58 12,363.19VR 0.84 0.84 0.80 0.84 0.82 0.80 0.85 0.81 0.87 0.81
62 CHAPTER 1.
Table A.14: REGARCH-SplineThis table reports full-sample estimated parameters, information criteria as well as full maxi-mized log-likelihood value for the REGARCH-Spline. Results are for K = 6.
AA AIG AXP BA CAT DD DIS GE IBM INTC
µ -0.017 0.048 0.059 0.088 0.075 0.047 0.068 0.023 0.035 0.017β 0.933 0.864 0.943 0.936 0.941 0.932 0.942 0.935 0.948 0.913α 0.359 0.588 0.389 0.342 0.376 0.417 0.355 0.428 0.440 0.487ξ -0.510 -0.274 -0.379 -0.399 -0.591 -0.199 -0.328 -0.326 -0.375 -0.275σ2
u 0.134 0.195 0.144 0.133 0.128 0.144 0.143 0.150 0.128 0.127τ1 -0.058 -0.075 -0.088 -0.069 -0.058 -0.079 -0.081 -0.065 -0.073 -0.055τ2 0.039 0.045 0.040 0.043 0.016 0.023 0.024 0.031 0.013 0.018δ1 -0.063 -0.039 -0.064 -0.055 -0.069 -0.069 -0.078 -0.045 -0.062 -0.037δ2 0.062 0.048 0.058 0.076 0.042 0.047 0.046 0.050 0.036 0.034φ 1.052 0.864 0.994 1.000 1.108 0.953 0.980 0.973 0.959 0.919ω 1.684 1.435 1.726 2.015 1.375 1.398 2.009 1.860 1.632 2.769
logL -7,854.94 -8,390.56 -7,086.92 -6,972.56 -7,214.19 -6,718.88 -6,931.05 -6,795.12 -6,156.17 -7,300.44AIC 15,745.89 16,817.12 14,209.84 13,981.11 14,464.37 13,473.76 13,898.10 13,626.24 12,348.33 14,636.88BIC 15,854.03 16,925.23 14,317.92 14,089.21 14,572.48 13,581.85 14,006.20 13,734.39 12,456.43 14,745.10
HQIC 15,998.18 17,069.35 14,462.00 14,233.30 14,716.59 13,725.94 14,150.30 13,878.54 12,600.52 14,889.32VR 0.51 0.79 0.75 0.62 0.50 0.56 0.63 0.68 0.40 0.61
JNJ KO MMM MRK MSFT PG VZ WHR WMT XOM
µ 0.035 0.033 0.047 0.039 0.050 0.021 0.033 0.067 0.020 0.049β 0.950 0.932 0.938 0.926 0.917 0.930 0.953 0.905 0.947 0.947α 0.353 0.404 0.353 0.283 0.449 0.371 0.343 0.307 0.310 0.343ξ -0.123 -0.147 -0.351 -0.902 -0.383 -0.143 -0.177 -0.988 -0.252 -0.295σ2
u 0.148 0.142 0.144 0.190 0.134 0.149 0.152 0.167 0.135 0.121τ1 -0.068 -0.067 -0.081 -0.031 -0.047 -0.059 -0.064 -0.045 -0.036 -0.087τ2 0.040 0.028 0.012 0.002 0.013 0.025 0.040 0.027 0.028 0.043δ1 -0.030 -0.052 -0.070 -0.052 -0.037 -0.053 -0.060 -0.044 -0.028 -0.104δ2 0.057 0.064 0.040 0.011 0.032 0.055 0.058 0.069 0.057 0.050φ 0.964 0.941 1.077 1.349 1.002 1.087 1.004 1.273 1.113 1.094ω 0.962 0.738 0.936 1.168 2.001 0.169 1.366 1.490 1.164 1.040
logL -5,403.06 -5,663.08 -6,270.65 -7,434.19 -6,796.46 -5,629.11 -6,401.72 -8,162.50 -5,927.05 -6,090.50AIC 10,842.13 11,362.17 12,577.29 14,904.38 13,628.92 11,294.23 12,839.43 16,361.01 11,890.10 12,217.00BIC 10,950.23 11,470.28 12,685.37 15,012.46 13,737.14 11,402.34 12,947.52 16,469.08 11,998.23 12,325.13
HQIC 11,094.33 11,614.40 12,829.44 15,156.55 13,881.35 11,546.45 13,091.61 16,613.15 12,142.36 12,469.26VR 0.46 0.59 0.40 0.63 0.57 0.42 0.44 0.61 0.53 0.35
A.4. IN-SAMPLE RESULTS FOR INDIVIDUAL STOCKS 63
Table A.15: FloEGARCHThis table reports full-sample estimated parameters, information criteria as well as full maxi-mized log-likelihood value for the FloEGARCH.
AA AIG AXP BA CAT DD DIS GE IBM INTC
µ 0.017 -0.009 0.045 0.061 0.072 0.035 0.042 0.021 0.025 0.010β 0.195 0.121 0.195 0.133 0.104 0.179 0.080 0.117 0.161 0.131α 0.400 0.589 0.418 0.373 0.436 0.460 0.424 0.473 0.484 0.536ξ -0.476 -0.293 -0.389 -0.440 -0.566 -0.205 -0.327 -0.332 -0.378 -0.272σ2
u 0.134 0.192 0.144 0.132 0.127 0.144 0.142 0.149 0.128 0.126τ1 -0.065 -0.096 -0.098 -0.078 -0.066 -0.089 -0.095 -0.072 -0.084 -0.063τ2 0.041 0.048 0.045 0.045 0.020 0.025 0.029 0.037 0.015 0.020δ1 -0.063 -0.047 -0.067 -0.057 -0.069 -0.071 -0.082 -0.047 -0.064 -0.042δ2 0.060 0.048 0.060 0.076 0.043 0.047 0.045 0.050 0.036 0.033φ 1.036 0.872 0.999 1.041 1.092 0.964 0.985 0.979 0.975 0.923ω 1.393 1.076 1.299 1.365 0.950 0.961 1.510 0.980 0.825 1.898d 0.633 0.620 0.678 0.658 0.673 0.645 0.673 0.678 0.672 0.644
logL -7,843.29 -8,370.57 -7,082.26 -6,960.83 -7,195.15 -6,722.47 -6,922.21 -6,791.17 -6,159.11 -7,291.01AIC 15,710.58 16,765.13 14,188.52 13,945.66 14,414.29 13,468.95 13,868.42 13,606.34 12,342.22 14,606.02BIC 15,782.68 16,837.21 14,260.57 14,017.72 14,486.36 13,541.01 13,940.48 13,678.44 12,414.28 14,678.16
HQIC 15,878.78 16,933.28 14,356.63 14,113.79 14,582.44 13,637.07 14,036.55 13,774.54 12,510.34 14,774.31
JNJ KO MMM MRK MSFT PG VZ WHR WMT XOM
µ 0.030 0.029 0.042 0.029 0.029 0.024 0.028 0.088 0.014 0.040β 0.093 0.121 0.137 0.108 0.158 0.138 0.156 0.092 0.113 0.150α 0.430 0.458 0.413 0.298 0.498 0.425 0.396 0.317 0.361 0.413ξ -0.130 -0.147 -0.343 -1.010 -0.368 -0.159 -0.178 -1.062 -0.258 -0.292σ2
u 0.147 0.141 0.144 0.188 0.134 0.148 0.151 0.165 0.134 0.122τ1 -0.078 -0.072 -0.088 -0.041 -0.051 -0.068 -0.071 -0.051 -0.045 -0.101τ2 0.049 0.035 0.015 0.002 0.015 0.031 0.049 0.030 0.035 0.049δ1 -0.030 -0.053 -0.070 -0.057 -0.039 -0.055 -0.060 -0.044 -0.029 -0.106δ2 0.058 0.065 0.041 0.010 0.033 0.056 0.059 0.065 0.057 0.050φ 0.955 0.939 1.073 1.448 0.998 1.080 1.001 1.325 1.116 1.078ω 0.300 0.367 0.466 0.827 1.307 0.307 0.726 1.171 0.643 0.832d 0.692 0.674 0.655 0.666 0.641 0.643 0.671 0.618 0.681 0.656
logL -5,399.76 -5,660.47 -6,259.27 -7,442.64 -6,796.38 -5,620.41 -6,398.68 -8,152.11 -5,922.81 -6,097.09AIC 10,823.52 11,344.93 12,542.53 14,909.29 13,616.76 11,264.82 12,821.36 16,328.22 11,869.62 12,218.18BIC 10,895.59 11,417.01 12,614.58 14,981.34 13,688.90 11,336.89 12,893.42 16,400.27 11,941.70 12,290.26
HQIC 10,991.66 11,513.09 12,710.63 15,077.40 13,785.05 11,432.97 12,989.49 16,496.32 12,037.79 12,386.35
C H A P T E R 2REALIZED EGARCH MODELS WITH
TIME-VARYING UNCONDITIONAL VARIANCE
Bo LaursenAarhus University and CREATES
Johan Stax JakobsenAarhus University and CREATES
Abstract
In this paper, we suggest four new parametric alternatives to the Realized EGARCH
model by allowing the unconditional variance to exhibit time-variation. The con-
ditional variance is decomposed into a stationary and a non-stationary part. The
stationary part is specified by a zero mean standard Realized EGARCH model. The un-
conditional variance is modeled by the non-stationary component. We propose four
functional alternatives to the non-stationary part: a smooth time-varying structure, a
flexible Fourier form, a quadratic spline and a cubic spline.
An empirical application to the exchange-traded index fund SPY that tracks the
S&P 500 Index including both a forecasting and Value-at-Risk exercise illustrates the
performance of the models in practice. The results show that the introduction of a
time-varying unconditional variance improves the in-sample fit of the models, but
does generally not lead to any improvements out-of-sample.
65
66 CHAPTER 2.
2.1 Introduction
The introduction of the AutoRegressive Conditional Heteroskedasticity (ARCH) model
of Engle (1982) and the Generalized ARCH (GARCH) model of Bollerslev (1986)
sparked a huge and successful literature on modeling time-varying volatility. An
astronomic number of generalizations and new models have been suggested and
studied; see Teräsvirta (2009) for an overview. The popularity of GARCH type models
is mainly due to their ability to describe stylized facts of financial return series such
as volatility clustering.
It is a well-known stylized fact that most daily and high-frequency financial
time series exhibit quite persistent autocorrelation in their squared return, realized
variance, and other measures of volatility. This has motivated a strand of the GARCH
literature to model long range dependence explicitly using long-memory models such
as the Fractionally Integrated GARCH (FIGARCH) model of Baillie, Bollerslev, and
Mikkelsen (1996). However, another strand of the literature is essentially arguing that
long-memory and the so-called ’Integrated GARCH effect’ in volatility may merely
be a statistical artifact. The observed long-memory behaviour may be spurious and
caused by structural changes in the volatility dynamics bound to occur in sufficiently
long time series. Diebold (1986) was the first to suggest that occasional level shifts in
the intercept of the GARCH model can bias the estimates towards the parameters of
an integrated GARCH model; see also Mikosch and Starica (2004) and Lamoureux
and Lastrapes (1990). Theoretical explanations have been provided by Hillebrand
(2005) and Morana (2002). However, it may be noted that Morana and Beltratti (2004)
argue that both long-memory and structural changes are important.
The idea of structural changes in volatility dynamics has promoted the develop-
ment of new and more flexible GARCH type models allowing for changing parameters,
state-dependence and time-varying unconditional variance, etc. In this paper, we
build upon the literature that explicitly model structural changes in the parameters.
One possibility is to assume that the volatility process is smoothly non-stationary
and model it accordingly. There is an expanding literature taking this approach begin-
ning with Bellegem and von Sachs (2004) and Feng (2004). Dahlhaus and Subba Rao
(2006) introduced a smoothly time-varying parameter ARCH model. Bellegem and
von Sachs (2004), Engle and Rangel (2008) and Amado and Teräsvirta (2013) all con-
sidered a multiplicative decomposition of the volatility process into a stationary and
non-stationary component. However, while Bellegem and von Sachs (2004) nonpara-
metrically estimated the non-stationary component, Engle and Rangel (2008) and
Amado and Teräsvirta (2013) parametrically estimated the non-stationary compo-
nent using splines and generalized logistic functions respectively. See also Brownless
and Gallo (2010) who use splines to fit a time-varying component in the Multiplica-
tive Error Model (MEM) of Engle and Gallo (2006) and Mishra, Su, and Ullah (2010)
who correct potential misspecification due to a ’rough’ parametric GARCH specifica-
tion by a smooth nonparametric component. Baillie and Morana (2009) suggested
2.1. INTRODUCTION 67
the additively decomposed Adaptive FIGARCH model that accounts for both long-
memory and structural changes in the volatility processes using the Fourier flexible
form; see also Mazur and Pipien (2012). We refer the reader to Teräsvirta (2012) and
Van Bellegem (2012) for a more comprehensive survey.
Since French, Schwert, and Stambaugh (1987) and due to the availability of
databases providing intra daily prices of financial assets, a literature focusing on
using data sampled at very high frequency to compute ex-post measures of volatility
at a lower frequency has emerged. A wide range of realized measures of volatility has
been proposed in the literature since Andersen and Bollerslev (1998) showed that
such measures can be very useful for the evaluation of volatility models. Barndorff-
Nielsen and Shephard (2002) proposed the Realized Variance (RV). However, this
measure is sensitive to market microstructure noise. This has motivated the devel-
opment of robust estimators such as the two-scale and multi-scale estimator by
Zhang, Mykland, and Aït-Sahalia (2005) and Zhang (2006), the Realized Kernel (RK)
by Barndorff-Nielsen, Hansen, Lunde, and Shephard (2008), and the Realized Range
by Christensen and Podolskij (2007). The economic and statistical gains from in-
corporating realized measures in volatility models are typically found to be large;
see Christoffersen, Feunou, Jacobs, and Meddahi (2014) and Dobrev and Szerszen
(2010). Andersen et al. (2003) find that classical GARCH models are poorly suited for
situations where volatility jumps to a new level. The basic intuition is that realized
measures constitute a stronger and less noisy signal of latent volatility than squared
daily returns. Hence, models utilizing realized measures can adapt faster to a new
level of volatility.
A number of volatility models incorporating realized measures has been suggested
in the literature. Models such as the Heteroskedastic AutoRegressive of Realized
Variance (HAR-RV) model of Corsi (2009) and its extensions seek to model realized
variance directly. On the other hand, a part of the literature models latent return
volatility by utilizing realized measures in conventional GARCH type models. The
simplest way of incorporating realized measures is by using GARCH-X type models;
see e.g. Engle (2002). However, a GARCH-X model is incomplete as it does not specify
the dynamic properties of the realized measures. Thus, multi-period forecasting
is infeasible. This motivated the MEM of Engle and Gallo (2006) and the HEAVY
model of Shephard and Sheppard (2010) who operate with multiple latent volatility
processes. The realized measures are modeled with additional GARCH type models.
Another possibility is to consider the Realized (E)GARCH framework that is based
on measurement equations that tie the realized measure to the conditional return
variance; see Hansen, Huang, and Shek (2012) and Hansen and Huang (2016). In this
paper, we utilize the Realized EGARCH framework.
The purpose of this paper is to combine the two strands of the GARCH litera-
ture described above by introducing a low-frequency or non-stationary component
into the Realized EGARCH framework of Hansen and Huang (2016). We suggest four
68 CHAPTER 2.
different specifications of the low-frequency component inspired by the existing
GARCH literature. The low-frequency component is deterministically time-varying
and proxies all factors that affect the unconditional variance. The addition of this
component has the potential to capture the long-run dynamic behaviour of volatility
such as structural changes that are bound to happen in sufficiently long time se-
ries. Thus, the model keeps the attractive features of GARCH type models in fitting
and forecasting at high and medium frequency while allowing for a low frequency
component associated with volatility dynamics at a lower frequency.
It is important to note that the low-frequency component, of course, is esti-
mated based on the in-sample observations and that it needs to be extrapolated
out-of-sample. Typically, it is fixed for the forcasting horizon. In-sample, the de-
terministically time-varying component is a convenient tool to allow for changing
amplitudes of volatility clusters often observed in financial time series (see e.g. the
empirical section of this paper or Amado and Teräsvirta (2013)) and it should help us
get the level of volatility correct. However, in an out-sample exercise, the new models
are prone to level-shift in volatility in the same way as the benchmark counterpart.
It is an empirical question whether the attempt to get the level right will help the
modeller out-of-sample or an estimated benchmark without this feature, maybe with
spuriously induced persistence, will outperform.
The rest of the paper proceeds as follows. In Section 2.2, we introduce the four
new Realized EGARCH models. Estimation and inference are discussed in Section 2.3.
Section 2.4 and 2.5 describe techniques used for forecasting and Value-at-Risk (VaR)
calculations. In Section 2.6, an empirical application to the exchange-traded index
fund SPY is presented. Finally, concluding remarks are given in Section 2.7.
2.2 Four new Realized EGARCH models
In this paper, we seek to model the volatility of an asset return series by extending the
Realized EGARCH framework of Hansen and Huang (2016). Let Ft−1 be the informa-
tion set containing the historical information at time t −1 and define the conditional
mean, µt = E(rt |Ft−1
), and the conditional variance, σ2
t = Var(rt |Ft−1
), of the asset
returns{rt
}. The conditional variance,σ2
t = ht g t , is multiplicatively decomposed into
a stationary (high-) and a non-stationary (low-frequency) component as in Engle and
Rangel (2008) and Amado and Teräsvirta (2013), among others. The idea is to model
the low-frequency component, g t , as a deterministic function of time to take changes
in the unconditional variance into account. The general framework is defined by the
2.2. FOUR NEW REALIZED EGARCH MODELS 69
following equations
rt = µt +√σ2
t zt , (2.1)
loght = β loght−1 +τ(zt−1
)+αut−1, (2.2)
log xt = ξ+φ log(σ2
t
)+δ(
zt)+ut , (2.3)
log g t = ωt . (2.4)
We refer to the equations as the return equation, the GARCH equation, the measure-
ment equation, and the low-frequency component, respectively. It is assumed that
zt ∼ i.i.d.(0,1
)and ut ∼ i.i.d.
(0,σ2
u
)are mutually and serially independent. xt is a
realized measure, e.g. the realized variance, bipower variation, daily range, or a robust
measure such as the RK.
In GARCH applications, the return equation is standard. The conditional mean
can be modeled in different ways including popular approaches such as GARCH-in-
mean, a constant or an autoregressive process. However, as the focus is on modeling
volatility, we assume that µt = 0. It can be noted that Hansen and Huang (2016)
empirically find that this restriction may improve out-of-sample fit relative to a model
using an unrestricted conditional mean, when considering the exchange-traded index
fund SPY also used in the financial application in Section 2.6.
We see that the GARCH equation is an AR(1) model for loght with the innovation
τ(zt−1
)+αut−1. Thus, β < 1 is a measure of persistence of the stationary part of
the conditional variance. The parameter α tells us how the realized measure affects
future volatility. The intercept of the GARCH equation is normalized to zero such
that E[loght
] = 0 in order to deal with an identification problem that emerges if
both loght and log g t are allowed to a have a free constant. This implies that log g t =E[
logσ2t
]models the now potentially time-varying unconditional variance.
The measurement equation defines the process for the realized measure. The
realized measure, xt , is an ex-post estimator for σ2t . Hence, it is natural to assume a
link between this ex-post measure of volatility and the the ex-ante conditional vari-
ance. However, discrepancies between the two measures are expected for numerous
reasons. In this paper, the conditional variance is a measure of close-to-close return
volatility while the realized measure only measures the volatility during trading hours.
As the open-to-close volatility empirically accounts for approximately 75% of daily
volatility, it is necessary to include the intercept, ξ, in (2.3). The parameter φ is an
exponential scaling factor that often is estimated to be close to one. In addition, the
error term, ut , is included to account for discrepancies in the two measures stemming
from noise and sampling errors in realized measures.
The leverage functions τ(zt
)and δ
(zt
)are defined by
τ (z) = τ1z +τ2
(z2 −1
), (2.5)
δ (z) = δ1z +δ2
(z2 −1
). (2.6)
70 CHAPTER 2.
The choice of the leverage functions follows Hansen and Huang (2016) who find that
this specification makes the independence assumption between zt and ut realistic
in an application to the exchange-traded index fund SPY. This specification allows
for a polynomial effect of zt on the realized measure and future volatility. The four
equations fully define the dynamic properties of returns and the realized measure of
volatility.
The non-stationary component, log g t , is a deterministic function of time that
proxies all factors that affect the unconditional variance. The addition of this compo-
nent has the potential to capture the long-run dynamic behaviour of volatility such
as structural changes that are bound to happen in sufficiently long financial time
series. Thus, the model keeps the attractive features of GARCH type models in fitting
and forecasting at high and medium frequency while allowing for a low-frequency
component associated with volatility dynamics at longer horizons.
We consider five different specifications of ωt in (2.4) inspired by the existing
literature on GARCH models with time-varying unconditional variance. First, we
consider the special case of a standard Realized EGARCH model by setting
ωt =ω. (2.7)
Second, we consider the Realized TV-EGARCH model inspired by the TV-GJR-GARCH
model of Amado and Teräsvirta (2013) by defining
ωt =ω+K∑
k=1γkG
(t
T;λk ,ck
)(2.8)
as a linear combination of bounded transition functions. The so-called transition
functions, G(
tT ;λk ,ck
)for k = 1, ...,K , are generalized logistic functions
G
(t
T;λk ,ck
)=
1+exp
(−λk
(t
T− ck
))−1
(2.9)
satisfying the identification restrictions λk > 0 and 0 ≤ c1 < c2 < ... < cK ≤ 1. The
transition functions allow the unconditional variance to change smoothly as a func-
tion of time, t/T . It should be noted that the specification differs from the one in
Amado and Teräsvirta (2013) by restricting the transition functions to only have one
transition each. This is done to simplify model specification. The parameters ck and
λk determine respectively the location and speed of the transition between different
regimes. The larger λk , the faster the transition between states. When λk →∞, the
transition function becomes a step function. In general, (2.8) and (2.9) can generate
very flexible parametrizations capable of modeling changes in the unconditional
variance. A difference between this specification and the following specifications is
that the location parameters are estimated. Estimation of the location makes the nu-
merical optimization quite tedious, but the additional flexibility most likely reduces
the number of transition functions needed.
2.2. FOUR NEW REALIZED EGARCH MODELS 71
Third, we also suggest a Realized Adaptive-EGARCH model using the flexible
functional form
ωt =ω+K∑
k=1
(γk sin
(2πk
t
T
)+λk cos
(2πk
t
T
))(2.10)
used in the Adaptive-FIGARCH model of Baillie and Morana (2009); see also Andersen
and Bollerslev (1997) and Mazur and Pipien (2012). Baillie and Morana (2009) argue
that advantages of the Fourier flexible form, specified by Gallant (1984), are efficient
modeling of smooth structural changes without requiring pretesting to determine
the actual location of break points and relatively straightforward estimation. We also
find the model easily estimable. Although the functional form is smooth, it has been
shown to accommodate quite abrupt shifts.
Finally, inspired by the Spline-GARCH model of Engle and Rangel (2008), we
consider two spline specifications: the quadratic spline defined by
ωt =ω+γ0t
T+
K∑k=1
γ1k
((t
T− ck−1
)+
)2
, (2.11)
and the cubic spline defined by
ωt =ω+γ0t
T+γ1
(t
T
)2
+K∑
k=1γ2k
((t
T− ck−1
)+
)3
. (2.12)
The knot-points, ck , are assumed to be equidistant as in Engle and Rangel (2008). The
models are labelled the Realized Q-Spline-EGARCH and Realized C-Spline-
EGARCH model, respectively. The assumption about equidistant knot-points makes
the estimation quite straightforward, but may have the consequence that one sud-
denly needs to increase K when increasing the sample size merely because the
locations of the knot-points have changed.
Clearly, all the new models add a high degree of flexibility in fitting the level
of volatility in-sample. The larger K , the more flexibility. Thus in order to avoid
overfitting, determining K is an important part of modeling the deterministically
time-varying component. Amado and Teräsvirta (2017) propose a modeling strat-
egy for the TV-GARCH model of Amado and Teräsvirta (2013) based on Lagrange
Multiplier (LM) misspecification tests. The idea is to keep adding transition function
until the null of no additional time-variation in the unconditional variance cannot
be rejected. For the Adaptive-FIGARCH model of Baillie and Morana (2009) and the
Spline-GARCH of Engle and Rangel (2008), K was selected using information criteria.
In this paper, we will apply information criteria to select K , because we are covering
multiple specifications and want to have an uniform way of selecting K .
72 CHAPTER 2.
2.3 Estimation and inference
In this section, we briefly discuss estimation and inference within a Quasi-Maximum
Likelihood (QML) framework obtained by assuming that zt ∼ i.i.d.N(0,1
)and ut ∼
i.i.d.N(0,σ2
u
). Write the leverage functions as τ
(zt
) = τ′a(zt
)and δ
(zt
) = δ′b(zt
)with a
(zt
)= b(zt
)= (zt , z2
t −1)′
. The initial value of the logarithm of the conditional
variance, log h1, is set equal to its unconditional mean, E[loght ] = 0. Define the
parameter vector
θ1 =(β,τ′,α,ξ,φ,δ′,σ2
u
), (2.13)
and let the parameter vector θ2 contain the parameters used to model the low-
frequency component, log g t . The log-likelihood function reads
L(r, x;θ1,θ2
)= T∑t=1
`t(rt , xt ;θ1,θ2
)(2.14)
with the log-likelihood contribution at time t given by
`t(rt , xt ;θ1,θ2
)=−1
2
[2log(2π)+ log
(ht
)+ log(g t
)+ z2t + log
(σ2
u
)+ u2
t
σ2u
], (2.15)
where zt = zt(θ1,θ2
)= rt /√
ht g t and ut(θ1,θ2
)= log(xt
)−ξ−φ log(σ2
t
)−δ(
zt). The
QML Estimator (QMLE), θ =(θ′1, θ′2
)′, is obtained by maximizing the log-likelihood
function. We now derive the score that defines the first-order conditions for the
QMLE. Key components are the derivatives stated in the following lemma.
Lemma 1. The derivatives of loght+1 with respect to loght and log g t are given by
respectively,
∂ loght+1
∂ loght= (
β−αφ)+ 1
2
(αδ
∂b(zt
)∂zt
−τ∂a(zt
)∂zt
)zt , (2.16)
∂ loght+1
∂ log g t=−αφ+ 1
2
(αδ
∂b(zt
)∂zt
−τ∂a(zt
)∂zt
)zt , (2.17)
and the derivatives of `t with respect to loght and log g t are given by respectively,
∂`t
∂ loght=−1
2
(1− z2
t
)+ ut
σ2u
(δ∂b
(zt
)∂zt
zt −2φ
) , (2.18)
∂`t
∂ log g t=−1
2
(1− z2
t
)+ ut
σ2u
(δ∂b
(zt
)∂zt
zt −2φ
) . (2.19)
Proof: See Appendix A.1.
2.3. ESTIMATION AND INFERENCE 73
Next, we define the score of loght+1 with respect to θ.
Lemma 2. The derivatives of loght+1 with respect to θ are given from the stochastic
recursions
∂ loght+1
∂(β,τ′,α
)′ = ∂ loght+1
∂ loght
∂ loght
∂(β,τ′,α
)′ + (loght , zt , z2
t −1,ut
)′, (2.20)
∂ loght+1
∂(ξ,φ,δ′
)′ = ∂ loght+1
∂ loght
∂ loght
∂(ξ,φ,δ′
)′ +α(1, log
(σ2
t
), zt , z2
t −1
)′, (2.21)
∂ loght+1
∂θ2= ∂ loght+1
∂ loght
∂ loght
∂θ2+ ∂ loght+1
∂ log g t
∂ log g t
∂θ2, (2.22)
where ∂ log g t∂θ2
differs between the specifications for the non-stationary component and
is therefore presented in Appendix A.1.
Proof: See Appendix A.1.
Finally, we present a theorem summarizing the score.
Theorem 1. The score with respect to the parameters in θ1 =(β,τ′,α,ξ,φ,δ′,σ2
u
)is
given by the components
∂`t
∂(β,τ′,α
)′ = ∂`t
∂ loght
∂ loght
∂(β,τ′,α
)′ (2.23)
∂`t
∂(ξ,φ,δ′
)′ = ∂`t
∂ loght
∂ loght
∂(ξ,φ,δ′
)′ + ut
σ2u
(1, log
(σ2
t
), zt , z2
t −1
)′(2.24)
∂`t
∂σ2u= 1
2
u2t −σ2
u
σ4u
(2.25)
and the score with respect to θ2 is given by
∂`t
∂θ2= ∂`t
∂ loght
∂ loght
∂θ2+ ∂`t
∂ log g t
∂ log g t
∂θ2, (2.26)
where ∂ log g t∂θ2
differs between the specifications for the non-stationary component; see
Appendix A.1.
Proof: See Appendix A.1.
Following Hansen and Huang (2016), we note that the score is a martingale differ-
ence sequence if E(zt |Ft−1
)= 0, E(z2
t |Ft−1
)= 1, E
(ut |zt ,Ft−1
)= 0, and E(u2
t |zt ,Ft−1
)=
σ2u . The first two conditions are related to the correct specification of the conditional
mean and variance of rt . The third condition is basically a requirement for δ(zt
)being flexible enough to model the conditional mean of ut . The last condition is a
homoskedasticity assumption on ut .
74 CHAPTER 2.
2.3.1 Asymptotic distribution of estimators
The development of asymptotic theory for conventional GARCH models has been
and continue to be a demanding journey stretching more than two decades. So far, we
only have a limited number of asymptotic results for the simplest models; see Francq
and Zakoïan (2010) and references therein. It is only recently that the asymptotic
theory for the EGARCH(1,1) model was established by Wintenberger (2013) and to
our knowledge nothing has been established for Realized EGARCH models yet. The
introduction of a low frequency component complicates the matter further. Chen
and Hong (2016) develop the asymptotic theory for a time-varying parameter GARCH
model by imposing a smoothness condition such that the process displays locally
stationary behaviour; see also Dahlhaus (1996a), Dahlhaus (1996b), and Dahlhaus
(1997). Likeωt in this paper, the parameters in Chen and Hong (2016) are functions of
rescaled time, t/T , rather than t . A similar approach is taken in Amado and Teräsvirta
(2013), who develop the asymptotic theory for the TV-GJR-GARCH model. This is a
common scaling scheme in the time series literature; see Robinson (1989), Hillebrand,
Medeiros, and Xu (2013) and Dahlhaus and Subba Rao (2006), among others. To
understand the necessity of this approach, we refer to Hillebrand et al. (2013), who
develop the asymptotic theory in Smooth Transition Regression (STR) models with
rescaled time as the state variable. However, the basic idea of the rescaling is to keep
the parameters fixed as a fraction of the sample size when T →∞ such as the amount
of local information increases suitably. For instance, the transitions are assumed to
happen after a certain fraction of the sample and not at a certain point in time. We
leave the development of the asymptotic theory for the estimators of the four new
Realized EGARCH models for future research, but conjecture as in Hansen and Huang
(2016) that
pT
(θ−θ0
)d→N
(0,T I−1JI−1
), (2.27)
where J is the limit of the outer-product of the scores and I is (minus) the limit of the
Hessian matrix for the log-likelihood functions.
2.3.2 Partial log-likelihood function
The partial log-likelihood function is defined by the time t contribution
`pt
(rt , xt ;θ1,θ2
)=−1
2
[log(2π)+ log
(ht
)+ log(g t
)+ z2t
], (2.28)
and it is the Kullback-Leibler measure associated with the conditional distribution
of returns. Consequently, it is directly comparable with the log-likelihood obtained
from conventional GARCH models, such as the EGARCH model.
2.4. FORECASTING 75
2.3.3 Numerical issues related to the Realized TV-EGARCH model
For the TV-GARCH model, Amado and Teräsvirta (2013) find that parameter estima-
tion often is numerically difficult. We encounter similar problems in the Realized
TV-EGARCH model. In practice, it can be very useful to improve the reliability of the
estimates by using non-gradient based optimization routines and relevant parameter
transformations. As in the case of STR models, see Teräsvirta, Tjøstheim, and Granger
(2010), one problem is related to the estimation of the slope parameters, λk , when
it is very large. In that case, the switches in the intercept are rather abrupt which in
turn implies that large changes of λk only affect the transition function in a close
neighbourhood of the location parameter, ck . Therefore, determining the curvature
of the transition function requires a large number of observations of the transition
variable, t/T , in a small neighbourhood of ck . To alleviate numerical issues, we simply
restrict λk to not exceed a certain threshold. In the empirical application, we set the
threshold to 250. In addition, as originally suggested by Goodwin, Holt, and Preste-
mon (2011), we apply the transformation λk = exp(νk
)and estimate νk . Writing the
parameter as an exponential mapping scales the parameter used in the optimization
to a smaller range that is comparable to the other parameters. It is well known that
badly scaled parameters, i.e. parameters that differ considerably in magnitude, can
result in numerical convergence issues. An additional advantage is that no positivity
constraint is needed since the transformation is positively monotone.
As a result of the numerical difficulties often encountered, the choice of initial
values is of utmost importance. We apply a strategy similar to the one suggested
by Teräsvirta et al. (2010) in the context of STR models. In the case of the Realized
TV-EGARCH model, we consider a grid of starting values for the location parameters,
ck ,k = 1, ...,K . Then, conditional on the combination of parameter values in the grid,
we estimate the remaining parameters and choose the starting values that maximize
the QML criterion. Finally, all parameters are estimated based on the chosen starting
values. This strategy is computationally feasible when the number of transitions is
limited and the required identification restrictions, 0 ≤ c1 < ... < cK ≤ 1, are imposed.
2.4 Forecasting
The Realized EGARCH and the extensions presented in this paper can easily be used
to perform multi-period ahead forecasting of both the conditional return variance
and the realized measure. Recursive substitution of the autoregressive equation (2.2)
implies that
loght+h =βh loght +h∑
i=1βi−1
(τ(zt+h−i
)+αut+h−i
). (2.29)
76 CHAPTER 2.
As the logarithm of the conditional return variance is the sum of its stationary part,
loght , and its non-stationary part, log g t , we have that
logσ2t+h =ωt+h +βh loght +
h∑i=1
βi−1(τ(zt+h−i
)+αut+h−i
). (2.30)
We are not required to generate auxiliary future values of the realized measure, xt ,
when the objective is to forecast future values of logσ2t . The reason is that the innova-
tions zt and ut are sufficient for generating the volatility path. Due to the assumption
E[ut
] = E[zt
] = 0 and the specification of the leverage functions, we may obtain a
forecast of logσ2t+h , h > 0 by using
logσ2t+h|t = E
[logσ2
t+h |Ft
]=ωt+h +βh loght +βh−1
(τ(zt
)+αut
). (2.31)
One of the major issues, when forecasting using models with a time-varying structure,
is how to forecast the low-frequency component out-of-sample. As the low-frequency
component models the unconditional variance and may be assumed to be relatively
stable, we have chosen to set it equal to its value at time t , i.e. ωt+h =ωt . This is the
standard approach taken in the literature.
It should be noted that the deterministic component is based on in-sample ob-
servations. Therefore, the new models with a deterministic component are prone
to sudden changes in the level of volatility in the same way as the benchmark coun-
terparts. The idea is to get the level of volatility ’right’ at the end of the sample. As
forecast converge to this level with the forecast horizon, it may be crucial for the
out-of-sample performance. However, it may be empirical preferred to get the level
wrong, but have more persistent (β close on 1) volatility dynamics such that the
convergence to the unconditional volatility is slower.
One issue related to forecasting logσ2t+h is the need to account for the distribu-
tional aspects of logσ2t+h in order to produce an unbiased forecast of σ2
t+h . To deal
with this problem in our forecasting application, we resort to simulations. Thus, we
consider
σ2t+h|t =
1
S
S∑s=1
exp(logσ2
t+h|t ,s
), (2.32)
where logσ2t+h|t ,s is a simulated path obtained using (2.30) and S denotes the num-
ber of simulated paths. In our application, we use the empirical distribution of the
innovations.
2.5 A VaR framework
A key application for volatility modeling is in the field of financial risk management.
New financial regulation, such as the Basel Accords, has dramatically increased the
need for risk measurement. According to Basel Accords, banks are required to hold
an amount of regulatory capital that can be thought of as the amount of capital that
2.5. A VAR FRAMEWORK 77
must be held to make the risk acceptable to regulators. Value at Risk is arguably the
most widely used risk measure in financial institutions and hundreds of academic
papers on VaR have been published.
Numerous methods for forecasting VaR are found in the literature; see Kuester,
Mittnik, and Paolella (2006) for a comparison. In this paper, the 100(1−p
)% VaR is
simply defined as the p-quantile of the return distribution. We consider a method
called "filtered historical simulation" suggested by Barone-Adesi, Bourgoin, and
Giannopoulos (1998) that is easily applicable in a GARCH framework. Let F denote the
zero mean, unit variance and possibly skewed and leptokurtic cumulative distribution
function of the i.i.d. innovations{
zt}
. Our modeling framework implies that the close-
to-close return, rt =√σ2
t zt , conditional on information available at time t − 1 is
distributed as
rt /√σ2
t |Ft−1 ∼ F(0,1
). (2.33)
Thus, the one-day ahead 100(1−p
)% VaR is simply defined by
VaRpt |t−1 ≡ F−1 (
p)√
σ2t . (2.34)
In a QML framework, F is, however, generally unknown and needs to be estimated
either parametrically or nonparametrically. A simple nonparametric method involves
taking for F the empirical distribution of the standardized residuals, rt /√σ2
t . Another
issue is the lack of an explicit formula for the h-day ahead return distribution condi-
tional on the information available at time t −1. This implies that we need to resort
to simulations. We apply the following recipe suggested by Barone-Adesi et al. (1998);
see also Francq and Zakoïan (2010):
1. Fit a model using the observed returns rt , t = 1, ...,n, and compute the esti-
mated volatility σ2t for t = 1, ...,n +1.
2. Simulate a large number S of scenarios for rn+1, ...,rn+h by iterating the follow-
ing three steps for each simulation, s:
a) simulate the i.i.d. innovations zn+1,s , ..., zn+h,s using the empirical distri-
bution F ;
b) set σn+1,s =σn+1 and rn+1,s =σn+1,s zn+1,s ;
c) for k = 2, ...,h, set σn+k,s equal to the value obtained from recursively
applying the volatility model of interest.
3. Determine the h-period VaR as the p-quantile of the simulations rn+1,s + ...+rn+h,s , s = 1, ...,S.
78 CHAPTER 2.
2.6 Empirical application to stock market volatility
In this section, we present an empirical application using returns and realized mea-
sures for the exchange-traded index fund SPY, which tracks the S&P 500 Index. The
series was also considered in Hansen and Huang (2016) and Hansen et al. (2012).
We apply the restriction φ= 1 also used in Hansen and Huang (2016). Generally, the
empirical results indicate that introducing a low-frequency component helps us to
more accurately model volatility in-sample. The computational and practical burden
of estimating the new models and the comparable forecasting and VaR performance
limit the desire to implement the new models unless explicitly called for. However, as
the results from the forecasting and the VaR application seem to differ between sub-
samples, it will be of interest to consider more empirical applications and investigate
forecast combination.
2.6.1 Data and realized measures
Our full sample spans the time period from January 2, 1998 to December 31, 2013,
which amounts to 4025 daily observations. We consider the model fit on the full
sample, but we do also reserve the period until December 31, 2004 (1760 observations)
for initial estimation of the models in the forecasting and VaR application. The only
realized measure that we consider is the RK.1 The motivation for this is twofold. First,
the main goal of this paper is to examine the effect of including a low-frequency
component in Realized EGARCH models. Second, the RK is found to perform very
well in Hansen and Huang (2016), where a range of realized measures are considered,
and in the VaR application by Brownless and Gallo (2010); see also the comparison of
realized measures in Gatheral and Oomen (2010).
The RK by Barndorff-Nielsen et al. (2008) is one of several robust measures of
volatility. In this paper, we consider the variant derived in Barndorff-Nielsen, Hansen,
Lunde, and Shephard (2011) which is given by RK =∑Hh=−H k
(h
H+1
)κh ,where k(x) is
the Parzen kernel and κh =∑ni=|h|+1 r i ,nri−h,n . Here ri ,n = pti ,n −pti−1,n is the intraday
return. See also Barndorff-Nielsen, Hansen, Lunde, and Shephard (2009).
Figure A.1 depicts returns, squared returns (annualized), RK (annualized) and the
autocorrelation function of the RK. The dotted line separates the initial estimation
window from the window used for the forecasting and VaR application. As common
for financial return series, we recognize volatility clustering that has motivated the
GARCH literature. There is a period of relative tranquillity in the middle of the sample
while volatility seems to be higher in the beginning and end of the sample. We
also observe the well-known stylized fact of long-memory looking behaviour in
the different measures of volatility; the autocorrelation function of the RK is slowly
decaying and significant for very high lags. The latter observation motivates either an
1The data was kindly provided by Asger Lunde.
2.6. EMPIRICAL APPLICATION TO STOCK MARKET VOLATILITY 79
explicit modeling of the long-memory property or allowing for structural changes
that can generate spurious long-memory behaviour if not accounted for.
¿ Insert Figure A.1 about here À
2.6.2 Full sample estimates
When applying the models to the full period from January 2, 1998 to December
31, 2013, we simply select the number of transitions, trigonometric functions and
splines using the Hannan-Quinn information criteria.2 It should be noted that the
number of parameters selected in a given model is quite sensitive to the choice of
information criteria. However, we leave a thorough investigation of model selection
for future research. Table A.1 contains the parameter estimates for the full period. The
parameters in ωt are generally not comparable between the specifications and not
particularly interesting, so we leave them out. The parameter estimates for the five
models are quite similar, but in line with our expectations the persistence of volatility
measured by β is reduced when allowing for a time-varying unconditional variance.
¿ Insert Table A.1 about here À
The estimated conditional variance processes, σ2t = g t ht , for the five models are very
hard to discriminate from each other based on a visual inspection. The same picture
is seen from the information contained in Table A.2, which summarizes some basic
descriptive statistics and presents the correlation between the estimated conditional
variance processes. This is interesting compared to the differences between the TV-
GJR-GARCH and GJR-GARCH model found in Amado and Teräsvirta (2013). In this
case, there is a substantial difference between the conditional variance processes
due to the extreme persistence required in classical GARCH model to fit the observed
long-memory behaviour. This supports the usefulness of using realized measures in
GARCH type models as it helps the model to adapt quickly to a new level of volatility.
We only plot the process for the Realized EGARCH in Figure A.2. Evidently, there
is a period of high baseline volatility in the beginning of the sample followed by a
period of relative tranquillity. In the second part of the sample, we see a big spike
in volatility around the financial crisis and some smaller spikes afterwards that may
be contributed to increased uncertainty in financial markets due to the European
sovereign debt crisis.
¿ Insert Table A.2 about here À
Figure A.2 also depicts the unconditional variance for the five Realized EGARCH
models. The Realized TV-, Adaptive-, and Q-Spline-EGARCH models seem to capture
2The Hannan-Quinn information criteria penalizes additional parameters less than the Bayesianinformation criteria, but more than the Akaike information criteria.
80 CHAPTER 2.
similar movements in the baseline volatility, i.e. the high volatility in the beginning of
the crisis and spikes during the financial crisis and the sovereign debt crisis. The ωt
component of the Realized C-Spline-EGARCH model, however, is quite different from
the three other models and not intuitive. A similar picture is seen, when we in Figure
A.3 plot the stationary components for the four non-stationary Realized EGARCH
models. While ht looks ’stationary’ for the Realized TV-, Adaptive-, and Q-Spline-
EGARCH model, it does not seem to be the case for the Realized C-Spline-EGARCH
model. This indicates that using a cubic spline may be problematic, if we want to
separate the conditional variance into a stationary and non-stationary part.
¿ Insert Figure A.2 about here À
¿ Insert Figure A.3 about here À
Even though the QML approach allows for possible misspecification of the distribu-
tional assumptions about{
zt ,ut}
, it is of interest assessing whether the assumptions
seem reasonable. Our findings are very similar to Hansen and Huang (2016) across
all five models. Hence, we only present diagnostics for the Realized TV-EGARCH
model. In order for the score to be a martingale difference sequence, we require
the absence of autocorrelation in zt , z2t ,ut and u2
t . The first 40 autocorrelations are
reported in Figure A.4 with 95% confidence bands. In general, many autocorrelations
are significant for all four series. One concern is that the first order autocorrelations
all are significant at a 5% level. For the case of zt , z2t and u2
t , we do not find more
violations than may be explained by pure chance. In the case of ut , a large proportion
of the first 40 autocorrelations are significant. Hansen and Huang (2016) explain this
with time-variation in σ2u and examine a GARCH structure for σ2
u , but do find little
impact on parameter estimates of introducing this feature. However, based on the
autocorrelation functions, it is more likely that misspecification of the conditional
mean of ut is the issue.
The independence assumption between zt and ut is supported by the finding
that the empirical correlation is numerically equal to zero for all models.
¿ Insert Figure A.4 about here À
Now, we turn to the normality for zt and ut . Figure A.5 depicts QQ-plots for the
two series. The normality assumption for zt seems fairly reasonable, but negative
skewness is observed. One culprit is the extreme outlier that occurred on February 27,
2007. The outlier has been associated with a computer glitch on the New York Stock
Exchange on that day. On the other hand, the empirical distribution of ut exhibits
severe excess kurtosis. Again, this may be related to time-variation in the conditional
mean of ut or alternatively be explained by outliers in zt that affect ut through the
leverage function, δ (z). It should be noted that the degree of time-variation in the
mean of ut is more pronounced for the standard Realized EGARCH model which
2.6. EMPIRICAL APPLICATION TO STOCK MARKET VOLATILITY 81
is supporting the necessity of explicitly modeling a low-frequency component. In
fact, implementing the CUSUM test by Brown, Durbin, and Evans (1975) only results
in rejection of the null hypothesis of no break in the mean of ut in the case of the
Realized EGARCH model; see Appendix A.2.
To sum up, the Realized EGARCH model and the extensions presented in this
paper model the returns reasonably, but the models including a low-frequency com-
ponent are more in line with the assumptions about ut .
¿ Insert Figure A.5 about here À
2.6.3 Forecasting exercise
In this section, we present the results from the forecasting exercise. For both the
forecasting and the Value-at-Risk application we reserve the time period from January
2, 1998 to December 31, 2004 for initial estimation of the models. For every 20th
business day hereafter, we reestimate the model using all data up to that date. As
in the full sample, we select the number of transitions, trigonometric functions and
splines using the Hannan-Quinn information criteria. Reestimation of the model is
necessary as it allows the time-varying component to update. We believe that this
approach is appropriate from a practitioner’s perspective; it is reasonable to assume
that asset managers, etc. update models at least on a monthly basis.
In order to evaluate forecasting performance, we need a proxy for latent volatility.
As the RK is a measure of open-to-close volatility, it will be a biased measure of close-
to-close latent volatility. To deal with this issue, we follow a strategy similar to Sharma
and Vipul (2016). As a proxy for the latent close-to-close volatility, we use σ2t = ηRKt
with
η=1T
∑Tt=1 r 2
t1T
∑Tt=1 RK t
. (2.35)
We have chosen to only use data for the out-of-sample period to calculate η due to
observed time variation in the scaling factor.
It is common in the forecasting literature to implement a variety of evaluation
criteria or loss functions. However, Hansen and Lunde (2006) show that when the
target is observed with error, the choice of criteria becomes critical. In fact, some
criteria may be non-robust, i.e. the ranking based on such a criterion will depend
on whether the proxy or the true latent volatility are the target. Patton (2011) pro-
vides necessary and sufficient conditions on the functional form of the loss function
ensuring consistency of the ordering when using a proxy. Two robust measures are
the Squared Forecasting Error (SFE) and the Quasi-Likelihood (QLIKE) loss function.
82 CHAPTER 2.
Considering h-step ahead forecasts, the two loss functions are defined as respectively
SFEt+h =(σ2
t+h −σ2t+h|t
)2,
QLIKEt+h = σ2t+h
σ2t+h|t
− logσ2
t+h
σ2t+h|t
−1.
The SFE penalizes the forecasting errors in a symmetrical way while the QLIKE is
an asymmetric loss function that penalizes underprediction more heavily than over-
prediction. The latter may be suitable if the application involves risk management and
VaR forecasting, where under-prediction of volatility is more of a concern than over-
prediction. Since the QLIKE loss function is related to the Gaussian log-likelihood
function, it should be comparable with the likelihood based inference in Hansen et al.
(2012).
Tabel A.3 presents the results based on the full out-of-sample period from January
2, 2005 to December 31, 2013. By looking at the median forecast error, we see that
all models tend to overpredict latent volatility measured by rescaled RK. In general,
the Realized EGARCH model, followed by the EGARCH model, performs best using
both robust loss functions, i.e. the MSFE and the QLIKE loss function. However, it
should be noted that the Realized TV-, C-Spline-, and Q-Spline-Q-EGARCH model
perform better when considering the Median Squared Forecast Error (MedSFE).
This indicates that the poor performance of the new models may be related to a
limited number of large forecast errors. The upper panel of Figure A.6 depicts the
MSFE of the models relative to the MSFE of the Realized EGARCH model and a filled
marker indicates that the forecasting model is included in the Model Confidence
Set (MCS) by Hansen et al. (2011) at a 10% significance level. The Realized EGARCH,
the Realized Adaptive-EGARCH and the EGARCH model are all included in the MCS
at all forecasting horizons while the rest of the models are included only at some
horizons. As expected, it seems that using a Realized EGARCH model is preferred
at short horizons. However, a standard EGARCH model seems suitable at longer
horizons. The picture is the same when considering the QLIKE loss function in Figure
A.7.
We find that the forecasting performance is very sensitive to the choice of out-of-
sample period. In fact, the new models tend to perform much better before and after
the global financial and European sovereign debt crisis. This may be contributed to
estimation difficulties related to fitting the unconditional variance during extreme
market conditions. To investigate this issue further, we present in Table A.4 the
results from a forecasting exercise when excluding the one year period from August 1,
2008 to July 31, 2009 covering the most volatile parts of the financial crisis. Clearly,
the non-stationary models now perform much better and, in fact, better than the
Realized EGARCH and EGARCH model for many horizons. Considering the MCS
for the shortened sample presented in the lower panel of Figure A.6, the Realized
Adaptive-EGARCH model is now the only one included at all horizons.
2.6. EMPIRICAL APPLICATION TO STOCK MARKET VOLATILITY 83
¿ Insert Table A.3-A.4 about here À
¿ Insert Figure A.6-A.7 about here À
2.6.4 VaR application
We consider three different values for the VaR coverage rate p, 0.01, 0.05 and 0.1.
As a benchmark, we also consider VaR forecasts from the EGARCH model and a
Historical Simulation (HS) method. In HS, the VaR forecasts are made based solely
on the empirical distribution of historical returns. Figure A.8 shows the empirical
coverage rates compared to the theoretical coverage rates for all forecast horizons.
It is clear that all models incorporating time-varying volatility are generally very
close and tend to produce conservative VaR forecasts. Except for one-period VaR the
models generally underestimate the coverage rate. The overprediction of the latent
volatility presented in the previous section gives some support for this result. HS is
different since it implicitly assumes a constant volatility. The model overestimates
the coverage rate for 1% VaR and underestimates for 5% and 10% VaR.
¿ Insert Figure A.8 about here À
Methods for backtesting VaR forecasts are usually based on two key statistical proper-
ties: unconditional coverage and independence. The first property is obviously related
to whether the model provides an empirical coverage rate in line with the theoretical.
The independence property states that VaR violations should be independent over
time. A good model is expected to satisfy both properties. Generally, the tests are
based on the sequence
Hi tt+h|t =1{
r t+h|t < VaRpt+h|t
}(2.36)
of VaR violations where r t+h|t = rt+h+rt+h−1+...+rt+1. Coverage tests include Kupiec
(1995), Pearsons-Q in Cambell (2007), and Pérignon and Smith (2008)(multivariate).
Independence tests are presented in Christoffersen (1998), Berkowitz, Christoffersen,
and Pelletier (2011) and Christoffersen and Pelletier (2004). Joint tests for both cover-
age and independence include Engle and Manganelli (2004) and Hurlin and Tokpavi
(2007)(multivariate). Here multivariate refers to tests that consider several coverage
rates jointly. We have implemented all aforementioned tests using a 5% significance
level.3 Despite the conservative VaR forecasts presented in Figure A.8, the coverage
tests do not reject the null of a correctly specified model more often than what may
be explained by chance. The independence tests rarely reject the null for all mod-
els incorporating time-varying volatility. For HS, the null is rejected for almost all
3Generally the tests assume that the sequence of Hi t variables cannot be based on overlappingforecasts horizons. In our forecasting exercise this is not satisfied when h ≥ 2. In these cases, the tests arebased on Bonferroni sub samples of length T /h and we reject the null if the p-value from any sub sampleis smaller than 0.05/h.
84 CHAPTER 2.
coverage rates and forecast horizons. In periods of high volatility, HS often leads to
violation clusters and thereby violates the assumption of independence. This is a
general finding and supports the idea that time-varying volatility is crucial when
modeling VaR. The joint test for both coverage and independence rejects the null
more often. However, for the time-varying volatility models there are no systematic
rejections and it is impossible to announce a ’winner’ based on these tests.
One problem with many of the tests is that they may exhibit low power in realistic
finite sample sizes since the amount of information in a binary sequence is limited
(Cambell (2007)). An alternative is to specify a loss function that also depends on the
magnitude of VaR violations. The tick loss function
T Lpt =
(p −1
{r t+h|t < VaRp
t+h|t})(
r t+h|t −VaRpt+h|t
)(2.37)
examined in Giacomini and Komunjer (2005) is an example of such a function. The
function is motivated by the loss functions used in quantile regression. If a model
provides a VaR that is indeed the p-quantile, it would minimize the average tick loss
function. Figure A.9 presents the ratio of the average tick loss functions relative to
the standard Realized EGARCH model for the three coverage rates and all forecasts
horizons. Filled out markers indicate that the model is part of the MCS at a 10%
significance level. HS performs poorly and for visualization we choose omit it. It is
only a part of the MCS for long forecasting horizons when p = 0.01. The plots show
that we only rarely are able to exclude any of the time- varying volatility models from
the MCS especially for long forecasting horizons. The performance of the models
is generally very similar. However, for very short forecasting horizons, we find the
EGARCH model to be the worst performing model across all coverage rates. This
is expected since the Realized EGARCH models are adapting faster to a changing
volatility environment than conventional GARCH models. The effect is, however,
not always large enough to exclude the EGARCH model from the MCS. For medium
forecast horizons and a coverage rate of 0.05 or 0.1 the EGARCH model actually
performs better than the models based on realized measures.
Similar to the forecasting analysis, the VaR measures are also heavily affected
by the financial crises. Figure A.10 presents the ratio of average tick loss functions
excluding the period during the financial crises. For short term forecasting horizons,
we find no significant differences compared to the full sample. The EGARCH model
is still the worst performing model and all Realized EGARCH models generally show
similar behaviour. For long term horizons we observe a different picture. For many
horizons it is possible to exclude the Realized C-Spline-EGARCH model from the
MCS. For p = 0.01, the standard Realized EGARCH model can also be excluded. While
the Realized TV-EGARCH model was generally the worst performing model in the full
sample, it is now the best model closely followed by the Realized Q-Spline-EGARCH
model.
¿ Insert Figure A.9-A.10 about here À
2.7. CONCLUSION 85
2.7 Conclusion
In this paper, we suggest four new Realized EGARCH models by introducing a low-
frequency or non-stationary component into the Realized EGARCH framework of
Hansen and Huang (2016). We suggest four different specifications of the low-frequency
component inspired by the existing GARCH literature. The low-frequency component
is deterministically time-varying and proxies all factors that affect the unconditional
variance. The addition of this component has the potential to capture the long-run
dynamic behaviour of volatility such as structural changes that are bound to occur in
sufficiently long time series. Thus, the model keeps the attractive features of GARCH
type models in fitting and forecasting at high and medium frequency while allow-
ing for a low-frequency component associated with volatility dynamics at longer
horizons.
We consider an empirical application to the exchange-traded index fund SPY
examining both the performance in respect to forecasting and VaR calculations. In
general, models utilizing the RK outperform the standard EGARCH model at short
horizons but not necessarily at longer horizons. The new Realized EGARCH models
deliver a better in-sample fit, but in general fail to deliver superior performance
out-of-sample in regards to forecasting and VaR calculations. It seems that a Realized
EGARCH model does quite a good job. The initial suggestion for the practitioner is to
apply the Realized EGARCH model and only model a low-frequency component if
it seems reasonable, e.g. in a situation where the GFC is included in the estimation
window and not in the forecasting window. It will be of interest to consider more
empirical studies and see whether forecast combination can improve the out-of-
sample fit.
Acknowledgement
We thank Timo Teräsvirta, Asger Lunde, Peter Reinhardt Hansen, and Esther Ruiz
Ortega for insightful comments. We also extend our gratitude to the participant at the
27th New Zealand Econometric Study Group meeting and the Brown-bag seminar
at the School of Economics and Finance, Queensland University of Technology for
feedback on this paper.
The authors acknowledge support from CREATES - Center for Research in Econo-
metric Analysis of Time Series (DNRF78), funded by the Danish National Research
Foundation.
86 CHAPTER 2.
2.8 References
Amado, C., Teräsvirta, T., 2013. Modelling volatility by variance decomposition. Jour-
nal of Econometrics 175, 142–153.
Amado, C., Teräsvirta, T., 2017. Specification and testing of multiplicative time-
varying GARCH models with applications. Econometric Reviews 36 (4), 421–446.
Andersen, T., Bollerslev, T., 1997. Heterogeneous information arrivals and return
volatility dynamics: Uncovering the long-run in high frequency returns. Journal of
Finance 52, 975–1005.
Andersen, T., Bollerslev, T., 1998. Answering the skeptics: Yes, standard volatility
models do provide accurate forecasts. International Economic Review 39, 885–905.
Andersen, T. G., Bollerslev, T., Diebold, F. X., Labys, P., 2003. Modeling and forecasting
realized volatility. Econometrica 71, 579–625.
Baillie, R. T., Bollerslev, T., Mikkelsen, H. O., 1996. Fractionally Integrated Generalized
Autoregressive Conditional Heteroskedasticity. Journal of Econometrics 74, 3–30.
Baillie, R. T., Morana, C., 2009. Modelling long memory and structural breaks in condi-
tional variances: An Adaptive FIGARCH approach. Journal of Economic Dynamics
and Control 33, 1577–1592.
Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N., 2008. Designing
realized kernels to measure the ex post variation of equity prices in the presence of
noise. Econometrica 76, 1481–1536.
Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N., 2009. Realized kernels
in practice: Trades and quotes. The Econometrics Journal 12, 1–32.
Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N., 2011. Multivariate
realised kernels: Consistent positive semi-definite estimators of the covariation of
equity prices with noise and non-synchronous trading. Journal of Econometrics
162, 149 – 169.
Barndorff-Nielsen, O. E., Shephard, N., 2002. Econometric analysis of realized volatil-
ity and its use in estimating stochastic volatility models. Journal of the Royal Statis-
tical Society: Series B (Statistical Methodology) 64, 253–280.
Barone-Adesi, G., Bourgoin, F., Giannopoulos, K., 1998. Don’t Look Back. Risk 11,
100–104.
Bellegem, S. V., von Sachs, R., 2004. Forecasting economic time series with uncondi-
tional time-varying variance. International Journal of Forecasting 20, 611–627.
2.8. REFERENCES 87
Berkowitz, J., Christoffersen, P., Pelletier, D., 2011. Evaluating Value-at-Risk models
with desk-level data. Management Science 57, 2213–2227.
Bollerslev, T., 1986. Generalized Autoregressive Conditional Heteroscedasticity. Jour-
nal of Econometrics 31, 207–327.
Brown, R., Durbin, J., Evans, J., 1975. Techniques for testing the constancy of re-
gression relationships over time. Journal of the Royal Statistical Society. Series B
(Methodological) 37, 149–192.
Brownless, C. T., Gallo, G. M., 2010. Comparison of volatility measures: A risk man-
agement perspective. Journal of Financial Econometrics 8, 29–56.
Cambell, S. D., 2007. A review of backtesting and backtesting procedures. Journal of
Risk 9, 1–17.
Chen, B., Hong, Y., 2016. Detecting for smooth structural changes in GARCH models.
Econometric Theory 32, 740–791.
Christensen, K., Podolskij, M., 2007. Realized range-based estimation of integrated
variance. Journal of Econometrics 141, 323 – 349.
Christoffersen, P., 1998. Evaluating interval forecasts. International Economic Review
30, 841–862.
Christoffersen, P., Feunou, B., Jacobs, K., Meddahi, N., 2014. The economic value of
realized volatility: Using high-frequency returns for option valuation. Journal of
Financial and Quantitative Analysis 49, 663–697.
Christoffersen, P., Pelletier, D., 2004. Backtesting Value-at-Risk: A duration-based
approach. Journal of Empirical Finance 2, 84–108.
Corsi, F., 2009. A simple approximate long-memory model of realized volatility. Jour-
nal of Financial Econometrics 7, 174–196.
Dahlhaus, R., 1996a. Maximum likelihood estimation and model selection for locally
stationary processes. Journal of Nonparametric Statistics 6, 171–191.
Dahlhaus, R., 1996b. On the Kullback-Leibler information divergence of locally sta-
tionary processes. Stochastic Processes and their Applications 62, 139–168.
Dahlhaus, R., 1997. Fitting time series models to nonstationary processes. Annals of
Statistics 25, 1–37.
Dahlhaus, R., Subba Rao, S., 2006. Statistical inference for time-varying ARCH pro-
cesses. Annals of Statistics 34, 1075–1114.
88 CHAPTER 2.
Diebold, F. X., 1986. Modeling the persistence of conditional variances: A comment.
Econometric Reviews 5, 51–56.
Dobrev, D., Szerszen, P., 2010. The information content of high-frequency data for
estimating equity return models and forecasting risk. Finance and economics
discussion series. Federal Reserve Board.
Engle, R., 2002. New frontiers for ARCH models. Journal of Applied Econometrics 17,
425–446.
Engle, R., Manganelli, S., 2004. CAViaR: Conditional autoregressive Value at Risk by
regression quantiles. American Statistical Association 22, 367–381.
Engle, R. F., 1982. Autoregressive conditional heteroscedasticity with estimates of the
variance of United Kingdom inflation. Econometrica 50, 987–1007.
Engle, R. F., Gallo, G. M., 2006. A multiple indicators model for volatility using intra-
daily data. Journal of Econometrics 131, 3–27.
Engle, R. F., Rangel, J. G., 2008. The Spline-GARCH model for low-frequency volatility
and its global macroeconomic causes. Review of Financial Studies 21, 1187–1222.
Feng, Y., 2004. Simultaneously modeling conditional heteroskedasticity and scale
change. Econometric Theory 20, 563–596.
Francq, C., Zakoïan, J.-M., 2010. GARCH models: Structure, statistical inference and
financial applications. John Wiley & Sons, Ltd.
French, K., Schwert, G., Stambaugh, R., 1987. Expected stock returns and volatility.
Journal of Financial Economics 19, 3–29.
Gallant, R., 1984. The Fourier flexible form. American Journal of Agricultural Eco-
nomics 66, 204–208.
Gatheral, J., Oomen, R. C. A., 2010. Zero-intelligence realized variance estimation.
Finance and Stochastics 14, 249–283.
Giacomini, R., Komunjer, I., 2005. Evaluating and combination of conditional quantile
forecasts. American Statistical Association 23, 416–431.
Goodwin, B. K., Holt, M. T., Prestemon, J. P., 2011. North american oriented strand
board markets, arbitrage activity, and market price dynamics: A smooth transition
approach. American Journal of Agricultural Economics 93, 993–1014.
Hansen, P. R., Huang, Z., 2016. Exponential GARCH modeling with realized measures
of volatility. Journal of Business and Economic Statistics 34:2, 269–287.
2.8. REFERENCES 89
Hansen, P. R., Huang, Z., Shek, H. H., 2012. Realized GARCH: A joint model for returns
and realized measures of volatility. Journal of Applied Econometrics 27, 877–906.
Hansen, P. R., Lunde, A., 2006. Consistent ranking of volatility models. Journal of
Econometrics 131, 97 – 121.
Hansen, P. R., Lunde, A., Nason, J. M., 2011. The model confidence set. Econometrica
79, 453–497.
Hillebrand, E., 2005. Neglecting parameter changes in GARCH models. Journal of
Econometrics 129, 121 – 138.
Hillebrand, E., Medeiros, M. C., Xu, J., 2013. Asymptotic theory for regressions with
smoothly changing parameters. Journal of Time Series Econometrics 5, 133–162.
Hurlin, C., Tokpavi, S., 2007. Backtesting Value-at-Risk accuracy: A simple new test.
Journal of Risk 9, 19–37.
Kuester, K., Mittnik, S., Paolella, M. S., 2006. Value-at-Risk prediction: A comparison
of alternative strategies. Journal of Financial Econometrics 4, 53–89.
Kupiec, P., 1995. Techniques for verifying the accuracy of risk management models.
Journal of Derivatives 3, 73–84.
Lamoureux, C. G., Lastrapes, W. D., 1990. Persistence in variance, structural change,
and the GARCH Model. Journal of Business & Economic Statistics 8, 225–234.
Mazur, B., Pipien, M., 2012. On the empirical importance of periodicity in the volatility
of financial returns - time varying GARCH as a second order APC(2) process. Central
European Journal of Economic Modelling and Econometrics 4, 95–116.
Mikosch, T., Starica, C., 2004. Nonstationarities in financial time series, the long-
range dependence, and the IGARCH effects. Review of Economics and Statistics 86,
378–390.
Mishra, S., Su, L., Ullah, A., 2010. Semiparametric estimator of time series conditional
variance. Journal of Business & Economic Statistics 28, 256–274.
Morana, C., 2002. IGARCH effects: An interpretation. Applied Economics Letters 9,
745–748.
Morana, C., Beltratti, A., 2004. Structural change and long-range dependence in
volatility of exchange rates: Either, neither or both? Journal of Empirical Finance
11, 629–658.
Patton, A. J., 2011. Volatility forecast comparison using imperfect volatility proxies.
Journal of Econometrics 160, 246 – 256, realized Volatility.
90 CHAPTER 2.
Pérignon, C., Smith, D. R., 2008. A new approach to comparing VaR estimation meth-
ods. The Journal of Derivatives 16, 54–66.
Robinson, P. M., 1989. Nonparametric estimation of time-varying parameters.
Springer Berlin Heidelberg.
Sharma, P., Vipul, 2016. Forecasting stock market volatility using Realized GARCH
model: International evidence. The Quarterly Review of Economics and Finance
59, 222 – 230.
Shephard, N., Sheppard, K., 2010. Realising the future: forecasting with high-
frequency-based volatility (HEAVY) models. Journal of Applied Econometrics 25,
197–231.
Teräsvirta, T., 2009. An introduction to univariate GARCH models. In: Handbook of
Financial Time Series. Springer Berlin Heidelberg.
Teräsvirta, T., 2012. Nonlinear models for autoregressive conditional heteroskedastic-
ity. John Wiley & Sons, Inc.
Teräsvirta, T., Tjøstheim, D., Granger, C. W. J., 2010. Modelling nonlinear economic
time series. Oxford University Press.
Van Bellegem, S., 2012. Locally stationary volatility modeling. John Wiley & Sons, Inc.
Wintenberger, O., 2013. Continuous invertibility and stable QML estimation of the
EGARCH(1,1) model. Scandinavian Journal of Statistics 40, 846–867.
Zhang, L., 2006. Efficient estimation of stochastic volatility using noisy observations:
A multi-scale approach. Bernoulli 12, 1019–1043.
Zhang, L., Mykland, P. A., Aït-Sahalia, Y., 2005. A tale of two time scales: Determin-
ing integrated volatility with noisy high-frequency data. Journal of the American
Statistical Association 100, 1394–1411.
A.1. DERIVATION OF SCORES 91
Appendix
A.1 Derivation of scores
A.1.1 Proof of Lemma 1
First, consider ∂ loght+1/∂ loght . Using zt = rt
h1/2t g 1/2
t, it can easily be shown that
∂zt
∂ loght=−1
2zt . (A.1)
For ut = log xt −φ logσ2t −δ
(zt
), we find that
∂ut
∂ loght=− ∂δ
(zt
)∂ loght
−φ, (A.2)
where∂δ
(zt
)∂ loght
= δ′ ∂b(zt
)∂zt
∂zt
∂ loght. (A.3)
Similarly, we have∂τ
(zt
)∂ loght
= τ′ ∂a(zt
)∂zt
∂zt
∂ loght. (A.4)
Inserting the above components in the following expression for the ∂ loght+1/∂ loght
∂ loght+1
∂ loght=β+ ∂τ
(zt
)∂ loght
+α ∂ut
∂ loght(A.5)
yields∂ loght+1
∂ loght= (
β−αφ)+ 1
2
(αδ
∂b(zt
)∂zt
−τ∂a(zt
)∂zt
)zt . (A.6)
Using the exactly the same arguments, we can derive
∂ loght+1
∂ log g t=−αφ+ 1
2
(αδ
∂b(zt
)∂zt
−τ∂a(zt
)∂zt
)zt . (A.7)
Next, we turn to the derivates of `t with respect to loght and log g t . As the two
expressions will be identical, we only derive it with respect to loght . loght enters the
log-likelihood contribution at time t directly due to the loght term and indirectly
through z2t and u2
t . Thus, we have
∂`t
∂ loght=−1
2
[1+ ∂z2
t
∂ loght+ 1
σ2u
2ut∂ut
∂ loght
]. (A.8)
We note that∂z2
t
∂ loght=−z2
t . (A.9)
92 CHAPTER 2.
Combing the different expressions yields,
∂`t
∂ loght=−1
2
(1− z2
t
)+ ut
σ2u
(δ∂b
(zt
)∂zt
zt −2φ
) . (A.10)
�
A.1.2 Proof of Lemma 2
First, consider ∂ loght+1/∂(β,τ′,α
)′. We note that
∂ loght+1
∂(β,τ′,α
)′ =β ∂ loght
∂(β,τ′,α
) + ∂τ(zt
)∂(β,τ′,α
) +α ∂ut
∂(β,τ′,α
) + (loght , zt , z2
t −1,ut
)′. (A.11)
However, we remember that τ(zt
)and ut only depends on
(β,τ′,α
)′ through loght
such that the three first terms can be collapsed to one. Thus, we can instead write
∂ loght+1
∂(β,τ′,α
)′ = ∂ loght+1
∂ loght
∂ loght
∂(β,τ′,α
) + (loght , zt , z2
t −1,ut
)′. (A.12)
Exactly, what we wanted to show. For ∂ loght+1/∂(ξ,φ,δ′
)′, we have
∂ loght+1
∂(β,τ′,α
)′ =β ∂ loght
∂(ξ,φ,δ′
) + ∂τ(zt
)∂(ξ,φ,δ′
) +α(1, log
(σ2
t
), zt , z2
t −1
)′. (A.13)
Using the same arguments as before, we obtain
∂ loght+1
∂(ξ,φ,δ′
)′ = ∂ loght+1
∂ loght
∂ loght
∂(ξ,φ,δ′
)′ +α(1, log
(σ2
t
), zt , z2
t −1
)′. (A.14)
Finally, for ∂ loght+1/∂θ2, we can write
∂ loght+1
∂θ2=β∂ loght
∂θ2+ ∂τ
(zt
)∂θ2
+α∂ut
∂θ2. (A.15)
We note that∂zt
∂θ2=−1
2zt
[∂ loght
∂θ2+ ∂ log g t
∂θ2
](A.16)
and that θ2 does not enter directly in loght+1, but only indirectly through loght and
log g t . Thus, we can additively separate the score in the following two contributions
∂ loght+1
∂θ2= ∂ loght+1
∂ loght
∂ loght
∂θ2+ ∂ loght+1
∂ log g t
∂ log g t
∂θ2. (A.17)
Exactly, what we wanted to show.
�
A.1. DERIVATION OF SCORES 93
A.1.3 Proof of Theorem 1
As(β,τ′,α
)′ only enters the log-likelihood contribution at time t indirectly through
loght , an application of the chain-rule yields
∂`t
∂(β,τ′,α
)′ = ∂`t
∂ loght
∂ loght
∂(β,τ′,α
)′ , (A.18)
and as(ξ,φ,δ′
)′ enters through loght and u2t , an application of the chain-rule yields
∂`t
∂(ξ,φ,δ′
)′ = ∂`t
∂ loght
∂ loght
∂(ξ,φ,δ′
)′ + ut
σ2u
(1, log
(σ2
t
), zt , z2
t −1
)′. (A.19)
The parameter σ2u only enters directly into the log-likelihood contribution such that
we obtain
∂`t
∂σ2u= 1
2
u2t −σ2
u
σ4u
. (A.20)
Finally, θ2 enters the log-likelihood contribution both through loght and log g t . We
obtain∂`t
∂θ2= ∂`t
∂ loght
∂ loght
∂θ2+ ∂`t
∂ log g t
∂ log g t
∂θ2, (A.21)
where ∂ log g t∂θ2
differs between the different specifications for the non-stationary com-
ponent.
�
Derivatives of log g t with respect to θ2
In the case of the Realized EGARCH model, we have
∂ log g t
∂ω= 1. (A.22)
In the case of the Realized TV-EGARCH model, we have
∂ log g t
∂ω= 1, (A.23)
∂ log g t
∂γi=G
(t
T;λi ,ci
)i = 1, ...,K , (A.24)
∂ log g t
∂λi= γi G
(t
T;λi ,ci
)(1−G
(t
T;λi ,ci
))(t/T − ci
)i = 1, ...,K , (A.25)
∂ log g t
∂ci=−γi G
(t
T;λi ,ci
)(1−G
(t
T;λi ,ci
))i = 1, ...,K . (A.26)
94 CHAPTER 2.
In the case of the Realized Adaptive-EGARCH model, we have
∂ log g t
∂ω= 1, (A.27)
∂ log g t
∂γi= sin
(2πi
t
T
)i = 1, ...,K , (A.28)
∂ log g t
∂λi= cos
(2πi
t
T
)i = 1, ...,K . (A.29)
In the case of the Realized Spline-Q-EGARCH model, we have
∂ log g t
∂ω= 1, (A.30)
∂ log g t
∂γ0= t i = 1, ...,K , (A.31)
∂ log g t
∂γ1i=
((t − tk−1
)+)2
i = 1, ...,K . (A.32)
In the case of the Realized Spline-Q-EGARCH model, we have
∂ log g t
∂ω= 1, (A.33)
∂ log g t
∂γ0= t , (A.34)
∂ log g t
∂γ1= t 2, (A.35)
∂ log g t
∂γ2k=
((t − tk−1
)+)3
i = 1, ...,K . (A.36)
A.2. FIGURES 95
A.2 Figures
1998 2002 2006 2009 2013
-0.1
0
0.1
1998 2002 2006 2009 20130
2
4
6
1998 2002 2006 2009 20130
0.5
1
1.5
2
20 40 60 80 1000
0.2
0.4
0.6
0.8
1
Figure A.1: Data plotsThe upper-left panel: SPY daily return. The lower-left panel: SPY daily RK (annualized). Theupper-right panel: SPY daily squared returns (annualized). The lower-right panel: SPY daily RKautocorrelation function.
96 CHAPTER 2.
1998 2002 2006 2009 20130
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure A.2: Illustration of non-stationery componentAnnualized volatility for Realized EGARCH (grey). Annualized unconditional volatility gtfor Realized EGARCH (black), Realized TV-EGARCH (red), Realized Adaptive-EGARCH(yellow), Realized C-Spline-EGARCH (green) and Realized Q-Spline-EGARCH (purple).
A.2. FIGURES 97
1998 2002 2006 2009 20130
5
10
15
20
25
30
1998 2002 2006 2009 20130
5
10
15
20
25
30
1998 2002 2006 2009 20130
5
10
15
20
25
30
1998 2002 2006 2009 20130
5
10
15
20
25
30
Figure A.3: illustration of stationary part of volatility processIllustration of ht for Realized TV EGARCH (red), Realized Adaptive-EGARCH (yel-low), Realized C-Spline-EGARCH (green) and Realized Q-Spline-EGARCH (purple).
98 CHAPTER 2.
Lag10 20 30 40
Autocorrelationofzt
-0.1
-0.05
0
0.05
0.1
Lag10 20 30 40
Autocorrelationofz2 t
-0.1
-0.05
0
0.05
0.1
Lag10 20 30 40
Autocorrelationofut
-0.1
-0.05
0
0.05
0.1
Lag10 20 30 40
Autocorrelationofu2 t
-0.1
-0.05
0
0.05
0.1
Figure A.4: Autocorrelations for z, z2,u and u2
A.2. FIGURES 99
Standard Normal Quantiles-4 -2 0 2 4
EmpiricalQuantilesofzt
-8
-6
-4
-2
0
2
4
Standard Normal Quantiles-4 -2 0 2 4
EmpiricalQuantilesofut
-3
-2
-1
0
1
2
3
Figure A.5: QQ-plots for z and u
100 CHAPTER 2.
Full sample
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.90
1.00
1.10
1.20
1.30
Excl. Financial Crisis
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.90
1.00
1.10
1.20
1.30
Figure A.6: Ratio of MSFE relative to the stationary Realized EGARCHFilled out markers indicate that the model is part of the MCS at a 10% significance level.Realized EGARCH (black), Realized TV-EGARCH (red), Realized Adaptive-EGARCH (yellow),Realized C-Spline-EGARCH (green), Realized Q-Spline-EGARCH (purple) and EGARCH (blue).
A.2. FIGURES 101
Full sample
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.75
1.00
1.25
1.50
1.75
Excl. Financial Crisis
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.75
1.00
1.25
1.50
1.75
Figure A.7: Ratio of MSFE relative to the stationary Realized EGARCHFilled out markers indicate that the model is part of the MCS at a 10% significance level.Realized EGARCH (black), Realized TV-EGARCH (red), Realized Adaptive-EGARCH (yellow),Realized C-Spline-EGARCH (green), Realized Q-Spline-EGARCH (purple) and EGARCH (blue).
102 CHAPTER 2.
Forecast Horizon - h
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Em
piric
al C
over
age
Rat
e
0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
Figure A.8: Empirical coverage ratesSignificance levels are indicated in the following way: p = 0.01 (circle), p = 0.05 (triangle) andp = 0.1 (square). Realized EGARCH (black), Realized TV-EGARCH (red), Realized Adaptive-EGARCH (yellow), Realized C-Spline-EGARCH (green), Realized Q-Spline-EGARCH (purple),EGARCH (blue) and Historical Simulation (grey).
A.2. FIGURES 103
Full sample
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.92
0.97
1.03
1.09
1.14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.97
0.98
0.99
1.00
1.01
1.02
1.03
1.04
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.97
0.98
0.99
1.01
1.02
1.03
Figure A.9: Ratio of average tick loss functions relative to the stationary Realized EGARCHFilled out markers indicate that the model is part of the MCS at a 10% significance level.Realized EGARCH (black), Realized TV-EGARCH (red), Realized Adaptive-EGARCH (yellow),Realized C-Spline-EGARCH (green), Realized Q-Spline-EGARCH (purple) and EGARCH (blue).
104 CHAPTER 2.
Excl. Financial Crisis
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.92
0.97
1.03
1.09
1.14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.97
0.98
0.99
1.00
1.01
1.02
1.03
1.04
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.97
0.98
0.99
1.01
1.02
1.03
Figure A.10: Ratio of average tick loss functions relative to the stationary Realized EGARCHFilled out markers indicate that the model is part of the MCS at a 10% significance level.Realized EGARCH (black), Realized TV-EGARCH (red), Realized Adaptive-EGARCH (yellow),Realized C-Spline-EGARCH (green), Realized Q-Spline-EGARCH (purple) and EGARCH (blue).
A.2. FIGURES 105
Stationary
500 1000 1500 2000 2500 3000 3500 4000
Iteration
-200
-150
-100
-50
0
50
100
150
CU
SU
M
Figure A.11: CUSUM test
TV Adaptive
1000 2000 3000 4000
Iteration
-150
-100
-50
0
50
100
150
CU
SU
M
1000 2000 3000 4000
Iteration
-150
-100
-50
0
50
100
150
CU
SU
M
Q-Spline C-Spline
1000 2000 3000 4000
Iteration
-150
-100
-50
0
50
100
150
CU
SU
M
1000 2000 3000 4000
Iteration
-150
-100
-50
0
50
100
150
CU
SU
M
Figure A.12: CUSUM test.
106 CHAPTER 2.
A.3 Tables
Table A.1: Maximum likelihood parameter estimates for the five different modelsStandard errors are given in parenthesis. p denotes the total number of parameters in the givenmodel.
Stationary TV Adaptive Q-Spline C-Spline
ω -9.105 (0.0717) -8.788 (0.0553) -9.126 (0.0424) -8.837 (0.2069) -8.797 (0.1482)β 0.969 (0.0028) 0.921 (0.0053) 0.936 (0.0042) 0.943 (0.0039) 0.966 (0.0026)α 0.301 (0.0113) 0.284 (0.0116) 0.289 (0.0113) 0.290 (0.0114) 0.281 (0.0106)ξ -0.278 (0.0193) -0.274 (0.0198) -0.277 (0.0199) -0.277 (0.0198) -0.280 (0.0194)σ2
u 0.160 (0.0026) 0.154 (0.0025) 0.156 (0.0025) 0.156 (0.0026) 0.158 (0.0026)τ1 -0.149 (0.0053) -0.154 (0.0056) -0.154 (0.0056) -0.153 (0.0056) -0.147 (0.0053)τ2 0.032 (0.0026) 0.033 (0.0024) 0.033 (0.0023) 0.033 (0.0024) 0.032 (0.0024)δ1 -0.124 (0.0061) -0.122 (0.0061) -0.123 (0.0061) -0.122 (0.0061) -0.121 (0.0061)δ2 0.047 (0.0031) 0.047 (0.0031) 0.048 (0.0030) 0.048 (0.0030) 0.045 (0.0031)
p 9 27 21 18 14logL 10,683.87 10,758.99 10,737.32 10,728.02 10,698.84
logLp 12,705.82 12,712.61 12,707.88 12,706.45 12,705.93
Table A.2: Descriptive statistics for the annualized volatility of the five models
Stationary TV Adaptive Q-Spline C-Spline
Mean 18.04 18.02 18.02 18.01 18.06Std. 9.47 9.61 9.37 9.32 9.45Min 5.93 6.11 6.34 5.72 5.68Max 97.68 98.41 94.55 94.17 95.68
CorrelationsStationary 1.000
TV 0.996 1.000Adaptive 0.998 0.998 1.000Q-Spline 0.998 0.997 0.999 1.000C-Spline 1.000 0.996 0.998 0.999 1.000
A.3. TABLES 107
Table A.3: Full sample forecasting results
Realized EGARCH
Horizon MedFE MSFE MedSFE QLIKE Lp
h = 1 −8.145 ·10−6 1.078 ·10−7 4.293 ·10−10 1.084 ·10−1 7331.76h = 5 −1.587 ·10−5 1.509 ·10−7 1.079 ·10−9 2.150 ·10−1
h = 10 −2.559 ·10−5 1.885 ·10−7 1.772 ·10−9 3.030 ·10−1
h = 20 −4.270 ·10−5 2.149 ·10−7 3.522 ·10−9 4.315 ·10−1
Realized TV-EGARCH
Horizon MedFE MSFE MedSFE QLIKE Lp
h = 1 −6.622 ·10−6 1.181 ·10−7 4.001 ·10−10 1.100 ·10−1 7326.89h = 5 −1.289 ·10−5 1.757 ·10−7 9.336 ·10−10 2.284 ·10−1
h = 10 −1.682 ·10−5 2.090 ·10−7 1.251 ·10−9 3.350 ·10−1
h = 20 −2.165 ·10−5 2.389 ·10−7 1.775 ·10−9 4.986 ·10−1
Realized Adaptive-EGARCH
Horizon MedFE MSFE MedSFE QLIKE Lp
h = 1 −9.237 ·10−6 1.137 ·10−7 4.503 ·10−10 1.089 ·10−1 7327.28h = 5 −1.967 ·10−5 1.654 ·10−7 1.178 ·10−9 2.217 ·10−1
h = 10 −2.841 ·10−5 2.013 ·10−7 1.919 ·10−9 3.188 ·10−1
h = 20 −4.164 ·10−5 2.270 ·10−7 3.230 ·10−9 4.501 ·10−1
Realized Q-Spline-EGARCH
Horizon MedFE MSFE MedSFE QLIKE Lp
h = 1 −6.430 ·10−6 1.118 ·10−7 4.277 ·10−10 1.093 ·10−1 7326.36h = 5 −1.254 ·10−5 1.616 ·10−7 9.827 ·10−10 2.246 ·10−1
h = 10 −1.595 ·10−5 1.979 ·10−7 1.379 ·10−9 3.142 ·10−1
h = 20 −1.902 ·10−5 2.293 ·10−7 1.812 ·10−9 4.499 ·10−1
Realized C-Spline-EGARCH
Horizon MedFE MSFE MedSFE QLIKE Lp
h = 1 −7.255 ·10−6 1.121 ·10−7 4.329 ·10−10 1.098 ·10−1 7329.44h = 5 −1.347 ·10−5 1.619 ·10−7 1.009 ·10−9 2.273 ·10−1
h = 10 −1.784 ·10−5 1.989 ·10−7 1.509 ·10−9 3.274 ·10−1
h = 20 −2.176 ·10−5 2.306 ·10−7 1.981 ·10−9 4.715 ·10−1
EGARCH
Horizon MedFE MSFE MedSFE QLIKE Lp
h = 1 −7.214 ·10−6 1.212 ·10−7 6.967 ·10−10 1.624 ·10−1 7275.93h = 5 −1.177 ·10−5 1.599 ·10−7 1.252 ·10−9 2.560 ·10−1
h = 10 −1.795 ·10−5 1.879 ·10−7 1.748 ·10−9 3.284 ·10−1
h = 20 −2.846 ·10−5 2.128 ·10−7 2.588 ·10−9 4.363 ·10−1
108 CHAPTER 2.
Table A.4: Forecasting results excl. financial crises
Realized EGARCH
Horizon MedFE MSFE MedSFE QLIKE Lp
h = 1 −9.281 ·10−6 9.644 ·10−9 3.455 ·10−10 1.097 ·10−1 6738.80h = 5 −1.751 ·10−5 1.800 ·10−8 8.925 ·10−10 2.122 ·10−1
h = 10 −2.732 ·10−5 1.910 ·10−8 1.512 ·10−9 2.764 ·10−1
h = 20 −4.572 ·10−5 2.153 ·10−8 3.267 ·10−9 3.669 ·10−1
Realized TV-EGARCH
Horizon MedFE MSFE MedSFE QLIKE Lp
h = 1 −7.011 ·10−6 9.820 ·10−9 2.966 ·10−10 1.084 ·10−1 6737.02h = 5 −1.294 ·10−5 1.729 ·10−8 7.028 ·10−10 2.141 ·10−1
h = 10 −1.615 ·10−5 1.956 ·10−8 9.310 ·10−10 2.848 ·10−1
h = 20 −2.080 ·10−5 2.318 ·10−8 1.316 ·10−9 3.766 ·10−1
Realized Adaptive-EGARCH
Horizon MedFE MSFE MedSFE QLIKE Lp
h = 1 −1.055 ·10−5 9.464 ·10−9 3.779 ·10−10 1.097 ·10−1 6736.28h = 5 −2.106 ·10−5 1.684 ·10−8 9.811 ·10−10 2.142 ·10−1
h = 10 −3.008 ·10−5 1.843 ·10−8 1.656 ·10−9 2.798 ·10−1
h = 20 −4.417 ·10−5 2.093 ·10−8 2.902 ·10−9 3.570 ·10−1
Realized Q-Spline-EGARCH
Horizon MedFE MSFE MedSFE QLIKE Lp
h = 1 −6.857 ·10−6 9.550 ·10−9 3.050 ·10−10 1.087 ·10−1 6735.88h = 5 −1.238 ·10−5 1.729 ·10−8 7.416 ·10−10 2.156 ·10−1
h = 10 −1.576 ·10−5 1.928 ·10−8 1.044 ·10−9 2.776 ·10−1
h = 20 −1.932 ·10−5 2.361 ·10−8 1.308 ·10−9 3.635 ·10−1
Realized C-Spline-EGARCH
Horizon MedFE MSFE MedSFE QLIKE Lp
h = 1 −7.508 ·10−6 9.661 ·10−9 3.072 ·10−10 1.091 ·10−1 6739.19h = 5 −1.346 ·10−5 1.812 ·10−8 7.382 ·10−10 2.188 ·10−1
h = 10 −1.743 ·10−5 2.043 ·10−8 1.074 ·10−9 2.935 ·10−1
h = 20 −2.194 ·10−5 2.614 ·10−8 1.469 ·10−9 4.019 ·10−1
EGARCH
Horizon MedFE MSFE MedSFE QLIKE Lp
h = 1 −1.028 ·10−5 1.213 ·10−8 5.517 ·10−10 1.616 ·10−1 6691.30h = 5 −1.451 ·10−5 1.783 ·10−8 1.011 ·10−9 2.510 ·10−1
h = 10 −2.037 ·10−5 1.932 ·10−8 1.340 ·10−9 3.052 ·10−1
h = 20 −3.274 ·10−5 2.120 ·10−8 2.330 ·10−9 3.688 ·10−1
C H A P T E R 3INTRODUCING MACRO-FINANCE VARIABLES INTO
THE REALIZED EGARCH FRAMEWORK
Johan Stax JakobsenAarhus University and CREATES
Abstract
We propose two ways of including macro-finance variables into the Realized EGARCH
model of Hansen and Huang (2016). Firstly, an additive decomposition, where the
exogenous variables are added directly to the GARCH equation. Secondly, a multi-
plicative component model that separates the latent volatility into a part describing
the conditional heteroskedasticity and a part modeling the baseline volatility as a
function of exogenous variables. An empirical application to the exchange-traded
index fund SPY that tracks the S&P 500 Index and 20 individual stocks involving
three macro-finance variables shows promising in-sample gains but only modest
out-of-sample gains. Furthermore, our results corroborate that the additional infor-
mation content from including exogenous covariates is much smaller when working
in a framework utilizing realized measures of volatility compared to one that solely
exploits squared returns.
109
110 CHAPTER 3.
3.1 Introduction
The seminal papers of Engle (1982) and Bollerslev (1986) introduced the GARCH
framework able to model the time-varying conditional heteroskedasticity inherent to
financial return series. The GARCH model and a plethora of extensions are capable
of describing salient features of financial returns series such as volatility clustering
(Mandelbrot, 1963) and the leverage effect (Black, 1976). Quickly, a big literature uti-
lizing in particular the GARCH framework for research in linking macroeconomic and
financial variables to volatility emerged and numerous studies have since explored
different macro-finance variables’ ability to improve in-sample and out-of-sample
volatility predictions.1 Although much research in this area continues to use clas-
sical GARCH models that utilize squared returns as the signal about volatility, the
benchmark volatility model arguably ought to exploit realized measures that have
revolutionized the modeling of latent volatility of financial markets in the last couple
of decades.
In this paper, we are interested in specifying a model that extracts information
from exogenous covariates while exploiting realized measures of volatility. To this end,
we apply the successful Realized EGARCH (REGARCH) model of Hansen and Huang
(2016). The REGARCH model is basically an EGARCH-X model with the realized mea-
sure as the covariate combined with a measurement equation. By linking the realized
measure and the conditional variance, the measurement equation ensures dynami-
cal completeness and thereby facilitates multi-step ahead volatility predictions. We
consider both an additive and a multiplicative extension of the REGARCH model.
The former adds the exogenous variables directly to the GARCH equation while the
latter is a multiplicative component model that separates the latent volatility into a
part describing the conditional heteroskedasticity and a part modeling the baseline
volatility as a function of the exogenous variables.
We present an empirical application using the exchange-traded index fund SPY
tracking the S&P 500 Index and 20 individual stocks. We consider three daily macro-
finance indicators commonly applied in the literature: the Chicago Board Options
Exchange’s volatility index typically known by its ticker symbol VIX, the Economic
Policy Uncertainty index (EPU), and the Arouba-Diebold-Scotti business condition
index (ADS). The variables are selected since they cover three interesting cases. VIX
is the risk neutral expectation of the integrated variance of the S&P 500 Index and
1A non-exhaustive list of considered variables includes interest levels (Glosten, Jagannathan, andRunkle, 1993; Brenner, Harjes, and Kroner, 1996; Gray, 1996), interest spreads (Dominguez, 1998; Hagiwaraand Herce, 1999), bid-ask spread (Bollerslev and Melvin, 1994), forward-premiums (Hodrick, 1989), volume(Lamoureux and Lastrapes, 1990; Wagner and Marsh, 2005; Fleming, Kirby, and Ostdiek, 2008; Gallo andPacini, 2000; Girma and Mougoué, 2002), period effects such as day-of-week effect (Connolly, 1989; Baker,Rahman, and Saadi, 2008; Alagidede, 2008; Charles, 2010), realized measures (Engle, 2002), VIX (Day andLewis, 1992; Amado and Laakkonen, 2014; Kambouroudis and McMillan, 2016), the Policy UncertaintyIndex (Asgharian, Christiansen, and Hou, 2015; Liu and Zhang, 2015), the Arouba-Diebold-Scotti businesscondition index (Dorion, 2016), low-frequency macro-economic time series (Engle and Rangel, 2008;Engle, Ghysels, and Sohn, 2013; Conrad and Loch, 2015; Paye, 2012).
3.2. REALIZED MEASURES OF VOLATILITY 111
therefore a natural predictor of realized variance. ADS tells us about the state of the
economy and helps us examining whether financial market volatility is countercycli-
cal. EPU allows us to investigate whether economic policy uncertainty can explain
financial market volatility. Furthermore, the variables are available at daily frequency
such that there is no need for mixed data sampling or alternative approaches. Several
interesting results emerge from the in- and out-of-sample evaluation of the models
including exogenous covariates and the comparison with their benchmark counter-
parts. For the multiplicative decomposition, we realize large in-sample and modest
short horizon out-of-sample gains from including VIX as a covariate, while somewhat
smaller gains for the additive specification. This stipulates that a multiplicative speci-
fication may be the preferred avenue when incorporating implied volatility in GARCH
type models. For the ADS and EPU, we also find more modest evidence of superior
in-sample performance, but close to non out-of-sample gains. Expectedly, we find
that including additional covariates is less beneficial in the REGARCH model than
in the EGARCH model, which does not include realized measures. In the estimated
EGARCH models, we find that VIX subsumes the information in squared returns.
The paper is structured as follows. Section 3.2 introduces the concept of Realized
Volatility. The modeling framework and estimation strategy are introduced in Section
3.3 and Section 3.4, respectively. We discuss forecasting methodology and evaluation
in Section 3.5 and Section 3.6, respectively. The empirical applications are presented
in Section 3.7. Finally, concluding remarks are given in Section 3.8.
3.2 Realized measures of volatility
Assume that there exists a representation of the log-price of an asset or index, Yt ,
such that for all t ∈ [0,T
]Yt =
T∫0
µudu +T∫
0
σudWu , (3.1)
where µu is the instantaneous drift, σu is the instantaneous volatility, and dWu is a
standard Brownian motion.
We are interested in daily compounded returns defined by rt = Yt−Yt−1. Andersen
et al. (2003) and Barndorff-Nielsen and Shephard (2002) showed that rt is Gaussian
conditional on Ft =σ(Ys , s ≤ t
), the σ-algebra generated by the sample paths of Y .
In particular,
rt |Ft ∼ N
t∫t−1
µt−1+udu,
t∫t−1
σ2udu
, (3.2)
where the term
IVt =t∫
t−1
σ2udu, (3.3)
112 CHAPTER 3.
is known as the Integrated Variance (IV) and measures the ex-post variance of day t .
The integrated variance is latent. Thus, in empirical applications, we need to esti-
mate the quantity using prices observed at discrete and possibly irregularly spaced
intervals. One possibility is to sample n intra-daily observations equidistant in calen-
dar time and calculate the Realized Variance (RV):
RVt =n∑
i=1r 2
t ,i , (3.4)
where rt ,i = Yt−1+i 1n−Yt−1+(i−1) 1
n. Realized volatility is the square root of RVt . Under
the stated assumption of a semi-martingale in (3.1), Barndorff-Nielsen and Shephard
(2002) showed that the RV is ap
n consistent estimator of the IV. However, this as-
sumption is often at odds with empirical evidence due to issues related to market
microstructure noise. Research in estimation of IV in the presence of microstructure
noise has become an flourishing area in financial econometrics. The early literature,
e.g. Andersen, Bollerslev, Diebold, and Ebens (2001), focused on sparse sampling. The
idea is to sample at an arbitrary lower frequency, e.g. at 5 min intervals, to balance
the accuracy and bias introduced by market microstructure noise. The more recent
literature focuses on the development of estimators robust to market microstructure
noise. Consistent estimation methods for IV include the Realized Kernel (RK) esti-
mators of Barndorff-Nielsen et al. (2008), the modified MA filter of Hansen, Large,
and Lunde (2008), the two time scales realized volatility estimator of Zhang et al.
(2005), the Multi-Scale approach of Zhang (2006), and the Range-Based estimator of
Christensen and Podolskij (2007).
In the empirical applications of this paper, we only consider the RK defined by
RKt =H∑
h=−HK
(h
H +1
)γh , γh =
n∑i=|h|+1
rt ,i rt ,i−|h|, (3.5)
where K (·) is the Parzen kernel function. The RK was found to perform very well in
Hansen and Lunde (2006), where a range of measures was considered, and in the
VaR application in Brownless and Gallo (2010); see also the comparison of realized
measures in Gatheral and Oomen (2010).
3.3 Modeling framework
In this paper, we model the latent volatility of the daily return series of an asset or
index{rt
}by including exogenous variables in the REGARCH framework of Hansen
and Huang (2016). We start by introducing the original framework. Let Ft−1 be the
information set containing the historical information at time t −1. Define the condi-
tional mean, µt = E[rt |Ft−1], and the conditional variance, σ2t = Var
[rt |Ft−1
], of the
return series.
3.3. MODELING FRAMEWORK 113
3.3.1 The Realized EGARCH model
The REGARCH model of Hansen and Huang (2016) with a single realized measure is
defined by the following three equations:
rt =µt +σt zt , (3.6)
logσ2t =ω+β logσ2
t−1 +τ(zt−1
)+αut−1, (3.7)
log xt = ξ+φ logσ2t +δ
(zt
)+ut . (3.8)
The equations are known as the return equation, the GARCH equation, and the
measurement equation, respectively. It is typically assumed that zt ∼ i.i.dN(0,1
)and
ut ∼ i.i.d.N(0,σ2
u
)are mutually and serially independent. The realized measure, xt ,
is an estimate of IV such as RV or RK.
The return equation, which decomposes the returns into a time-varying condi-
tional mean and an error term with conditional heteroskedasticity, and the GARCH
equation, modeling the conditional heteroskedasticity, are standard in GARCH type
models. The focus is typically on modeling σ2t as daily returns often are close to
serially uncorrelated for many financial assets. Therefore, we simply model the con-
ditional mean as a constant, i.e. µt =µ. The GARCH equation has an autoregressive
structure for logσ2t with the innovation term τ
(zt−1
)+αut−1. Thus, β< 1 measures
the persistence of the conditional variance and α tells us how shocks to the realized
measure affect the conditional variance. One reason for choosing a log specification
is that it guarantees positivity of the conditional variance. Another advantages is a
more reasonable fit to the distributional assumption applied in the quasi-maximum-
likelihood estimation. However, one disadvantage to be noted is the necessity to
account for distributional aspects when performing multi-period ahead forecasting
due to a Jensen’s inequality term.
The idea of a measurement equation discriminates the REGARCH model from
other volatility models incorporating realized measures such as the GARCH-X model
of Engle (2002), the HEAVY model of Shephard and Sheppard (2010), and the MEM
model of Engle and Gallo (2006). The measurement equation makes the model com-
plete in the sense that it specifies the dynamic structure of the realized measure. As
the conditional variance is an ex-ante measure of volatility and the realized measure
is an ex-post measure of volatility, it is natural to assume a link between the two.
However, some discrepancies between the two measures are expected. One source
is that we are modeling close-to-close return volatility while the realized measure
is calculated using open-to-close information. Thus, we would expect the realized
measure to be lower than the conditional variance on average. Therefore, it is neces-
sary to include the parameters ξ and φ. If the two measures are almost proportional,
we should expect to find ξ≈ 0 and φ≈ 1. Another way to motivate the link between
the conditional variance and the realized measure follows from the mean equation
which implies log(rt −µt
)2 = logσ2t + log z2
t . Because xt is a realized measure just as
114 CHAPTER 3.
(rt −µt
)2, albeit less noisy, we would expect that log xt ≈ logσ2t + f
(zt
)+et , where et
is an error term. This also motivates the logarithmic measurement equation.
The so-called leverage functions τ (z) and δ (z) are defined by
τ (z) = τ1z +τ2
(z2 −1
),
δ (z) = δ1z +δ2
(z2 −1
).
The notion of leverage is related to the well-known leverage effect (Black, 1976). The
term τ (z) can be related to the news impact curve introduced by Engle and Ng
(1993) as it models how positive and negative returns impact the log conditional
variance. Hansen et al. (2012) considered a class of leverage functions constructed
using Hermite polynomials. They and later Hansen and Huang (2016) found the
quadratic functional form to be satisfactory in different empirical applications. One
advantage of this particular leverage function is that E[τ (z)
] = E[δ (z)
] = 0 when
E [z] = 0 and Var[z] = 1. Together with the logarithmic volatility specification, this
particular choice of leverage function is empirically found to make the assumption
about independence between zt and ut realistic.
3.3.2 Additive decomposition
We now turn to the inclusion of exogenous variables in the volatility dynamics speci-
fied by (3.6)-(3.8). One possibility often considered in classical GARCH models is to
additively include the exogenous variables in the GARCH equation:
rt =µt +σt zt , (3.9)
logσ2t =ω+β logσ2
t−1 +τ(zt−1
)+γ′vt−1 +αut−1, (3.10)
log xt = ξ+φ logσ2t +δ
(zt
)+ut , (3.11)
where vt−1 is a vector of exogenous variables that must be lagged one period to be
included in the information set at time t −1. We note that the latent volatility can be
written as
logσ2t+1 =βk+1 logσ2
t−k +k∑
i=0βi [
ω+γ′vt−i +τ(zt−i )+αut−i]
. (3.12)
This implies that weight put on the information of the exogenous variables is decaying
at the same speed as the weight put on the realized measure, which is the main
difference compared to the multiplicative specification presented below. Thus, if the
persistence is large and the coefficients on the exogenous covariates are large, we also
put large weight on past values of the exogenous covariates. This may be problematic,
if only the latest values are relevant for predicting latent volatility while we need a
large persistence for incorporating the information from the realized measure.
From the literature on classical GARCH models, we know that the additive inclu-
sion of exogenous covariates may help to overcome shortcominings of the benchmark
3.3. MODELING FRAMEWORK 115
counterparts. Han (2015) discusses the asymptotic properties of GARCH-X processes
and shows that such processes more adequately explain stylized facts of financial time
series such as the long-memory and leptokurtosis. The properties of the GARCH-X
process heavily depend on the degree of persistence of the exogenous covariate.
3.3.3 Multiplicative decomposition
Another way to include covariates is to specify a multiplicative component model by
assuming that the latent volatility can be decomposed into a component describing
the conditional heteroskedasticity, ht , and a component describing the baseline
volatility, g t : σ2t = ht g t . This modeling approach has become increasingly popular in
the GARCH literature (see e.g. Engle and Lee (1999), Engle and Rangel (2008), Amado
and Teräsvirta (2013), and Engle et al. (2013)). The reason is that component models
offer a parsimonious way to model the often complex dynamics of financial time se-
ries and explain stylized facts that cannot be captured by classical, stationary GARCH
models. Conrad and Kleen (2016) discuss the statistical properties of multiplicative
GARCH models and show that these models are better able to match stylized facts of
financial return series than their benchmark GARCH model.
One particular empirical issue is the presence of structural breaks that may cause
spurious long-memory; see Mikosch and Starica (2004) and Hillebrand (2005), among
others. Several component models in the literature address non-stationarity by de-
composing the volatility into a high-frequency (short-run) and low-frequency (long-
run) component. In the GARCH literature, Engle and Rangel (2008), Morana (2002),
and Amado and Teräsvirta (2013), among others, model the low-frequency compo-
nent as a deterministic function of time. One appealing feature of these models is
that the amplitude of volatility clusters now is allowed to vary over time in line with
empirical observations but in contradiction to stationary GARCH models. Laursen
and Jakobsen (2017) extend the idea to the REGARCH framework.
In order to link, in particular low-frequency, macroeconomic variables to financial
market volatility, Engle et al. (2013) propose the GARCH-MIDAS model that specifies
the low-frequency component as a MIDAS-filter of the macroeconomic variables.
This modeling framework has been applied in numerous empirical applications
(Asgharian, Hou, and Javed, 2013; Asgharian et al., 2015; Conrad and Loch, 2015;
Dorion, 2016). In the Realized GARCH framework, Dominicy and Vander Elst (2015)
include a MIDAS component.
In this paper, we restrict ourselves to daily macro-finance indicators. Thus, there
is no explicit need for a MIDAS type specification. Actually, Conrad and Schienle
(2015) admit that their attempt to estimate a GARCH-MIDAS specification using VIX
and RV resulted in weights decaying so fast to zero that it resembled a GARCH-X
specification. However, this should not discourage the use of MIDAS specifications
with daily indicators in general. It is likely that a MIDAS filter may help smooth noisy
signals such as the EPU.
116 CHAPTER 3.
In this paper, we consider the following specification:
rt =µt +σt zt , (3.13)
loght =β loght−1 +τ(zt−1
)+αut−1, (3.14)
log xt = ξ+φ logσ2t +δ
(zt
)+ut , (3.15)
log g t =ω+γ′vt−1. (3.16)
Due to identification issues, a constant is only included in the baseline volatility
component such that E[loght
]= 0.2
For the multiplicative specification, we have
logσ2t+1 = log g t+1 + loght+1
= ω+γ′vt +βk+1 loght−k +k∑
i=0βi [
αut−i +τ(zt−i )]
. (3.17)
Thus, compared to the additive case, the information content in the most recent
observations of the exogenous covariates is used to model the time-varying baseline
volatility and not as an additional variable in the GARCH filter.
3.3.4 The EGARCH model
The EGARCH model of Nelson (1991) is defined by the return equation in (3.6) with
logσ2t =ω+β logσ2
t−1 +α(|zt−1|−
p2/π
)+τ1zt−1. (3.18)
In the empirical application, we consider both an additive and a multiplicative version
with exogenous covariates similar to the ones specified in the REGARCH framework.
3.4 Estimation
We now turn to estimation and inference within a Quasi-Maximum-Likelihood
(QML) framework. We assume that zt ∼ i.i.d.N(0,1
)and ut ∼ i.i.d.N
(0,σ2
u
)and write
the leverage functions as τ(zt
) = τ′a(zt
)and δ
(zt
) = δ′b(zt
)with a
(zt
) = b(zt
) =(zt , z2
t −1)′
. Define the parameter vector
θ =(µ,ω,β,τ′,α,ξ,φ,δ′,σ2
u ,γ′)′
. (3.19)
The log-likelihood function reads
L(r, x, v ;θ
)= T∑t=1
`t(rt , xt , vt ;θ
), (3.20)
2The GARCH equation implies that loght = β j loght− j +∑ j−1
i=0 βi [τ(zt−1−i )+αut−1−i
]such that
loght has a stationary representation if |β| < 1. Hence, the result immediately follows.
3.4. ESTIMATION 117
where `t(rt , xt , vt ;θ
)is the log-likelihood contribution at time t . As the log-likelihood
contribution and therefore also the score differ depending on whether an additive or
a multiplicative decomposition are applied, we present the two cases separately. We
now derive the score since it defines the first order conditions for the QML Estimator
(QMLE) and is necessary to obtain standard errors. For notional convenience, let
azt = ∂a(zt )/∂zt and bzt = ∂b(zt )/∂zt .
3.4.1 Additive decomposition
The log-likelihood contribution at time t is given by
`t(rt , xt , vt ;θ
)=−1
2
[2log2π+ logσ2
t + z2t + logσ2
u + u2t
σ2u
], (3.21)
with zt = zt(θ)= (
rt −µ)
/σt and ut(θ)= log xt −ξ−φ logσ2
t −δ(zt
). Key components
are the derivatives stated in the following lemma.
Lemma 1. The derivatives A(zt ) = ∂ logσ2t+1/∂ logσ2
t and B(zt ,ut ) = ∂`t /∂ logσ2t are
given from respectively
A(zt ) = (β−αφ)+ 1
2
(αδ′bzt −τ′azt
)zt , (3.22)
and
B(zt ,ut ) =−1
2
[(1− z2
t
)+ ut
σ2u
(δ′bzt zt −2φ
)]. (3.23)
Proof: See Appendix A.1.
Next, we define the score of logσ2t+1 with respect to θ.
Lemma 2. Let θ1 = (ω,β,τ′,α)′ and θ2 = (ξ,φ,δ′)′. Furthermore, define mt =(1, logσ2
t , zt ,
z2t −1,ut
)′and nt =
(1, logσ2
t , zt , z2t −1
)′. The derivatives ∂ logσ2
t+1/∂θ are given from
the stochastic recursions
hµ,t+1 = ∂ logσ2t+1
∂µ= A(zt )hµ,t +
(αδ′bzt −τ′azt
) 1
σt, (3.24)
hθ1,t+1 = ∂ logσ2t+1
∂θ1= A(zt )hθ1,t +mt , (3.25)
hθ2,t+1 = ∂ logσ2t+1
∂θ2= A(zt )hθ2,t +αnt , (3.26)
hγ,t+1 = ∂ logσ2t+1
∂γ= A(zt )hγ,t +vt . (3.27)
Proof: See Appendix A.1.
118 CHAPTER 3.
Finally, we present the score.
Theorem 1. Utilizing components from Lemma 1 and Lemma 2, the score with respect
to the parameters, θ =(µ,ω,β,τ′,α,ξ,φ,δ′,σ2
u ,γ′)′
, is given from
∂`t
∂µ= B(zt ,ut )hµ,t +
[zt −δ′ ut
σ2u
bzt
]1
σt, (3.28)
∂`t
∂θ2= B(zt ,ut )hθ1,t , (3.29)
∂`t
∂θ3= B(zt ,ut )hθ2,t +
ut
σ2u
nt , (3.30)
∂`t
∂γ= B(zt ,ut )hγ,t , (3.31)
∂`t
∂σ2u
= 1
2
u2t −σ2
u
σ4u
. (3.32)
Proof: See Appendix A.1.
3.4.2 Multiplicative decomposition
The initial value of the logarithm of the conditional variance, log h0, is set equal to its
unconditional mean, E[loght ] = 0.3
`t(rt , xt , vt ;θ
)=−1
2
[2log2π+ loght + log g t + z2
t + logσ2u + u2
t
σ2u
], (3.33)
with zt = zt(θ)= (
rt −µ)
/σt and ut(θ)= log xt −ξ−φ
[loght + log g t
]−δ(zt
).
Lemma 3. The derivatives C (zt ) = ∂ loght+1/∂ loght , D(zt ) = ∂ loght+1/∂ log g t , and
E(zt ,ut ) = ∂`t /∂ loght = ∂`t /∂ log g t are given from respectively
C (zt ) = (β−αφ)+ 1
2
(αδ′bzt −τ′azt
)zt , (3.34)
D(zt ) = −αφ+ 1
2
(αδ′bzt −τ′azt
)zt , (3.35)
and
E(zt ,ut ) = −1
2
[(1− z2
t
)+ ut
σ2u
(δ′bzt −2φ
)]. (3.36)
Proof: See Appendix A.1.
Next, we define the score of loght+1 with respect to θ.
3Alternatively, one could estimate it like Hansen and Huang (2016)
3.4. ESTIMATION 119
Lemma 4. Let θ1 = (β,τ′,α)′, θ2 = (ξ,φ,δ′)′, and θ3 = (ω,γ)′. Furthermore, define mt =(loght , zt , z2
t −1,ut
)′and nt =
(1, logσ2
t , zt , z2t −1
)′. The derivatives ∂ loght+1/∂θ are
given from the stochastic recursions
hµ,t+1 = ∂ loght+1
∂µ=C (zt )hµ,t +
(αδ′bzt −τ′azt
) 1
σt, (3.37)
hθ1,t+1 = ∂ loght+1
∂θ1=C (zt )hθ1,t +mt , (3.38)
hθ2,t+1 = ∂ loght+1
∂θ2=C (zt )hθ2,t +αnt , (3.39)
hθ3,t+1 = ∂ loght+1
∂θ3=C (zt )hθ3,t +D(zt )gθ3,t+1, (3.40)
where
gθ3,t+1 =∂ log g t
∂θ3= (
1, v ′t
)′ .
Proof: See Appendix A.1.
Finally, we present the score.
Theorem 2. Utilizing components from Lemma 3 and Lemma 4, the score with respect
to the parameters, θ =(µ,ω,β,τ′,α,ξ,φ,δ′,σ2
u ,γ′)′
, is given from
∂`t
∂µ= E(zt ,ut )hµ,t +
[zt −δ′ ut
σ2u
bzt
]1
σt, (3.41)
∂`t
∂θ1= E(zt ,ut )hθ1,t , (3.42)
∂`t
∂θ2= E(zt ,ut )hθ2,t +
ut
σ2u
nt , (3.43)
∂`t
∂σ2u
= 1
2
u2t −σ2
u
σ4u
, (3.44)
∂`t
∂θ3= E(zt ,ut )hθ3,t +E(zt ,ut )gθ3,t . (3.45)
Proof: See Appendix A.1.
3.4.3 Partial log-likelihood function
The conditional density of(rt , xt
)can be factorized as
f(rt , xt |Ft−1
)= f(rt |Ft−1
)f(xt |rt ,Ft−1
).
120 CHAPTER 3.
Implying that the Gaussian log-likelihood can be decomposed as
T∑t=1
`t(rt , xt , vt ;θ
)=−1
2
T∑t=1
[log2π+ logσ2
t + z2t
]− 1
2
T∑t=1
[log2π+ logσ2
u + u2t
σ2u
].
The first part is known as the partial log-likelihood function since it only measures
the goodness of fit of the return distribution. The partial log-likelihood is in particu-
lar relevant when comparing models utilizing realized measures with GARCH type
models.
3.4.4 Asymptotic properties of the estimators
To our knowledge, the asymptotic properties of even the standard REGARCH model
have not yet been established. Neither has the asymptotic properties of the EGARCH-
X model, but only the special case of an EGARCH(1,1) model (Wintenberger, 2013).
The asymptotic analysis of the REGARCH model and the extensions presented in this
paper is a complicated task and it is beyond the scope of this paper. However, we will
point the reader towards some of the contributions in the literature that may justify a
conjecture regarding consistency and asymptotic normality of the estimators. The
proof of consistency and asymptotic normality for stationary GARCH models can
be found in Francq and Zakoïan (2010) and references therein. Han and Kristensen
(2014) established the asymptotic theory for the QMLE in GARCH-X models with
stationary and non-stationary covariates. Han and Kristensen (2015) perform an
asymptotic analysis of a multiplicative GARCH-X model where a non-linear function
of exogenous covariates constitutes the ’long-run’ component.
Following Corollary 1 and the arguments in Hansen and Huang (2016), the score
function of both the additive and multiplicative decomposition is a martingale differ-
ence sequence provided that E[zt |Ft−1
]= 0, E[
z2t |Ft−1
]= 1, E
[ut |zt ,Ft−1
]= 0 and
E[
u2t |zt ,Ft−1
]=σ2
u . Based on this result, they conjecture that the distribution of the
QMLE is asymptotically normal:p
T(θ−θ0
)d→N
(0,T I−1JI−1
), (3.46)
where J is the limit of the outer-product of the scores and I is (minus) the limit of
the Hessian matrix for the log-likelihood functions. We make the same conjecture.
It can be noted that Vander Elst (2015) and Borup and Jakobsen (2017) use a para-
metric bootstrap to investigate the validity of this conjecture in their extension of the
REGARCH model. They found the asymptotic approximation to be reasonable with
sample sizes similar to the ones used in this paper.
3.5 Forecasting methodology
We seek to forecast the conditional variance k ≥ 1 days into the future. One-step ahead
forecasts are easily obtained directly from the GARCH equation (3.10) in the additive
3.6. FORECAST EVALUATION 121
case or by combining the GARCH equation (3.14) and the new component (3.16) in
the multiplicative case. However, some care must be taken when considering multi-
step ahead forecasting. First, we note that the introduction of exogenous variables
makes the dynamic structure incomplete unless we specify a dynamic model for
the exogenous variables. It is standard in the literature to circumvent this issue by
a random walk assumption such that vt+k|t = E[
vtk |Ft]= vt or by keeping g t fixed
(Engle et al., 2013; Dominicy and Vander Elst, 2015). We assume that the exogenous
variables are martingales. Secondly, we note that the dynamic structure for the models
are specified in terms of logσ2t (additive model) or loght and log g t (multiplicative
model). Jensen’s inequality implies that exp
(E[
logσ2t+k |Ft
])6= E
[exp
(logσ2
t+k
)|Ft
].
To obtain an unbiased forecast we therefore need to account for distributional aspects.
If M denotes the number of simulations, we obtain the conditional variance forecast
as the average
σ2t+k|t =
1
M
M∑m=1
exp(logσ2
t+k|t ,m
),
where logσ2t+k|t ,m is a simulated value of the conditional volatility at time t +k condi-
tional on the information available at time t . In the simulation procedure, we utilize
the empirical distributions of zt and ut .4 In the additive case, we note that recursive
substitution of the GARCH equation (3.14) implies
logσ2t+k =βk logσ2
t +k∑
i=1βi−1
(ω+τ(
zt+k−i)+αut+k−i +γ′vt+k−i
). (3.47)
In the multiplicative case, we have similarly
loght+k =βk loght +k∑
i=1βi−1
(τ(zt+k−i
)+αut+k−i
), (3.48)
and therefore,
logσ2t+k = loght+k + log g t+k =βk loght +
k∑i=1
βi−1(τ(zt+k−i
)+αut+k−i
)+ log g t+k .
(3.49)
3.6 Forecast evaluation
In order to evaluate forecasting performance in the empirical applications, we need a
proxy for latent volatility. As the RK only is a measure of open-to-close volatility, it will
not be an unbiased measure of the close-to-close volatility used in this paper. To deal
with this issue, we follow a strategy similar to Sharma and Vipul (2016) and Huang,
4Using the empirical distribution is standard in the literature, see e.g. Brownless and Gallo (2010).Alternatively, one can use the distribution applied in the MLE.
122 CHAPTER 3.
Liu, and Wang (2016), among others. Hence, as a proxy for the latent close-to-close
volatility, we use σ2t = ηRKt with
η=1T
∑Tt=1 r 2
t1T
∑Tt=1 RKt
. (3.50)
We have chosen to only use data for the out-of-sample period to calculate η due to
observed time-variation in the scaling factor.
A variety of evaluation criteria or loss functions has been suggested in the litera-
ture in order to ascertain the quality of forecasts. However, only a few are applicable
when the target is observed with error as in the case with latent volatility; see Hansen
and Lunde (2006). If the ranking of forecasting models based on a given criteria
depends on whether the proxy or true latent volatility are the target, the criteria is
said to be non-robust. Patton (2011) provides necessary and sufficient conditions on
the functional form of the loss functions ensuring consistency of the ordering when
using a proxy. Two robust measures are the Squared Forecasting Error (SFE) and the
Quasi-Likelihood (QLIKE) loss function defined as respectively
SFEt+k =(σ2
t+k −σ2t+k|t
)2,
QLIKEt+k = σ2t+k
σ2t+k|t
− logσ2
t+k
σ2t+k|t
−1,
where σ2t+k|t denotes the model based forecast. The main difference between the loss
functions is that the SFE solely depends on the forecast error σ2t+k −σ2
t+k|t while the
QLIKE solely depends on the ratio σ2t+k /σ2
t+k|t . Following the arguments in Patton
(2011), this implies that the average QLIKE loss will be less affected by the most
extreme observations while the MSFE will be sensitive to extreme observations and
the level of return volatility. Another important feature is that the SFE is symmetric
whereas the QLIKE penalizes underprediction more heavily than overprediction.
Patton and Sheppard (2009) show that QLIKE exhibits more statistical power in
differentiating between volatility forecasts.
3.6.1 Model Confidence Set
To statistically discriminate between competing forecasting models, we implement
the Model Confidence Set (MCS) of Hansen et al. (2011). The procedure starts with
the full set of candidate models M0 ={1, ...,m0
}. Hereafter, the MCS is obtained by
iteratively reducing the number of models in M0 to m < m0. The sequential testing
procedure is based on the loss differential between forecasts i and j , di j ,t+k , for
i > j constructed using the evaluation criteria listed above. At each step, the null
hypothesis of Equal Predictive Ability (EPA)
H0 : E(di j ,t+k
)= 0, ∀i > j ∈M
3.7. EMPIRICAL APPLICATION 123
is tested for a set of models M ∈M0 with M=M0 at the initial step. If H0 is rejected
at the significance level α, the worst performing model is removed and the process is
continued until no rejections occur. The surviving models constitute the MCS, M∗α. If
a fixed significance level α is used at each step, M∗α contains the best model from M0
with (1−α) confidence.
The test of EPA is based on the t-statistics
ti j =di j√
V(di j
) ,
where di j = 1H
∑Hh=1 di j ,h+k is the mean forecast differential over the H periods where
forecasts are available. The (m −1)m/2 unique t-statistics for the set M need to be
combined into one test statistic. Two possibilities are the range statistic,
TR = maxi> j∈M
|ti j | = maxi> j∈M
|di j |√V
(di j
) ,
and the semi-quadratic statistic,
TSQ = ∑i> j∈M,i< j
t 2i j =
∑i> j∈M,i< j
(di j
)2
√V
(di j
) .
Both test statistics indicate a rejection of the EPA hypothesis for large values. It is
necessary to obtain p-values of the test statistics using a bootstrap distribution as the
actual distribution is complicated and depends on the covariance structure between
the forecast models. In this paper, we showcase the results obtained using the semi-
quadratic statistic, but the results are qualitatively similar for the range statistic.
3.7 Empirical application
In this section, we investigate the empirical performance of the aforementioned
models on returns and realized measures for the exchange-traded index fund SPY,
which tracks the S&P 500 Index, and 20 individual stocks. The same series were also
investigated using the RGARCH model in Hansen et al. (2012) and the REGARCH
model in Hansen and Huang (2016) and Banulescu et al. (2014).
3.7.1 Data
The full dataset covers the period from January 2, 2002 to December 31, 2013 and
consists of daily close-to-close returns and daily RK of SPY and the 20 individual
stocks. In the computation of the RK, we restrict attention to the official trading
124 CHAPTER 3.
hours 9:30:00 and 16:00:00 New York time. For each stock, we remove short trading
days where trading spanned less than 20,000 seconds compared to typically 23,400
seconds for a full trading day.5
Table A.1 reports descriptive statistics for the daily returns and the RK. We com-
pute outlier-robust estimates of return skewness and kurtosis (Kim and White, 2004;
Teräsvirta and Zhao, 2011) along with their conventional estimates. The robust mea-
sures point to negligible skewness and quite mild kurtosis in the return series. This
stands in contrast to the moderately skewed, severely fat-tailed distributions sug-
gested by the conventional measures, corroborating the findings in Kim and White
(2004) that stylized facts of returns series change by the use of robust estimators.
We estimate the fractional integrated parameter d with the two-step exact local
Whittle estimator of Shimotsu (2010). Over the full sample all series have d > 0.5,
suggesting that log(RK) is highly persistent and non-stationary.6
Figure A.1 contains different financial time series related to the SPY: returns,
squared returns (annualized), log(RK) (annualized), and the autocorrelation of the
RK. The dotted line separates the estimation period from the out-of-sample period in
the forecasting exercise. As commonly observed in financial returns series and in line
with classical, stationary GARCH models, volatility clusters are observed throughout
the sample. However, the amplitude or baseline volatility seems to change over time.
In particular, there are a volatile period in the beginning of the sample, a tranquil
period in the middle of the sample, a volatile period around the Global Financial
Crisis (GFC), and finally a less volatile period at the end of the sample. The long-range
dependence in different measures of volatility is corroborated by the slowly decaying
autocorrelation of the log(RK) for the SPY.
Following the arguments in Han (2015) and Conrad and Kleen (2016), the long-
range dependence may motivate the inclusion of, probably quite persistent, exoge-
nous variables in the volatility dynamics. In the following, we present the exogenous
variables employed in this application.
¿ Insert Table A.1 about here À
¿ Insert Figure A.1 about here À
Exogenous information
In this application, we consider three macro-finance variables observed at daily
frequency: VIX, EPU, and ADS. In the following, we describe the variables and some
previous applications.
Since Fleming, Ostdiek, and Whaley (1995), VIX has been widely used to forecast
financial market volatility. VIX is based on implied volatilities of the S&P 500 Index
5The data was kindly provided by Asger Lunde.6We estimated the parameters with m = bT q c for q =∈ {0.5,0.55, . . . ,0.8}, leading to no alterations of
the conclusions obtained for q = 0.65.
3.7. EMPIRICAL APPLICATION 125
options and is therefore the risk-neutral expectation of the integrated variance of
the S&P 500 Index for the next 30 calender days.7 If options markets are efficient,
implied volatility should be an efficient forecast of future volatility (Christensen
and Prabhala, 1998). Hence, VIX is a natural predictor of future realized volatility
of the S&P 500 Index. Following Bekaert and Hoerova (2014), we however note that
Et
[RV22
t+1
]= VIX2
t −VPt , where VPt is (the negative of) the variance risk premium
and that the variance premium will be increasing in the risk aversion of the economy
and most likely time-varying with a mean different from zero. Due to its embedded
risk premium VIX will not be an unbiased predicator of future realized volatility.
Furthermore, VIX index is often viewed as an indicator of global market sentiment or
the investors’ fear gauge. Thus, it should also be relevant when forecasting volatility
of individual stocks or other indices than the S&P 500 Index. In the GARCH mod-
eling framework, implied volatility and VIX are often found to improve volatility
forecasts (Day and Lewis, 1992; Martens and Zein, 2004; Amado and Laakkonen,
2014; Kambouroudis and McMillan, 2016).8
The EPU of Baker et al. (2016) is a measure of uncertainty based on the appear-
ance of certain words in news articles. Extensive research has focused on linking
stock market return or volatility to economic uncertainty. Utilizing a GARCH-MIDAS
approach, Asgharian et al. (2015) find that macro economic uncertainty influences
long-run stock and bond volatility, but their out-of-sample results are fairly weak. In
a HAR framework, Liu and Zhang (2015) conclude that EPU contains significantly
predictive power of market volatility both in-sample as well as out-of-sample.
The ADS developed by Aruoba et al. (2009) and published by the Federal Re-
serve (Fed) Bank of Philadelphia proxies business conditions by combining six widely
followed macroeconomic indicators, namely weekly initial jobless claims, monthly
log-growth of non-agricultural payroll employment, industrial production, real man-
ufacturing and trade sales, real personal income minus transfers, and quarterly real
GDP. Dorion (2016) introduces a Macro-GARCH model by employing the ADS in a
GARCH-MIDAS model. The Macro-GARCH model is found to significantly reduce
option pricing errors.
Figure A.2 depicts the variables together with National Bureau of Economic Re-
search (NBER) recession periods and presents scatter plots of the relationship be-
tween the variables lagged one period and the realized variance measured by log(RK).
We have chosen to consider the following transformations: log(VIX2
t−1/250)
and
log(
EPUt−1100
). This is natural as we consider a log-volatility specification, but it also
7See Whaley (2000, 2009) and Mencía and Sentana (2013) for a more detailed description. See alsowww.cboe.com/VIX for a detailed index description.
8We note that there exists a large literature discussing the information content of implied volatilityand realized measures in predicting volatility in- and out-of-sample. A range of modeling approaches hasbeen employed on different data sets, resulting in very different conclusions. See e.g. Kruse et al. (2016),Han and Park (2013), Kambouroudis and McMillan (2016), and references therein. We do not investigatewhether VIX is an efficient forecast of future volatility, but only if it is a relevant covariate when taking themodeling framework as given.
126 CHAPTER 3.
makes the relationship between log(RK) and the exogenous variables more linear.
The intuition of the plots are clear. For the transformed version of VIX, we see, as
expected, an almost linear relationship with log(RK). It seems that EPU is a much
more noisy signal about future volatility. Economic uncertainty is positively corre-
lated with future volatility. Comparing the time series plot of EPU with the one for
RK in A.2 may indicate a structural change in the relationship around the GFC since
the EPU has not reverted to the same extent as volatility. Intuitively, ADS is negatively
correlated with future volatility. Clearly, volatility spikes and ADS plummets during
the GFC. The scatter plot may indicate that the relationship is somewhat non-linear.
¿ Insert Figure A.2 about here À
3.7.2 Results for the S&P 500 Index
The results pertaining to the in-sample fit of the models are based on the full sample.
In the forecasting exercise, we estimate the models using the sample period until
December 31, 2010, while the remaining part of the sample is reserved for the out-of-
sample exercise.
Estimation results
Table A.2 contains the results from estimating EGARCH and REGARCH models. Natu-
rally, the results when not including exogenous variables are identical for the additive
and multiplicative decomposition except for the magnitude of ω. The REGARCH
models dominate the EGARCH models in terms of partial log-likelihood value with
a large increase of approximately 60 log-likelihood points. This illustrates the use-
fulness of including realized measures in a framework purely based on the financial
time series’ own history, i.e. the RK is a much stronger signal of volatility than squared
returns. The parameter estimates are in line with our intuition, e.g. β is close to one
indicating a high degree of persistence in the volatility dynamics and the leverage
functions show that negative information affects future volatility more than positive
information of the same magnitude.
¿ Insert Table A.2 about here À
We now include the exogenous covariates one by one and finally jointly in the
volatility models: the results are presented in Table A.3-A.6. First, we consider VIX. For
both the additive and the multiplicative version of the EGARCH model, the inclusion
of VIX leads to very large increase of approximately 60 points in the (partial) log-
likelihood value. The coefficient α is equal to zero for both EGARCH models which
shows that VIX dominates daily squared returns as a predictor of future volatility. For
both REGARCH models, VIX is also highly significant with an increase close to 100
points for the additive version and close to 200 points for the multiplicative version.
3.7. EMPIRICAL APPLICATION 127
However, the partial log-likelihood values only increase marginally indicating that
the largest gains can be attributed to improved modeling of the realized measure. In
fact, the multiplicative EGARCH model now dominates the REGARCH models based
on the partial log-likelihood.
In Figure A.3, we depict the components of the multiplicative REGARCH models
for all covariates. For VIX, the story is quite clear. Since VIX is a measure of the
risk-neutral expectation of the integrated variance of the S&P 500 Index for the
next 30 calender days, its movements are less erratic than the conditional volatility.
The component g t can been seen as the forward looking measure that explains the
majority of conditional volatility, while ht , updated using the innovations to the
realized measure, helps explain the remaining part.
¿ Insert Figure A.3 about here À
The second exogenous variable, EPU, is significant in the EGARCH models. Again,
with the largest improvement for the multiplicative model. The values of the coef-
ficient α are comparable in magnitude with the values in the benchmark EGARCH
models indicating that the variable not just subsumes the information in the squared
return. The picture differs for the REGARCH models. Here, the EPU seems to be most
relevant in the additive model. The positive coefficient on EPU is intuitive since it
is reasonable to believe that economic uncertainty drives financial market volatility.
From Figure A.3, it is clear that the majority of conditional volatility is explained by
ht in the multiplicative REGARCH model.
The ADS is significant for all considered models, but the log-likelihood gain is sub-
stantially larger for the additive models. As volatility is known to be countercyclical,
the sign of the coefficients is again intuitive. Furthermore, examining A.3, we see that
g t explains a small fraction of conditional volatility in the multiplicative REGARCH
model.
For all variables jointly, we observe highly significant log-likelihood improve-
ments, but EPU is generally insignificant. The value of the partial log-likelihood is
generally largest when including all exogenous variables. The log-likelihood improve-
ments are higher for the multiplicative models than the additive models primarily
driven by the inclusion of VIX.
To sum up, the inclusion of exogenous variables is found to be highly relevant for
modeling financial market volatility in-sample. This holds true also in a framework
employing realized measures in a dynamically complete fashion. One general point
to be made is that the relevance of including exogenous variables is dependent on
whether an additive or a multiplicative specification are considered.
¿ Insert Table A.3-A.6 about here À
128 CHAPTER 3.
Forecasting results
In this section, we present the results from a forecasting exercise. We reestimate the
models using data covering the period from January 2, 2002 to December 31, 2010
(2267 observations). The remaining part of the sample (754 observations) is used to
evaluate the forecasting performance of the different models.
We start by considering the four models one by one. Figure A.4 presents a compar-
ison of the forecasting performance when including exogenous variables relative to
the benchmark of no exogenous variables. For the EGARCH models, we find evidence
of statistically significant superior forecasting performance up to 10 days ahead when
including VIX or all variables by applying a Diebold-Mariano test (Diebold and Mari-
ano, 1995) using the QLIKE loss function.9 For the multiplicative REGARCH model,
we find superior one and two day ahead forecasting performance for VIX and all
variables jointly. For the additive REGARCH models, we find statistically significant
one and two day ahead outperformance when including ADS.
¿ Insert Figure A.4 about here À
Now, we look at the overall picture by applying the MCS of Hansen et al. (2011) to
test for superior forecasting performance.10 Table A.7 contains mean QLIKE ratios rel-
ative to the EGARCH model and the MCS coverage based on the QLIKE loss function
for different forecasting horizons. Based on one day ahead forecasting only models
with VIX as an exogenous variable are included in the MCS. Although the mean QLIKE
is much smaller for the models including VIX for multi day ahead forecasting, it is
very hard to statically discriminate between the models.
¿ Insert Table A.7 about here À
The conclusion of this forecasting exercise is that the use of exogenous covari-
ates, in particular VIX, is beneficial for forecasting in both additive and multiplicative
EGARCH models. Although we realize short horizon gains, the benefits from including
VIX in the more sophisticated REGARCH model are dramatically smaller. A potential
explanation for this is that the REGARCH extensions with covariates are compet-
ing with benchmarks that in a dynamically complete way utilize realized measures
whereas the added covariates is assumed to follow a random walk. A possible way
to improve the multi day ahead forecasting results is to specify a more reasonable
dynamic structure for the covariates. We leave this for future research.
9The Diebold-Mariano test statistics, S = T 1/2d/p
V →d N(0,1), is based on the loss differentials, d ,defined in Subsection 3.6.1 of Section 3.6 and an HAC estimator of the long-run variance of loss differentials√
V . We note that the Diebold-Marino test formally cannot be used to compare nested models (see e.g.Clark and McCracken (2001)). Thus, the test can only be viewed as an approximation.
10We employ the code of Kevin Sheppard to calculate the MCS. The code is available at https://www.kevinsheppard.com/MFE_Toolbox.
3.7. EMPIRICAL APPLICATION 129
3.7.3 Results for individual stocks
In this section, we present an empirical application to 20 stocks: AA (Alcoa), AIG
(American International Group), AXP (American Express), BA (Boing, CAT (Cater-
pillar), DD (DuPont), DIS (Walt Disney), GE (General Electric), IBM (International
Business Machines), INTC (Intel), JNJ (Johnson & Johnson), KO (Coca-Cola), MMM
(3M), MRK (Merck), MSFT (Microsoft), PG (Procter & Gamble), WHR (Whirlpool),
WMT (Walmart), and XOM (Exxon Mobil). The sample period is identical to the sam-
ple period in the previous example. Due to space considerations, we will not present
the results for each single stock but only in an aggregate form.
Figures A.5-A.8 present LR tests for the 20 stocks considered. The overall picture is
similar to the results for SPY. VIX is significant in most volatility specifications and the
gains are substantially larger in the multiplicative models. In fact, in a few cases VIX
is insignificant in the additive EGARCH and REGARCH specification. For all models
including VIX, the log-likelihood gains are smaller than for SPY. EPU is rarely or only
borderline significant in most models. ADS is significant in a large proportion of the
additive models, but rarely so in the multiplicative models. For almost all models, the
variables are jointly significant.
¿ Insert Figure A.5-A.8 about here À
In Figure A.9, we depict g t for four stocks: AA, DD, GE and KO. Again, there is a
clear resemblance with g t for SPY depicted in A.3, but for the specifications including
VIX a smaller degree of the variation is now explained by the g t component. This is
naturally since we are using the implied volatility of the market and not the particular
assets. For the specifications with VIX, we can think of the g t component as capturing
the aggregate level of volatility while ht is modeling the idiosyncratic part of volatility.
¿ Insert Figure A.9 about here À
For different forecast horizons, Table A.8 shows the percentage of times the dif-
ferent models is included in the 10% MCS. Again, multiplicative REGARCH models
including VIX seem to outperform at short horizons. At longer horizons, it is again
very hard to discriminate between models.
¿ Insert Table A.8 about here À
3.7.4 Discussion
The number of empirical studies on the ability of VIX, and implied volatility in general,
to forecast future volatility is vast. Some find that implied volatility dominates realized
volatility, while other in contrast find that implied volatility is a biased and inefficient
forecast of future volatility and contains little or no incremental information beyond
that in past realized volatility. The comparison between these studies is complicated
130 CHAPTER 3.
by differences in data, sampling frequency, forecasting horizon, econometric method-
ology, benchmark models, and testing procedures. A part of the literature is using
encompassing tests to compare the forecast obtained from GARCH models and other
volatility models with implied volatility. Fleming et al. (1995) find that VIX performs
well as a volatility forecast (one month ahead), but that it is not an unbiased forecast.
Also, Fleming (1998) finds that implied volatility is an upward biased forecast, but
also that it contains relevant information regarding future volatility. A linear model
using only the implied volatility appears to deliver a quality forecast of ex post volatil-
ity (one month ahead). Christensen and Prabhala (1998) find that implied volatility
works better than realized volatility on a monthly horizon. The studies of Szakmary,
Ors, Kim, and Davidson (2003), Corrado and Miller (2005), Carr and Wu (2006), Giot
and Laurent (2007), and Yu, Lui, and Wang (2010) also favor implied volatility and
some of the studies even find implied volatility or VIX to be close to unbiased. In
contrast, Becker, Clements, and White (2006) find that historical models may contain
additional information, when shorter forecasting horizons are considered. Martens
and Zein (2004) find that volatility forecasts based on historical intra day returns do
provide good volatility forecasts that can compete with and even outperform implied
volatility. The difference from previous studies is the use of an long-memory volatility
model with realized measures of volatility rather than plain vanilla GARCH models.
In GARCH type models with VIX added directly to the GARCH equation, Blair, Poon,
and Taylor (2001) find that VIX performs very well and subsumes the information in
squared returns. In an application only focusing on one step ahead volatility predic-
tions, Kambouroudis and McMillan (2016) find rather mixed results using a range of
different GARCH type specifications. In general, their forecasting models including
VIX performs marginally better. Our results confirm that the inclusion of VIX in a
plain vanilla volatility model using only squared return leads to both large in-sample
and out-of-sample gains. We also, corroborate the finding in Martens and Zein (2004)
that the use of dynamically complete volatility models employing realized measures
may alter results.
In the Macro-GARCH model Dorion (2016), ADS is significant in-sample and
found to significantly reduce option pricing errors in an out-of-sample exercise for
the S&P 500 Index. We obtain qualitatively similar in-sample results, but we do not
find out-of-sample gains. One issue with the analysis conducted in this paper is the
limited number of recession periods. The advantage of a model using macroeconomic
variables is to allow for different states in the form of expansions and recessions and
such a model is likely to only show its true potential in an out-of-sample exercise with
a change in states. In fact, Dorion (2016) also finds that the Macro-GARCH model
is particular useful in recessions. This may explain the difference in out-of-sample
performance. Therefore, the out-of-sample results of this paper should not discourage
the use of macroeconomic variables to improve GARCH type models. However, it
should be stressed that in many application with a limited span of data available, it is
3.8. CONCLUSION 131
likely unfruitful to try harvesting the explanatory power of macroeconomic variables
for volatility forecasting in GARCH type models.
EPU is the variable with the worst performance in our analysis. We find like
the existing literature, see Liu and Zhang (2015) and reference therein, that EPU
is statistically significant in-sample. However, we cannot obtain the encouraging
out-of-sample results of Liu and Zhang (2015).
3.8 Conclusion
In this paper, we present two different ways of including exogenous variables in the
Realized EGARCH framework. The first approach includes the exogenous variables
in the GARCH equation while the second approach disentangles the conditional
heteroskedasticity and the baseline volatility into two components.
The empirical application consider three exogenous variables: VIX, EPU, and ADS.
The exogenous variables seem to add information both in-sample and out-of-sample.
Especially, the multiplicative decompositions including VIX as a exogenous variable
perform very well. In line with the literature, we observe large forecasting gains for
the EGARCH model with VIX as a covariate. Superior forecasting performance is
restricted to short horizons in the REGARCH model. This may be explained by strong
performance of the benchmark REGARCH model and the likely unreasonable random
walk assumption for the exogenous variables. Proper forecasting may require explicit
modeling of the dynamic structure of the exogenous variables just as we explicitly
model the dynamics of the realized measure.
Interesting work remains to be undertaken. In the GARCH literature, we have
seen recent attempts to improve the information content from exogenous variables.
Examples include the GARCH-MIDAS model of Engle et al. (2013) and the Semi-
parametric Multiplicative GARCH-X model of Han and Kristensen (2015). Similar
approaches may be considered in a Realized EGARCH framework. Another avenue of
research is to allow for non-linear dynamics such as smooth transitions in a fashion
similar to Amado and Laakkonen (2014). The development of the asymptotic theory
and issues related to model specification, evaluation and estimation are also very
interesting research areas.
Acknowledgement
We thank Timo Teräsvirta, Asger Lunde, Peter Reinhard Hansen and Esther Ruiz Or-
tega for useful comments and suggestions. We also thank Asger Lunde for providing
cleaned high-frequency data. The authors acknowledge support from CREATES -
Center for Research in Econometric Analysis of Time Series (DNRF78), funded by the
Danish National Research Foundation. Some of this research was carried out while
Johan S. Jakobsen was visiting the School of Economics and Finance, Queensland
132 CHAPTER 3.
University of Technology. He would like to acknowledge the generosity and hospi-
tality of the faculty at the department and in particular Stan Hurn and Annastiina
Silvennoinen.
3.9. REFERENCES 133
3.9 References
Alagidede, P., 2008. Day of the week seasonality in african stock markets. Applied
Financial Economics Letters 4 (2), 115–120.
Amado, C., Laakkonen, H., 2014. Modelling time-varying volatility in financial returns:
Evidence from the bond markets. Oxford University Press, 249–268.
Amado, C., Teräsvirta, T., 2013. Modelling volatility by variance decomposition. Jour-
nal of Econometrics 175, 142–153.
Andersen, T. G., Bollerslev, T., Diebold, F. X., Ebens, H., 2001. The distribution of
realized stock return volatility. Journal of Financial Economics 61 (1), 43 – 76.
Andersen, T. G., Bollerslev, T., Diebold, F. X., Labys, P., 2003. Modeling and forecasting
realized volatility. Econometrica 71, 579–625.
Aruoba, S. B., Diebold, F. X., Scotti, C., 2009. Real-time measurement of business
conditions. Journal of Business & Economic Statistics 27 (4), 417–427.
Asgharian, H., Christiansen, C., Hou, A. J., 2015. Effects of macroeconomic uncertainty
on the stock and bond markets. Finance Research Letters 13, 10–16.
Asgharian, H., Hou, A. J., Javed, F., 2013. The importance of the macroeconomic
variables in forecasting stock return variance: A GARCH-MIDAS approach. Journal
of Forecasting 32 (7), 600–612.
Baker, H. K., Rahman, A., Saadi, S., 2008. The day-of-the-week effect and conditional
volatility: Sensitivity of error distributional assumptions. Review of Financial Eco-
nomics 17 (4), 280–295.
Baker, S. R., Bloom, N., Davis, S. J., 2016. Measuring economic policy uncertainty. The
Quarterly Journal of Economics.
Banulescu, G. D., Hansen, P. R., Huang, Z., Matei, M., 2014. Volatility during the finan-
cial crisis through the lens of high frequency data: A Realized EGARCH approach.
Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N., 2008. Designing
realized kernels to measure the ex post variation of equity prices in the presence of
noise. Econometrica 76, 1481–1536.
Barndorff-Nielsen, O. E., Shephard, N., 2002. Econometric analysis of realized volatil-
ity and its use in estimating stochastic volatility models. Journal of the Royal Statis-
tical Society: Series B (Statistical Methodology) 64, 253–280.
Becker, R., Clements, A. E., White, S. I., 2006. On the informational efficiency of
S&P500 implied volatility. The North American Journal of Economics and Finance
17 (2), 139 – 153.
134 CHAPTER 3.
Bekaert, G., Hoerova, M., 2014. The VIX, the variance premium and stock market
volatility. Journal of Econometrics 183, 181–192.
Black, F., 1976. Studies of stock market volatility changes. Proceedings of the American
Statistical Association, Business and Economics Statistics Section, 177–181.
Blair, B. J., Poon, S.-H., Taylor, S. J., 2001. Forecasting S&P 100 volatility: The incremen-
tal information content of implied volatilities and high-frequency index returns.
Journal of Econometrics 105 (1), 5 – 26.
Bollerslev, T., 1986. Generalized autoregressive conditional heteroscedasticity. Journal
of Econometrics 31, 207–327.
Bollerslev, T., Melvin, M., 1994. Bid-ask spreads and volatility in the foreign exchange
market: An empirical analysis. Journal of International Economics 36, 355–372.
Borup, D., Jakobsen, J. S., 2017. Volatility persistence in the Realized Exponential
GARCH model, working Paper.
Brenner, R. J., Harjes, R. H., Kroner, K. F., 1996. Another look at models of the short-
term interest rate. The Journal of Financial and Quantitative Analysis 31 (1), 85–107.
Brownless, C. T., Gallo, G. M., 2010. Comparison of volatility measures. A risk man-
agement perspective. Journal of Financial Econometrics 8, 29–56.
Carr, P., Wu, L., 2006. A tale of two indices. Journal of Derivatives 13 (3), 13–29.
Charles, A., 2010. The day-of-the-week effects on the volatility: The role of the asym-
metry. European Journal of Operational Research 202 (1), 143 –152.
Christensen, B., Prabhala, N., 1998. The relation between implied and realized volatil-
ity. Journal of Financial Economics 50 (2), 125 –150.
Christensen, K., Podolskij, M., 2007. Realized range-based estimation of integrated
variance. Journal of Econometrics 141, 323 –349.
Clark, T. E., McCracken, M., 2001. Tests of equal forecast accuracy and encompassing
for nested models. Journal of Econometrics 105 (1), 85–110.
Connolly, R. A., 1989. An examination of the robustness of the weekend effect. The
Journal of Financial and Quantitative Analysis 24 (2), 133–169.
Conrad, C., Kleen, O., 2016. On the statistical properties of multiplicative GARCH
models, working Paper.
Conrad, C., Loch, K., 2015. Anticipating long-term stock market volatility. Journal of
Applied Econometrics 30 (7), 1090–1114.
3.9. REFERENCES 135
Conrad, C., Schienle, M., 2015. Misspecification testing in GARCH-MIDAS models.
Tech. rep., University of Heidelberg, Department of Economics.
Corrado, C. J., Miller, Jr., T. W., 2005. The forecast quality of CBOE implied volatility
indexes. Journal of Futures Markets 25 (4), 339–373.
Day, T. E., Lewis, C. M., 1992. Stock market volatility and the information content of
stock index options. Journal of Econometrics 52 (1), 267 – 287.
Diebold, F. X., Mariano, R. S., 1995. Comparing predictive accuracy. Journal of Busi-
ness and Economic Statistics 13 (3), 253–263.
Dominguez, K., 1998. Central bank intervention and exchange rate volatility. Journal
of International Money and Finance 17 (1), 161–190.
Dominicy, Y., Vander Elst, H., 2015. Macro-driven VaR forecasts: From very high to
very low-frequency data, working Paper.
Dorion, C., 2016. Option valuation with macro-finance variables. Journal of Financial
and Quantitative Analysis 51 (4), 1359–1389.
Engle, R., 2002. New frontiers for ARCH models. Journal of Applied Econometrics 17,
425–446.
Engle, R., Ghysels, E., Sohn, B., 2013. Stock market volatility and macroeconomic
fundamentals. The Review of Economics and Statistics 95, 776–797.
Engle, R., Lee, H., 1999. Cointegration, causality, and forecasting: A festschrift in
honour of Clive W.J. Granger. Oxford University Press, Ch. A long-run and short-
run component model of stock return volatility, 475–497.
Engle, R. F., 1982. Autoregressive conditional heteroscedasticity with estimates of the
variance of United Kingdom inflation. Econometria 50, 987–1007.
Engle, R. F., Gallo, G. M., 2006. A multiple indicators model for volatility using intra-
daily data. Journal of Econometrics 131, 3–27.
Engle, R. F., Ng, V. K., December 1993. Measuring and testing the impact of news on
volatility. Journal of Finance 48 (5), 1749–78.
Engle, R. F., Rangel, J. G., 2008. The Spline-GARCH model for low-frequency volatility
and its global macroeconomic causes. Review of Financial Studies 21, 1187–1222.
Fleming, J., 1998. The quality of market volatility forecasts implied by S&P 100 index
option prices. Journal of Empirical Finance 5 (4), 317 – 345.
Fleming, J., Kirby, C., Ostdiek, B., 2008. The specification of GARCH models with
stochastic covariates. Journal of Futures Markets 28 (10), 911–934.
136 CHAPTER 3.
Fleming, J., Ostdiek, B., Whaley, R. E., 1995. Predicting stock market volatility: A new
measure. Journal of Futures Markets 15 (3), 265–302.
Francq, C., Zakoïan, J.-M., 2010. GARCH models: Structure, statistical inference and
financial applications. John Wiley & Sons, Ltd, 311–340.
Gallo, G. M., Pacini, B., 2000. The effects of trading activity on market volatility. The
European Journal of Finance 6 (2), 163–175.
Gatheral, J., Oomen, R. C. A., 2010. Zero-intelligence realized variance estimation.
Finance and Stochastics 14, 249–283.
Giot, P., Laurent, S., 2007. The information content of implied volatility in light of the
jump/continuous decomposition of realized volatility. Journal of Futures Markets
27 (4), 337–359.
Girma, P. B., Mougoué, M., 2002. An empirical examination of the relation between
futures spreads volatility, volume, and open interest. Journal of Futures Markets
22 (11), 1083–1102.
Glosten, L. R., Jagannathan, R., Runkle, D. E., 1993. On the relation between the
expected value and the volatility of the nominal excess return on stocks. The
Journal of Finance 48 (5), 1779–1801.
Gray, S. F., 1996. Modeling the conditional distribution of interest rates as a regime-
switching process. Journal of Financial Economics 42 (1), 27–62.
Hagiwara, M., Herce, M. A., 1999. Endogenous exchange rate volatility, trading vol-
ume and interest rate differentials in a model of portfolio selection. Review of
International Economics 7 (2), 202–218.
Han, H., 2015. Asymptotic properties of GARCH-X processes. Journal of Financial
Econometrics 13 (1), 188–221.
Han, H., Kristensen, D., 2014. Asymptotic theory for the QMLE in GARCH-X models
with stationary and nonstationary covariates. Journal of Business & Economic
Statistics 32 (3), 416–429.
Han, H., Kristensen, D., 2015. Semparametric multiplicative GARCH-X model: Adapt-
ing economic variables to explain volatility. Tech. rep., University College London.
Han, H., Park, M. D., 2013. Comparison of realized measure and implied volatility in
forecasting volatility. Journal of Forecasting 32 (6), 522–533.
Hansen, P. R., Huang, Z., 2016. Exponential GARCH modeling with realized measures
of volatility. Journal of Business and Economic Statistics 34, 269–287.
3.9. REFERENCES 137
Hansen, P. R., Huang, Z., Shek, H. H., 2012. Realized GARCH: A joint model for returns
and realized measures of volatility. Journal of Applied Econometrics 27, 877–906.
Hansen, P. R., Large, J., Lunde, A., 2008. Moving average-based estimators of integrated
variance. Econometric Reviews 27 (1-3), 79–111.
Hansen, P. R., Lunde, A., 2006. Consistent ranking of volatility models. Journal of
Econometrics 131, 97 – 121.
Hansen, P. R., Lunde, A., Nason, J. M., 2011. The model confidence set. Econometrica
79, 453–497.
Hillebrand, E., 2005. Neglecting parameter changes in GARCH models. Journal of
Econometrics 129, 121–138.
Hodrick, R. J., 1989. Risk, uncertainty, and exchange rates. Journal of Monetary Eco-
nomics 23 (3), 433 – 459.
Huang, Z., Liu, H., Wang, T., 2016. Modeling long memory volatility using realized
measures of volatility: A realized HAR GARCH model. Economic Modelling 52,
812–821.
Kambouroudis, D. S., McMillan, D. G., 2016. Does VIX or volume improve GARCH
volatility forecasts? Applied Economics 48 (13), 1210–1228.
Kim, T.-H., White, H., 2004. On more robust estimation of skewness and kurtosis.
Finance Research Letters 1, 56–73.
Kruse, R., Leschinski, C., Will, C., 2016. Comparing predictive ability under long
memory - with an application to volatility forecasting, CREATES Working Paper.
Lamoureux, C. G., Lastrapes, W. D., 1990. Heteroskedasticity in stock return data:
Volume versus GARCH effects. The Journal of Finance 45 (1), 221–229.
Laursen, B., Jakobsen, J. S., 2017. Realized EGARCH models with time-varying uncon-
ditional variance, working Paper.
Liu, L., Zhang, T., 2015. Economic policy uncertainty and stock market volatility.
Finance Research Letters 15, 99 – 105.
Mandelbrot, B., 1963. The variation of certain speculative prices. The Journal of
Business 36, 394–394.
Martens, M., Zein, J., 2004. Predicting financial volatility: High-frequency time-series
forecasts vis-a-vis implied volatility. Journal of Futures Markets 24 (11), 1005–1028.
Mencía, J., Sentana, E., 2013. Valuation of VIX derivatives. Journal of Financial Eco-
nomics 108 (2), 367 – 391.
138 CHAPTER 3.
Mikosch, T., Starica, C., 2004. Nonstationarities in financial time series, the long-
range dependence, and the IGARCH effects. Review of Economics and Statistics 86,
378–390.
Morana, C., 2002. IGARCH effects: An interpretation. Applied Economics Letters 9,
745–748.
Nelson, D. B., 1991. Conditional heteroskedasticity in asset returns: A new approach.
Econometrica 59 (2), 347–370.
Patton, A. J., 2011. Volatility forecast comparison using imperfect volatility proxies.
Journal of Econometrics 160, 246 – 256.
Patton, A. J., Sheppard, K., 2009. Handbook of financial time series. Springer Berlin
Heidelberg, Berlin, Heidelberg, Ch. Evaluating volatility and correlation forecasts,
801–838.
Paye, B. S., 2012. Déja vol: Predictive regressions for aggregate stock market volatility
using macroeconomic variables. Journal of Financial Economics 106 (3), 527–546.
Sharma, P., Vipul, 2016. Forecasting stock market volatility using Realized Garch
model: International evidence. The Quarterly Review of Economics and Finance
59, 222 – 230.
Shephard, N., Sheppard, K., 2010. Realising the future: Forecasting with high-
frequency-based volatility (HEAVY) models. Journal of Applied Econometrics 25,
197–231.
Shimotsu, K., 2010. Exact local whittle estimation of fractional integration with un-
known mean and time trend. Econometric Theory 26 (2), 501–540.
Szakmary, A., Ors, E., Kim, J. K., Davidson, W. N., 2003. The predictive power of
implied volatility: Evidence from 35 futures markets. Journal of Banking & Finance
27 (11), 2151 – 2175.
Teräsvirta, T., Zhao, Z., 2011. Stylized facts of return series, robust estimates and three
popular models of volatility. Applied Financal Economics 21 (1-2), 67–94.
Vander Elst, H., 2015. FloGARCH: Realizing long memory and asymmetries in returns
volatility, working paper.
Wagner, N., Marsh, T. A., 2005. Surprise volume and heteroskedasticity in equity
market returns. Quantitative Finance 5 (2), 153–168.
Whaley, R. E., 2000. The investor fear gauge. Journal of Portfolio Management 26 (3),
12–17.
3.9. REFERENCES 139
Whaley, R. E., 2009. Understanding the VIX. Journal of Portfolio Management 35 (3),
98–105.
Wintenberger, O., 2013. Continuous invertibility and stable QML estimation of the
EGARCH(1,1) model. Scandinavian Journal of Statistics 40, 846–867.
Yu, W. W., Lui, E. C., Wang, J. W., 2010. The predictive power of the implied volatility of
options traded otc and on exchanges. Journal of Banking & Finance 34 (1), 1 – 11.
Zhang, L., 2006. Efficient estimation of stochastic volatility using noisy observations:
A multi-scale approach. Bernoulli 12, 1019–1043.
Zhang, L., Mykland, P. A., Aït-Sahalia, Y., 2005. A tale of two time scales: Determin-
ing integrated volatility with noisy high-frequency data. Journal of the American
Statistical Association 100, 1394–1411.
140 CHAPTER 3.
Appendix
A.1 Derivation of Scores
A.1.1 Proof of Lemma 1
First, consider A(zt ) = ∂ logσ2t+1/∂ logσ2
t . Using zt = rt−µσt
, it can easily be shown that
∂zt
∂ logσ2t
=−1
2zt . (A.1)
For ut = log xt −φ logσ2t −δ
(zt
), we find that
∂ut
∂ logσ2t
=− ∂δ(zt
)∂ logσ2
t
−φ=−δ′bzt
∂zt
∂ logσ2t
−φ, (A.2)
Similarly, we have
∂τ(zt
)∂ logσ2
t
= τ′azt
∂zt
∂ logσ2t
= τ′azt zzt
∂zt
∂ logσ2t
, (A.3)
Inserting the above components in the following expression for A(zt ),
A(zt ) =β+ ∂τ(zt
)∂ logσ2
t
+α ∂ut
∂ logσ2t
, (A.4)
yields
A(zt ) = (β−αφ)+ 1
2
(αδ′bzt −τ′azt
)zt . (A.5)
Next, we turn to B(zt ,ut ) = ∂`t /∂ logσ2t . The term logσ2
t enters the log-likelihood
contribution at time t directly due to the logσ2t term and indirectly through z2
t and
u2t . Thus, we have
B(zt ,ut ) =−1
2
[1+ ∂z2
t
∂ logσ2t
+ 1
σ2u
2ut∂ut
∂ logσ2t
]. (A.6)
where∂z2
t
∂ logσ2t
=−z2t . (A.7)
Combing the different expressions yields,
B(zt ,ut ) =−1
2
[(1− z2
t
)+ ut
σ2u
(δ′bzt zt −2φ
)]. (A.8)
�
A.1. DERIVATION OF SCORES 141
A.1.2 Proof of Lemma 2
For hµ,t+1 = ∂ logσ2t+1/∂µ, we have
hµ,t+1 =β∂ logσ2
t+1
∂σt
∂ logσ2t
∂µ+ ∂τ
(zt
)∂µ
+α∂ut
∂µ, (A.9)
with
∂τ(zt
)∂µ
= ∂τ(zt
)∂zt
∂zt
∂µ
= τ′ ∂a(zt
)∂zt
[−1
2zt∂ logσ2
t
logµ− 1
σt
]
= τ′azt
[−1
2zt hµ,t − 1
σt
], (A.10)
and
∂ut
∂µ= ∂ut
∂µ
=[−φ logσ2
t
∂µ− ∂δ
(zt
)∂zt
∂zt
∂µ
]
=[−φhµ,t −δ′bzt
[−1
2zt hµ,t − 1
σt
]]. (A.11)
Inserting (A.10) and (A.11) into (A.9) and manipulating yields
hµ,t+1 = A(zt )hµ,t +(αδ′bzt −τ′azt
) 1
σt.
For hθ1,t+1 = ∂ logσ2t+1/∂θ1, we have
hθ1,t+1 =β∂ logσ2
t
∂θ1+ ∂τ
(zt
)∂θ1
+α∂ut
∂θ1+
(1, logσ2
t , zt , z2t −1,ut
)′. (A.12)
However, we remember that τ(zt
)and ut only depends on θ1 through logσ2
t such
that the three first terms can be collapsed to one. Thus, we can instead write
hθ1,t+1 = A(zt )hθ1,t +mt . (A.13)
Exactly, what we wanted to show. For hθ2,t = ∂ logσ2t+1/∂θ2, we have
hθ2,t+1 =β∂ logσ2
t
∂θ2+ ∂τ
(zt
)∂θ2
+α∂ut
∂ut
(1, log
(σ2
t
), zt , z2
t −1
)′. (A.14)
Using the same arguments as before, we obtain
hθ2,t+1 = A(zt )hθ2,t +αnt . (A.15)
142 CHAPTER 3.
Finally, for hγ,t+1 = ∂ logσ2t+1/∂γ, we have
hγ,t+1 = A(zt )hγ,t +vt .
�
A.1.3 Proof of Theorem 1
The parameter µ enters the log-likelihood contribution at time t through logσ2t , zt ,
and u2t such that
∂`t
∂µ=−1
2
∂ logσ2t
∂µ− 1
2
∂z2t
∂µ− 1
2
1
σ2u
∂u2t
∂µ
= ∂`t
∂ logσ2t
∂ logσ2t
∂µ−
[zt −δ′ ut
σ2u
∂b(zt
)∂zt
]1
σt
= B(zt ,ut )hµ,t −[
zt +δ′ ut
σ2u
bzt
]1
σt.
As θ1,θ2, θ3, and γ only enter the log-likelihood contribution at time t indirectly
through logσ2t , an application of the chain-rule yields
∂`t
∂θ1= B(zt ,ut )hθ1,t , (A.16)
∂`t
∂θ3= B(zt ,ut )hθ3,t , (A.17)
∂`t
∂γ= B(zt ,ut )hγ,t , (A.18)
and as(ξ,φ,δ′
)′ enters through logσ2t and u2
t , an application of the chain-rule yields
∂`t
∂θ2= B(zt ,ut )hθ2,t +
ut
σ2u
nt . (A.19)
σ2u only enters directly into the log-likelihood contribution such that we obtain
∂`t
∂σ2u= 1
2
u2t −σ2
u
σ4u
. (A.20)
�
A.1.4 Proof of Lemma 3, Lemma 4, and Theorem 2
The proofs are almost identical to the additive case, so we do not present the deriva-
tions. See Lemma 1 and 2, and Theorem 1.
A.2. FIGURES 143
A.2 Figures
2002 2005 2008 2010 2013-0.15
-0.1
-0.05
0
0.05
0.1
0.15
2002 2005 2008 2010 20130
1
2
3
4
5
6
2002 2005 2008 2010 20130
0.5
1
1.5
2
50 100 150 2000
0.2
0.4
0.6
0.8
1
Figure A.1: Data plotsThe upper-left panel: SPY daily (close-to-close) return. The lower-left panel: SPY daily RK(annualized). The upper-right panel: SPY daily squared (close-to-close) returns (annualized).The lower-right panel: SPY daily log(RK) autocorrelation function.
144 CHAPTER 3.
2002 2005 2008 2010 20130
0.5
1
-12 -10 -8 -6 -4
-12
-10
-8
-6
-4
2002 2005 2008 2010 2013-4
-2
0
2
4
-4 -2 0 2
-12
-10
-8
-6
-4
2002 2005 2008 2010 2013
-4
-2
0
2
-4 -2 0 2
-12
-10
-8
-6
-4
Figure A.2: Plots of exogenous variablesThe upper-left panel: VIX. The middle-left panel: log(EPU/100). The lower-left panel: ADS.
The upper-right panel: logRKt plotted against log(VIX2
t−1/250). The middle-left panel: logRKt
plotted against log(
EPUt−1100
). The lower-right panel: logRKt plotted against ADSt−1.
A.2. FIGURES 145
2002 2005 2008 2010 20130
5
10
15
20
25
30
35
40
45
50
2002 2005 2008 2010 20130
5
10
15
20
25
30
35
40
2002 2005 2008 2010 20130
5
10
15
20
25
30
35
40
45
2002 2005 2008 2010 20130
5
10
15
20
25
30
35
40
45
50
Figure A.3: The components of the multiplicative REGARCH modelIllustrates ht (red), gt (blue) and σ2
t of the multiplicative REGARCH models with exogenouscovariates. The upper-left panel: VIX. The upper-right panel: ADS. The lower-left panel: EPU.The lower-right panel: all exogenous variables.
146 CHAPTER 3.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.70
0.80
0.90
1.00
1.10
1.20
1.30
Additive EGARCH
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.70
0.80
0.90
1.00
1.10
1.20
1.30
Multiplicative EGARCH
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.70
0.80
0.90
1.00
1.10
1.20
1.30
Additive REGARCH
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.70
0.80
0.90
1.00
1.10
1.20
1.30
Multiplicative REGARCH
Figure A.4: Forecast evaluation with Diebold-Mariano testOut-of-sample ratio of mean QLIKE for the four presented models with different exogenousvariables relative to their benchmark case without exogenous variables. For each model we
consider log(VIX2
t−1/250)
(blue), log(EPU/100) (yellow), ADSt−1 (red), and all (purple). A
filled marker indicates rejection of equal forecast at a 10% significance level using the Diebold-Mariano test (one-sided alternative).
A.2. FIGURES 147
AA AIG AXP BA CAT DD DIS GE IBM INTC JNJ KO MMM MRK MSFT PG VZ WHR WMT XOM0
50
100
150
200
250
Additve EGARCH
Multiplicative EGARCH
Additive REGARCH
Multiplicative REGARCH
Figure A.5: LR-test statistics for including VIXLR statistics for including VIX. The dashed line indicates the 5% critical value.
AA AIG AXP BA CAT DD DIS GE IBM INTC JNJ KO MMM MRK MSFT PG VZ WHR WMT XOM0
50
100
150
200
250
Additve EGARCH
Multiplicative EGARCH
Additive REGARCH
Multiplicative REGARCH
Figure A.6: LR-test statistics for including EPULR statistics for including EPU. The dashed line indicates the 5% critical value.
148 CHAPTER 3.
AA AIG AXP BA CAT DD DIS GE IBM INTC JNJ KO MMM MRK MSFT PG VZ WHR WMT XOM0
50
100
150
200
250
Additve EGARCH
Multiplicative EGARCH
Additive REGARCH
Multiplicative REGARCH
Figure A.7: LR-test statistics for including ADSLR statistics for including ADS. The dashed line indicates the 5% critical value.
AA AIG AXP BA CAT DD DIS GE IBM INTC JNJ KO MMM MRK MSFT PG VZ WHR WMT XOM0
50
100
150
200
250
Additve EGARCH
Multiplicative EGARCH
Additive REGARCH
Multiplicative REGARCH
Figure A.8: LR-test statistics for including exogenous variablesLR statistics for including all exogenous variables. The dashed line indicates the 5% criticalvalue.
A.2. FIGURES 149
2002 2005 2008 2010 20130
50
100
150
Cond. Vol. BenchmarkVIXADSEPUVIX, ADS, EPU
2002 2005 2008 2010 20130
10
20
30
40
50
60
70
Cond. Vol. BenchmarkVIXADSEPUVIX, ADS, EPU
2002 2005 2008 2010 20130
20
40
60
80
100
120
140
Cond. Vol. BenchmarkVIXADSEPUVIX, ADS, EPU
2002 2005 2008 2010 20130
5
10
15
20
25
30
35
Cond. Vol. BenchmarkVIXADSEPUVIX, ADS, EPU
Figure A.9: Exogenous components of the multiplicative REGARCH modelDepicts gt from the extension with exogenous covariates and σ2
t from the multiplicativeREGARCH model. The upper-left panel: AA. The upper-right panel: DD. The lower-left panel:GE. The lower-right panel: KO.
150 CHAPTER 3.
A.3 Tables
Table A.1: Descriptive statistics for daily returns and RKThis table reports descriptive statistics for the daily returns and the realized kernel. Dailyreturns and the RK are in percentages. Robust skewness and kurtosis are from Kim and White(2004). The fractional integrated parameter d is estimated using the two-step exact localWhittle estimator of Shimotsu (2010) and bandwith of m = [T 0.65].
Return Log(RK)
No. ofobs.
MeanStd.Dev.
SkewRobustSkew
Ex. Kurt.Robust
Ex. Kurt.Median Mean
Std.Dev.
Median d
SP500 3020 0.02 1.32 0.07 -0.08 11.67 1.03 0.08 -0.35 1.00 -0.50 0.66
AA 3004 0.00 2.73 0.23 0.02 8.92 0.99 0.00 1.13 0.86 0.98 0.64AIG 2999 -0.00 4.55 1.42 0.01 54.58 1.17 -0.03 1.07 1.30 0.88 0.64AXP 2994 0.07 2.44 0.55 0.04 11.12 1.07 0.02 0.60 1.18 0.38 0.70BA 2996 0.07 1.89 0.23 0.01 4.07 0.84 0.07 0.60 0.82 0.47 0.64CAT 2998 0.07 2.09 0.11 0.03 5.06 0.92 0.06 0.73 0.82 0.59 0.67DD 2995 0.04 1.78 -0.04 0.01 5.68 0.88 0.04 0.51 0.85 0.38 0.63DIS 2997 0.07 1.91 0.51 -0.02 6.76 0.88 0.06 0.55 0.88 0.38 0.66GE 3005 0.02 1.99 0.38 0.03 10.30 1.06 0.00 0.40 1.05 0.22 0.68IBM 2996 0.03 1.53 0.20 0.01 6.87 0.87 0.02 0.10 0.83 -0.05 0.65INTC 3016 0.03 2.20 -0.22 -0.00 6.09 0.90 0.04 0.85 0.80 0.74 0.63JNJ 2997 0.03 1.16 -0.28 0.03 20.06 0.95 0.02 -0.28 0.86 -0.43 0.68KO 2999 0.04 1.24 0.32 -0.02 11.96 0.92 0.04 -0.10 0.81 -0.22 0.63MMM 2992 0.05 1.45 -0.06 0.02 5.54 0.97 0.06 0.13 0.80 0.03 0.64MRK 2994 0.03 1.80 -1.21 0.04 24.18 0.87 0.03 0.38 0.85 0.26 0.61MSFT 3016 0.03 1.81 0.37 0.02 8.94 0.96 0.00 0.46 0.81 0.32 0.63PG 2998 0.04 1.14 -0.02 0.01 6.74 0.92 0.03 -0.18 0.77 -0.30 0.61VZ 2995 0.03 1.57 0.34 -0.03 7.37 0.90 0.05 0.31 0.89 0.14 0.67WHR 2992 0.07 2.52 0.40 0.03 5.14 0.96 -0.01 1.01 0.86 0.92 0.58WMT 3001 0.03 1.31 0.30 -0.03 5.57 0.88 0.02 0.05 0.80 -0.09 0.65XOM 3001 0.05 1.60 0.34 -0.01 13.37 0.81 0.07 0.24 0.86 0.14 0.68
Table A.2: Estimation results for S&P 500 Index for benchmark modelsMaximum likelihood parameter estimates for the four different models. Standard errors aregiven in parenthesis. p denotes the total number of parameters in the given model.
EGARCH EGARCH Realized EGARCH Realized EGARCH(additive) (multiplicative) (additive) (multiplicative)
µ 0.021(0.0144) 0.021(0.0144) 0.013(0.0127) 0.013(0.0127)ω 0.001(0.0030) 0.059(0.1521) −0.002(0.0035) −0.069(0.1237)β 0.980(0.0034) 0.980(0.0034) 0.972(0.0036) 0.972(0.0037)α 0.111(0.0163) 0.111(0.0163) 0.319(0.0199) 0.319(0.0199)ξ −0.281(0.0328) −0.281(0.0328)σ2
u 0.154(0.0055) 0.154(0.0055)τ1 −0.140(0.0119) −0.140(0.0119) −0.146(0.0076) −0.146(0.0076)τ2 0.029(0.0047) 0.029(0.0047)δ1 −0.115(0.0084) −0.115(0.0084)δ2 0.044(0.0052) 0.044(0.0052)φ 1.004(0.0330) 1.004(0.0330)
p 5 5 11 11logL -4,242.04 -4,242.04 -5,640.07 -5,640.07logLp -4,242.04 -4,242.04 -4,181.19 -4,181.19
A.3. TABLES 151
Table A.3: Estimation results for S&P 500 Index with VIXMaximum likelihood parameter estimates for the four different models when including
log(VIX2
t−1/250)
as an exogenous variable. Standard errors are given in parenthesis. p de-
notes the total number of parameters in the given model.
EGARCH EGARCH Realized EGARCH Realized EGARCH(additive) (multiplicative) (additive) (multiplicative)
µ 0.008(0.0133) 0.009(0.0107) 0.012(0.0147) 0.011(0.0091)ω −0.069(0.0061) −0.488(0.0000) −0.091(0.0098) 9.149(6.9455)β 0.860(0.0057) 0.923(0.0104) 0.814(0.0047) 0.894(0.0221)α 0.005(0.0244) 0.000(0.0000) 0.312(0.0194) 0.242(0.0022)ξ −0.280(0.0329) −0.282(2.2690)σ2
u 0.145(0.0054) 0.140(0.1373)τ1 −0.237(0.0181) −0.135(0.0125) −0.150(0.0084) −0.068(0.0379)τ2 0.028(0.0060) 0.022(0.0578)δ1 −0.121(0.0083) −0.120(0.3865)δ2 0.041(0.0048) 0.042(0.7918)φ 0.994(0.0199) 0.978(0.0247)γV I X 0.149(0.0056) 1.030(0.0000) 0.195(0.0057) 1.047(0.2693)
p 6 6 12 12logL -4,183.71 -4,175.25 -5,547.82 -5,497.26logLp -4,183.71 -4,175.25 -4,177.91 -4,179.33LR 116.66∗∗∗ 133.58∗∗∗ 184.49∗∗∗ 285.61∗∗∗
152 CHAPTER 3.
Table A.4: Estimation results for S&P 500 Index with EPUMaximum likelihood parameter estimates for the four different models when including
log EPUt−1100 as an exogenous variable. Standard errors are given in parenthesis. p denotes the
total number of parameters in the given model.
EGARCH EGARCH Realized EGARCH Realized EGARCH(additive) (multiplicative) (additive) (multiplicative)
µ 0.018(0.0141) 0.019(0.0148) 0.015(0.0132) 0.013(0.0126)ω 0.003(0.0032) 0.080(0.1415) −0.000(0.0038) −0.068(0.1219)β 0.974(0.0036) 0.978(0.0034) 0.964(0.0037) 0.972(0.0037)α 0.106(0.0150) 0.112(0.0159) 0.319(0.0198) 0.318(0.0198)ξ −0.281(0.0329) −0.280(0.0321)σ2
u 0.153(0.0055) 0.154(0.0055)τ1 −0.152(0.0121) −0.139(0.0122) −0.149(0.0077) −0.146(0.0075)τ2 0.029(0.0047) 0.030(0.0046)δ1 −0.116(0.0085) −0.115(0.0084)δ2 0.044(0.0051) 0.044(0.0050)φ 1.004(0.0323) 0.999(0.0319)γEPU 0.015(0.0051) 0.210(0.0762) 0.022(0.0054) 0.037(0.0172)
p 6 6 12 12logL -4,236.56 -4,233.45 -5,633.07 -5,636.90logLp -4,236.56 -4,233.45 -4,180.50 -4,178.28LR 10.95∗∗∗ 17.17∗∗∗ 13.99∗∗∗ 6.34∗∗
A.3. TABLES 153
Table A.5: Estimation results for S&P 500 Index with ADSMaximum likelihood parameter estimates for the four different models when including ADSt−1as an exogenous variable. Standard errors are given in parenthesis. p denotes the total numberof parameters in the given model.
EGARCH EGARCH Realized EGARCH Realized EGARCH(additive) (multiplicative) (additive) (multiplicative)
µ 0.019(0.0142) 0.016(0.0113) 0.010(0.0135) 0.009(0.0134)ω −0.005(0.0038) −0.033(0.1191) −0.010(0.0045) −0.164(0.1132)β 0.970(0.0033) 0.973(0.0044) 0.959(0.0036) 0.962(0.0044)α 0.109(0.0171) 0.116(0.0170) 0.316(0.0199) 0.325(0.0194)ξ −0.281(0.0324) −0.281(0.0326)σ2
u 0.152(0.0055) 0.153(0.0055)τ1 −0.152(0.0125) −0.147(0.0126) −0.147(0.0077) −0.148(0.0078)τ2 0.030(0.0048) 0.031(0.0049)δ1 −0.115(0.0084) −0.116(0.0084)δ2 0.044(0.0052) 0.044(0.0052)φ 1.001(0.0309) 1.004(0.0317)γADS −0.017(0.0029) −0.281(0.0981) −0.023(0.0039) −0.311(0.0839)
p 6 6 12 12logL -4,228.36 -4,238.29 -5,624.99 -5,635.31logLp -4,228.36 -4,238.29 -4,180.65 -4,182.06LR 27.35∗∗∗ 7.50∗∗∗ 30.15∗∗∗ 9.51∗∗∗
154 CHAPTER 3.
Table A.6: Estimation Results for S&P 500 Index with all exogenous variablesMaximum likelihood parameter estimates for the four different models when including
log(VIX2
t−1/250),(log EPUt−1
100
)2, and ADSt−1 as exogenous variables. Standard errors are given
in parenthesis. p denotes the total number of parameters in the given model.
EGARCH EGARCH Realized EGARCH Realized EGARCH(additive) (multiplicative) (additive) (multiplicative)
µ 0.009(0.0140) 0.009(0.0138) 0.011(0.0147) 0.012(0.0145)ω −0.074(0.0091) −0.480(0.0644) −0.100(0.0104) −0.507(0.0522)β 0.860(0.0055) 0.918(0.0104) 0.805(0.0047) 0.883(0.0129)α 0.003(0.0227) 0.000(0.0162) 0.306(0.0199) 0.243(0.0186)ξ −0.280(0.0326) −0.281(0.0309)σ2
u 0.144(0.0054) 0.139(0.0053)τ1 −0.232(0.0183) −0.141(0.0154) −0.149(0.0084) −0.075(0.0080)τ2 0.029(0.0061) 0.023(0.0048)δ1 −0.121(0.0083) −0.120(0.0081)δ2 0.041(0.0048) 0.042(0.0045)φ 0.992(0.0192) 0.975(0.0196)γV I X 0.141(0.0104) 0.921(0.0742) 0.193(0.0077) 0.973(0.0389)γEPU −0.008(0.0112) 0.095(0.0623) −0.006(0.0081) 0.022(0.0165)γADS −0.020(0.0079) −0.124(0.0481) −0.025(0.0060) −0.134(0.0350)
p 8 8 14 14logL -4,177.35 -4,167.78 -5,537.53 -5,487.50logLp -4,177.35 -4,167.78 -4,177.50 -4,177.10LR 129.38∗∗∗ 148.52∗∗∗ 205.07∗∗∗ 305.14∗∗∗
A.3. TABLES 155
Table A.7: MCS coverage for S&P 500 Index (QLIKE)This table reports the ratio of out-of-sample QLIKE relative to the EGARCH model and theMCS coverage.
Specification of volatility dynamicsEGARCH EGARCH REGARCH REGARCH(additive) (multiplicative) (additive) (Multiplicative)
1 day aheadBenchmark 1.00 1.00 0.70 0.70VIX 0.82 0.79 0.70 0.63EPU 1.12 1.04 0.70 0.70ADS 1.04 0.99 0.70 0.70VIX, EPU, ADS 0.83 0.82 0.70 0.63
5 day aheadBenchmark 1.00 1.00 0.85 0.85VIX 0.87 0.92 0.85 0.82EPU 1.09 1.01 0.86 0.85ADS 1.03 1.00 0.86 0.86VIX, EPU, ADS 0.88 0.90 0.86 0.82
10 day aheadBenchmark 1.00 1.00 0.93 0.94VIX 0.91 0.96 0.93 0.90EPU 1.11 0.98 0.95 0.94ADS 1.03 1.00 0.96 0.95VIX, EPU, ADS 0.91 0.93 0.95 0.90
15 day aheadBenchmark 1.00 1.00 0.93 0.95VIX 0.96 0.99 0.93 0.95EPU 1.19 1.02 1.04 0.95ADS 1.04 1.00 0.98 0.97VIX, EPU, ADS 0.95 0.97 1.03 0.95
20 day aheadBenchmark 1.00 1.00 0.98 1.00VIX 1.00 1.03 0.98 1.00EPU 1.21 1.00 1.10 1.00ADS 1.04 1.00 1.03 1.01VIX, EPU, ADS 0.98 0.99 1.06 1.00
MCS Coverage 99% 95% 90%
156 CHAPTER 3.
Table A.8: Out-of-sample evaluation for individual stocksThis table contains the percentage of times each model is included in the MCS at a 10% levelusing the QLIKE as the loss function, when looking at the 20 individual stocks.
Specification of volatility dynamicsEGARCH EGARCH REGARCH REGARCH
(additive) (multiplicative) (additive) (multiplicative)
1 day aheadBenchmark 0.00% 0.00% 10.00% 5.00%VIX 0.00% 5.00% 15.00% 75.00%EPU 0.00% 0.00% 10.00% 10.00%ADS 0.00% 0.00% 5.00% 5.00%VIX, EPU, ADS 5.00% 0.00% 25.00% 85.00%
5 day aheadBenchmark 5.00% 5.00% 75.00% 65.00%VIX 15.00% 30.00% 70.00% 85.00%EPU 10.00% 10.00% 75.00% 60.00%ADS 35.00% 15.00% 75.00% 70.00%VIX, EPU, ADS 30.00% 35.00% 80.00% 90.00%
10 day aheadBenchmark 65.00% 55.00% 75.00% 75.00%VIX 50.00% 70.00% 75.00% 85.00%EPU 50.00% 65.00% 75.00% 80.00%ADS 65.00% 60.00% 80.00% 75.00%VIX, EPU, ADS 75.00% 70.00% 80.00% 90.00%
15 day aheadBenchmark 55.00% 55.00% 70.00% 75.00%VIX 50.00% 60.00% 75.00% 80.00%EPU 50.00% 60.00% 65.00% 65.00%ADS 65.00% 50.00% 80.00% 75.00%VIX, EPU, ADS 75.00% 60.00% 70.00% 85.00%
20 day aheadBenchmark 60.00% 60.00% 75.00% 70.00%VIX 65.00% 70.00% 70.00% 80.00%EPU 55.00% 70.00% 65.00% 70.00%ADS 70.00% 60.00% 80.00% 80.00%VIX, EPU, ADS 85.00% 70.00% 75.00% 80.00%
DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY
SCHOOL OF BUSINESS AND SOCIAL SCIENCES www.econ.au.dk
PhD dissertations since 1 July 2011 2011-4 Anders Bredahl Kock: Forecasting and Oracle Efficient Econometrics 2011-5 Christian Bach: The Game of Risk 2011-6 Stefan Holst Bache: Quantile Regression: Three Econometric Studies 2011:12 Bisheng Du: Essays on Advance Demand Information, Prioritization and Real Options
in Inventory Management 2011:13 Christian Gormsen Schmidt: Exploring the Barriers to Globalization 2011:16 Dewi Fitriasari: Analyses of Social and Environmental Reporting as a Practice of
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Macroeconomic Forecasting 2012-2 Karina Hjortshøj Kjeldsen: Routing and Scheduling in Liner Shipping 2012-3 Soheil Abginehchi: Essays on Inventory Control in Presence of Multiple Sourcing 2012-4 Zhenjiang Qin: Essays on Heterogeneous Beliefs, Public Information, and Asset
Pricing 2012-5 Lasse Frisgaard Gunnersen: Income Redistribution Policies 2012-6 Miriam Wüst: Essays on early investments in child health 2012-7 Yukai Yang: Modelling Nonlinear Vector Economic Time Series 2012-8 Lene Kjærsgaard: Empirical Essays of Active Labor Market Policy on Employment 2012-9 Henrik Nørholm: Structured Retail Products and Return Predictability 2012-10 Signe Frederiksen: Empirical Essays on Placements in Outside Home Care 2012-11 Mateusz P. Dziubinski: Essays on Financial Econometrics and Derivatives Pricing
2012-12 Jens Riis Andersen: Option Games under Incomplete Information 2012-13 Margit Malmmose: The Role of Management Accounting in New Public Management Reforms: Implications in a Socio-Political Health Care Context 2012-14 Laurent Callot: Large Panels and High-dimensional VAR 2012-15 Christian Rix-Nielsen: Strategic Investment 2013-1 Kenneth Lykke Sørensen: Essays on Wage Determination 2013-2 Tue Rauff Lind Christensen: Network Design Problems with Piecewise Linear Cost
Functions
2013-3 Dominyka Sakalauskaite: A Challenge for Experts: Auditors, Forensic Specialists and the Detection of Fraud 2013-4 Rune Bysted: Essays on Innovative Work Behavior 2013-5 Mikkel Nørlem Hermansen: Longer Human Lifespan and the Retirement Decision 2013-6 Jannie H.G. Kristoffersen: Empirical Essays on Economics of Education 2013-7 Mark Strøm Kristoffersen: Essays on Economic Policies over the Business Cycle 2013-8 Philipp Meinen: Essays on Firms in International Trade 2013-9 Cédric Gorinas: Essays on Marginalization and Integration of Immigrants and Young Criminals – A Labour Economics Perspective 2013-10 Ina Charlotte Jäkel: Product Quality, Trade Policy, and Voter Preferences: Essays on
International Trade 2013-11 Anna Gerstrøm: World Disruption - How Bankers Reconstruct the Financial Crisis: Essays on Interpretation 2013-12 Paola Andrea Barrientos Quiroga: Essays on Development Economics 2013-13 Peter Bodnar: Essays on Warehouse Operations 2013-14 Rune Vammen Lesner: Essays on Determinants of Inequality 2013-15 Peter Arendorf Bache: Firms and International Trade 2013-16 Anders Laugesen: On Complementarities, Heterogeneous Firms, and International Trade
2013-17 Anders Bruun Jonassen: Regression Discontinuity Analyses of the Disincentive Effects of Increasing Social Assistance 2014-1 David Sloth Pedersen: A Journey into the Dark Arts of Quantitative Finance 2014-2 Martin Schultz-Nielsen: Optimal Corporate Investments and Capital Structure 2014-3 Lukas Bach: Routing and Scheduling Problems - Optimization using Exact and Heuristic Methods 2014-4 Tanja Groth: Regulatory impacts in relation to a renewable fuel CHP technology:
A financial and socioeconomic analysis 2014-5 Niels Strange Hansen: Forecasting Based on Unobserved Variables 2014-6 Ritwik Banerjee: Economics of Misbehavior 2014-7 Christina Annette Gravert: Giving and Taking – Essays in Experimental Economics 2014-8 Astrid Hanghøj: Papers in purchasing and supply management: A capability-based perspective 2014-9 Nima Nonejad: Essays in Applied Bayesian Particle and Markov Chain Monte Carlo Techniques in Time Series Econometrics 2014-10 Tine L. Mundbjerg Eriksen: Essays on Bullying: an Economist’s Perspective 2014-11 Sashka Dimova: Essays on Job Search Assistance 2014-12 Rasmus Tangsgaard Varneskov: Econometric Analysis of Volatility in Financial Additive Noise Models 2015-1 Anne Floor Brix: Estimation of Continuous Time Models Driven by Lévy Processes 2015-2 Kasper Vinther Olesen: Realizing Conditional Distributions and Coherence Across Financial Asset Classes 2015-3 Manuel Sebastian Lukas: Estimation and Model Specification for Econometric Forecasting 2015-4 Sofie Theilade Nyland Brodersen: Essays on Job Search Assistance and Labor Market Outcomes 2015-5 Jesper Nydam Wulff: Empirical Research in Foreign Market Entry Mode
2015-6 Sanni Nørgaard Breining: The Sibling Relationship Dynamics and Spillovers 2015-7 Marie Herly: Empirical Studies of Earnings Quality 2015-8 Stine Ludvig Bech: The Relationship between Caseworkers and Unemployed Workers 2015-9 Kaleb Girma Abreha: Empirical Essays on Heterogeneous Firms and International Trade 2015-10 Jeanne Andersen: Modelling and Optimisation of Renewable Energy Systems 2015-11 Rasmus Landersø: Essays in the Economics of Crime 2015-12 Juan Carlos Parra-Alvarez: Solution Methods and Inference in Continuous-Time Dynamic Equilibrium Economies (with Applications in Asset Pricing and Income
Fluctuation Models) 2015-13 Sakshi Girdhar: The Internationalization of Big Accounting Firms and the
Implications on their Practices and Structures: An Institutional Analysis 2015-14 Wenjing Wang: Corporate Innovation, R&D Personnel and External Knowledge
Utilization 2015-15 Lene Gilje Justesen: Empirical Banking 2015-16 Jonas Maibom: Structural and Empirical Analysis of the Labour Market 2015-17 Sylvanus Kwaku Afesorgbor: Essays on International Economics and Development 2015-18 Orimar Sauri: Lévy Semistationary Models with Applications in Energy Markets 2015-19 Kristine Vasiljeva: Essays on Immigration in a Generous Welfare State 2015-20 Jonas Nygaard Eriksen: Business Cycles and Expected Returns 2015-21 Simon Juul Hviid: Dynamic Models of the Housing Market 2016-1 Silvia Migali: Essays on International Migration: Institutions, Skill Recognition, and the Welfare State 2016-2 Lorenzo Boldrini: Essays on Forecasting with Linear State-Space Systems 2016-3 Palle Sørensen: Financial Frictions, Price Rigidities, and the Business Cycle 2016-4 Camilla Pisani: Volatility and Correlation in Financial Markets: Theoretical Developments and Numerical Analysis
2016-5 Anders Kronborg: Methods and Applications to DSGE Models 2016-6 Morten Visby Krægpøth: Empirical Studies in Economics of Education 2016-7 Anne Odile Peschel: Essays on Implicit Information Processing at the Point of Sale: Evidence from Experiments and Scanner Data Analysis 2016-8 Girum Dagnachew Abate: Essays in Spatial Econometrics 2016-9 Kai Rehwald: Essays in Public Policy Evaluation 2016-10 Reza Pourmoayed: Optimization Methods in a Stochastic Production Environment 2016-11 Sune Lauth Gadegaard: Discrete Location Problems – Theory, Algorithms, and Extensions to Multiple Objectives 2016-12 Lisbeth Palmhøj Nielsen: Empirical Essays on Child Achievement, Maternal Employment, Parental Leave, and Geographic Mobility 2016-13 Louise Voldby Beuchert-Pedersen: School Resources and Student Achievement: Evidence From Social and Natural Experiments 2016-14 Mette Trier Damgaard: Essays in Applied Behavioral Economics 2016-15 Andrea Barletta: Consistent Modeling and Efficient Pricing of Volatility Derivatives 2016-16 Thorvardur Tjörvi Ólafsson: Macrofinancial Linkages and Crises in Small Open Economies 2016-17 Carlos Vladimir Rodríguez Caballero: On Factor Analysis with Long-Range Dependence 2016-18 J. Eduardo Vera-Valdés: Essays in Long Memory 2016-19 Magnus Sander: Returns, Dividends, and Optimal Portfolios 2016-20 Ioana Daniela Neamtu: Wind Power Effects and Price Elasticity of Demand for the Nordic Electricity Markets 2016-21 Anne Brink Nandrup: Determinants of Student Achievement and Education Choice 2016-22 Jakob Guldbæk Mikkelsen: Time-Varying Loadings in Factor Models: Theory and Applications 2016-23 Dan Nguyen: Formidability and Human Behavior: An Interdisciplinary Approach
2016-24 Martin Petri Bagger: Attention and Decision-Making: Separating Top-Down from Bottom-Up Components
2016-25 Samira Mirzaei: Optimization Algorithms for Multi-Commodity Routing and Inventory Routing Problems 2017-1 Viktoryia Buhayenko: Determining Dynamic Discounts for Supply Chain Coordination 2017-2 Mikkel Bennedsen: Rough Continuous-Time Processes: Theory and Applications 2017-3 Max Weiss Dohrn: Business Cycle Costs for Finitely Living Individuals 2017-4 Oskar Knapik: Essays on Econometric Modelling and Forecasting of Electricity Prices 2017-5 Jesper Bo Pedersen: Essays on Financial Risk Management and Asset Allocation 2017-6 Bo Laursen: Econometric Analysis of Time-Varying Volatility in Financial Markets 2017-7 Federico Carlini: Essays on Fractional Filters and Co-Integration 2017-8 Jonas Juul Henriksen: International Trade and the Labour Market 2017-9 Christian Ellermann-Aarslev: History Dependent Unemployment Insurance 2017-10 Martin Alfaro: Essays on International Trade and Strategic Behavior 2017-11 Bastien Michel: Essays on the Economics of Crime and Development Economics 2017-12 Morten Holm Jacobsen Fenger: Essays on the Dynamics of Consumer Behavior 2017-13 Carsten P.T. Rosenskjold: Econometric Modelling of Mortality and its
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ISBN: 9788793195684