Handbook of Financial Time SeriesEditors Jens-Peter Krei Thomas MikoschHandbook of Financial Time SeriesTorben G. Andersen Richard A. Davis 2009 Springer-Verlag Berlin Heidelbergconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, 1965, in its current version, and permissions for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.Printed on acid-free paperspringer.comThe use of general descriptive names, registered names, trademarks, etc. in this publication does not or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 9 8 7 6 5 4 3 2 1This work is subject to copyright. All rights are reserved, whether the whole or part of the material is Cover design: WMXDesign GmbH, HeidelbergISBN 978-3-540-71296-1 reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication e-ISBN 978-3-540-71297-8Library of Congress Control Number: 2008943984Department of Finance2001 Sheridan RoadDepartment of StatisticsGermany38106 Braunschweig Kellogg School of ManagementUniversitetsparken 5Thomas MikoschDenmarkmikosch@math.ku.dk j.kreiss@tu-bs.deTorben G. AndersenUniversity of Copenhagen Northwestern University2100 CopenhagenEvanston, IL 60208t-andersen@kellogg.northwestern.eduU.S.A.Richard A. DavisColumbia University1255 Amsterdam AvenueNew York, NY 10027U.S.A.rdavis@stat.columbia.eduInstitut fr Mathematische StochastikTechnische Universitt BraunschweigPockelsstrasse 14Jens-Peter KreiDepartment MathematicsForewordThe Handbook of Financial Time Series, edited by Andersen, Davis, Kreissand Mikosch, is an impressive collection of survey articles by many of theleading contributors to the eld. These articles are mostly very clearly writ-ten and present a sweep of the literature in a coherent pedagogical manner.The level of most of the contributions is mathematically sophisticated, andI imagine many of these chapters will nd their way onto graduate readinglists in courses in nancial economics and nancial econometrics. In readingthrough these papers, I found many new insights and presentations even inareas that I know well.The book is divided into ve broad sections: GARCH-Modeling, Stochas-tic Volatility Modeling, Continuous Time Processes, Cointegration and UnitRoots, and Special Topics. These correspond generally to classes of stochas-tic processes that are applied in various nance contexts. However, there areother themes that cut across these classes. There are several papers that care-fully articulate the probabilistic structure of these classes, while others aremore focused on estimation. Still others derive properties of extremes for eachclass of processes, and evaluate persistence and the extent of long memory.Papers in many cases examine the stability of the process with tools to checkfor breaks and jumps. Finally there are applications to options, term struc-ture, credit derivatives, risk management, microstructure models and otherforecasting settings.The GARCH family of models is nicely introduced by Tersvirta and thenthe mathematical underpinning is elegantly and readably presented by Lind-ner with theorems on stationarity, ergodicity and tail behavior. In the samevein, Giraitis, Leipus and Surgailis examine the long memory properties ofinnite order ARCH models with memory decay slower than GARCH, andDavis and Mikosch derive tail properties of GARCH models showing thatthey satisfy a power law and are in the maximum domain of attraction ofthe Frchet distribution. The multivariate GARCH family is well surveyedby Silvennoinen and Tersvirta. Linton and ek and Spokoiny, respectively,specify models which are non- or semi-parametric or which are only constantover intervals of homogeneity.vi ForewordThe section on Stochastic Volatility Modelling (SV) brings us up to date onthe development of alternatives to GARCH style models. Davis and Mikoschin two chapters develop the somewhat easier underlying mathematical the-ory and tail properties of SV. They derive an important dierence fromGARCH models. While both stochastic volatility and GARCH processes ex-hibit volatility clustering, only the GARCH has clustering of extremes. Longmemory is conveniently described by SV models in Hurvich and Soulier. Chib,Omori and Asai extend these analyses to multivariate systems although theydo not envision very high dimensions. Estimation is covered in several chap-ters by Renault, Shephard and Andersen, and Jungbacker and Koopman.The continuous time analysis begins with familiar Brownian motion pro-cesses and enhances them with jumps, dynamics, time deformation, correla-tion with returns and Lvy process innovations. Extreme value distributionsare developed and estimation algorithms for discretely sampled processes areanalyzed. Lindner discusses the idea of continuous time approximations toGARCH and SV models showing that the nature of the approximation mustbe carefully specied. The continuous time framework is then applied to sev-eral nance settings such as interest rate models by Bjrk, option pricing byKallsen, and realized volatility by Andersen and Benzoni. The book then re-turns to analysis of rst moments with surveys of discrete time models withunit roots, near unit roots, fractional unit roots and cointegration.Finally, a remaining 13 chapters are collected in a section called SpecialTopics. These include very interesting chapters on copulas, non-parametricmodels, resampling methods, Markov switching models, structural breakmodels and model selection. Patton and Sheppard examine univariate andmultivariate volatility forecast comparisons. They show the advantages of aGLS correction, discuss multiple comparisons and economic loss functions.Bauwens and Hautsch survey a wide range of models for point processes thathave been used in the nance literature to model arrival times of trades andquotes. The survey is well grounded in the statistical literature and the eco-nomics literature. Embrechts, Furrer and Kaufmann discuss dierent typesof risk market, credit, operational and insuranceand some of the leadingapproaches to estimation. Christoersen applies the ltered historical simu-lation or FHS method to univariate and multivariate simulation based calcu-lation of VaR, Expected Shortfall and active portfolio risks. Lando surveysthe structural and reduced form approaches to modeling credit spreads. Hefocuses on CDS spreads and default dependence and gives a nice descriptionof tests between contagion and factor structures in formulating dependence.So make yourself a double cappuccino and relax in a comfortable chair, oradjust your headphones at 30,000 ft. over the Pacic, and dig in. There aretreats in lots of dierent areas just waiting to be discovered.New York, September 2008 Robert EngleContentsForeword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Torben G. Andersen, Richard A. Davis, Jens-Peter Kreiss and ThomasMikoschReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Part I Recent Developments in GARCH ModelingAn Introduction to Univariate GARCH Models . . . . . . . . . . . . . 17Timo Tersvirta1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 The ARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 The Generalized ARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . 193.1 Why Generalized ARCH? . . . . . . . . . . . . . . . . . . . . . . 193.2 Families of univariate GARCH models . . . . . . . . . . . 203.3 Nonlinear GARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 Time-varying GARCH . . . . . . . . . . . . . . . . . . . . . . . . . 263.5 Markov-switching ARCH and GARCH. . . . . . . . . . . 273.6 Integrated and fractionally integrated GARCH . . . 283.7 Semi- and nonparametric ARCH models . . . . . . . . . 303.8 GARCH-in-mean model . . . . . . . . . . . . . . . . . . . . . . . 303.9 Stylized facts and the rst-order GARCH model . . 314 Family of Exponential GARCH Models . . . . . . . . . . . . . . . . . 344.1 Denition and properties . . . . . . . . . . . . . . . . . . . . . . . 344.2 Stylized facts and the rst-order EGARCH model . 354.3 Stochastic volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 Comparing EGARCH with GARCH . . . . . . . . . . . . . . . . . . . . 376 Final Remarks and Further Reading . . . . . . . . . . . . . . . . . . . . 38References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Stationarity, Mixing, Distributional Properties and Momentsof GARCH(p, q)Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Alexander M. Lindner1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43viii Contents2 Stationary Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.1 Strict stationarity of ARCH(1) and GARCH(1, 1) . 452.2 Strict stationarity of GARCH(p, q) . . . . . . . . . . . . . . 492.3 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.4 Weak stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 The ARCH() Representation and the ConditionalVariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 Existence of Moments and the Autocovariance Function ofthe Squared Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1 Moments of ARCH(1) and GARCH(1, 1) . . . . . . . . . 564.2 Moments of GARCH(p, q) . . . . . . . . . . . . . . . . . . . . . . 574.3 The autocorrelation function of the squares . . . . . . 605 Strong Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 Some Distributional Properties . . . . . . . . . . . . . . . . . . . . . . . . . 647 Models Dened on the Non-Negative Integers . . . . . . . . . . . . 668 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67ARCH() Models and Long Memory Properties . . . . . . . . . . . . 71Liudas Giraitis, Remigijus Leipus and Donatas Surgailis1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712 Stationary ARCH() Process . . . . . . . . . . . . . . . . . . . . . . . . . 732.1 Volterra representations. . . . . . . . . . . . . . . . . . . . . . . . 732.2 Dependence structure, association, and centrallimit theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.3 Innite variance and integrated ARCH() . . . . . . . 773 Linear ARCH and Bilinear Model . . . . . . . . . . . . . . . . . . . . . . 79References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82A Tour in the Asymptotic Theory of GARCH Estimation . . . 85Christian Francq and Jean-Michel Zakoan1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852 LeastSquares Estimation of ARCH Models . . . . . . . . . . . . . 873 QuasiMaximum Likelihood Estimation . . . . . . . . . . . . . . . . . 893.1 Pure GARCH models . . . . . . . . . . . . . . . . . . . . . . . . . . 903.2 ARMAGARCH models . . . . . . . . . . . . . . . . . . . . . . . 944 Ecient Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955 Alternative Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.1 Selfweighted LSE for the ARMA parameters . . . . 1005.2 Selfweighted QMLE . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.3 Lpestimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.4 Least absolute deviations estimators . . . . . . . . . . . . . 1025.5 Whittle estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.6 Moment estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046 Properties of Estimators when some GARCH Coecientsare Equal to Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Contents ix6.1 Fitting an ARCH(1) model to a white noise . . . . . . 1056.2 On the need of additional assumptions . . . . . . . . . . . 1066.3 Asymptotic distribution of the QMLE on theboundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.4 Application to hypothesis testing . . . . . . . . . . . . . . . 1077 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Practical Issues in the Analysis of Univariate GARCH Models 113Eric Zivot1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1132 Some Stylized Facts of Asset Returns . . . . . . . . . . . . . . . . . . . 1143 The ARCH and GARCH Model . . . . . . . . . . . . . . . . . . . . . . . . 1153.1 Conditional mean specication. . . . . . . . . . . . . . . . . . 1183.2 Explanatory variables in the conditional varianceequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193.3 The GARCH model and stylized facts of assetreturns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193.4 Temporal aggregation . . . . . . . . . . . . . . . . . . . . . . . . . 1214 Testing for ARCH/GARCH Eects . . . . . . . . . . . . . . . . . . . . . 1214.1 Testing for ARCH eects in daily and monthlyreturns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1225 Estimation of GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . 1235.1 Numerical accuracy of GARCH estimates . . . . . . . . 1255.2 Quasi-maximum likelihood estimation . . . . . . . . . . . 1265.3 Model selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.4 Evaluation of estimated GARCH models . . . . . . . . . 1275.5 Estimation of GARCH models for daily andmonthly returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276 GARCH Model Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.1 Asymmetric leverage eects and news impact . . . . . 1316.2 Non-Gaussian error distributions . . . . . . . . . . . . . . . . 1357 Long Memory GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . 1377.1 Testing for long memory . . . . . . . . . . . . . . . . . . . . . . . 1397.2 Two component GARCH model . . . . . . . . . . . . . . . . . 1397.3 Integrated GARCH model . . . . . . . . . . . . . . . . . . . . . 1407.4 Long memory GARCH models for daily returns . . . 1418 GARCH Model Prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1428.1 GARCH and forecasts for the conditional mean . . . 1428.2 Forecasts from the GARCH(1,1) model . . . . . . . . . . 1438.3 Forecasts from asymmetric GARCH(1,1) models . . 1448.4 Simulation-based forecasts . . . . . . . . . . . . . . . . . . . . . 1458.5 Forecasting the volatility of multiperiod returns . . . 1458.6 Evaluating volatility predictions . . . . . . . . . . . . . . . . 146x Contents8.7 Forecasting the volatility of Microsoft and theS&P 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1509 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Semiparametric and Nonparametric ARCH Modeling . . . . . . . 157Oliver B. Linton1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1572 The GARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1573 The Nonparametric Approach. . . . . . . . . . . . . . . . . . . . . . . . . . 1583.1 Error density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1583.2 Functional form of volatility function . . . . . . . . . . . . 1593.3 Relationship between mean and variance . . . . . . . . . 1623.4 Long memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1633.5 Locally stationary processes . . . . . . . . . . . . . . . . . . . . 1643.6 Continuous time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1644 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165Varying Coecient GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . 169Pavel ek and Vladimir Spokoiny1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1692 Conditional Heteroscedasticity Models . . . . . . . . . . . . . . . . . . 1712.1 Model estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1732.2 Test of homogeneity against a changepointalternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1733 Adaptive Nonparametric Estimation . . . . . . . . . . . . . . . . . . . . 1753.1 Adaptive choice of the interval of homogeneity . . . . 1763.2 Parameters of the method and the implementationdetails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1764 RealData Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1794.1 Finitesample critical values for the test ofhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1794.2 Stock index S&P 500 . . . . . . . . . . . . . . . . . . . . . . . . . . 1805 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183Extreme Value Theory for GARCH Processes . . . . . . . . . . . . . . . 187Richard A. Davis and Thomas Mikosch1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1872 Strict Stationarity and Mixing Properties . . . . . . . . . . . . . . . 1883 Embedding a GARCH Process in a Stochastic RecurrenceEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1894 The Tails of a GARCH Process . . . . . . . . . . . . . . . . . . . . . . . . 1905 Limit Theory for Extremes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1945.1 Convergence of maxima . . . . . . . . . . . . . . . . . . . . . . . . 194Contents xi5.2 Convergence of point processes . . . . . . . . . . . . . . . . . 1955.3 The behavior of the sample autocovariance function 197References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199Multivariate GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201Annastiina Silvennoinen and Timo Tersvirta1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2012 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2032.1 Models of the conditional covariance matrix . . . . . . 2042.2 Factor models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2072.3 Models of conditional variances and correlations . . 2102.4 Nonparametric and semiparametric approaches . . . 2153 Statistical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2184 Hypothesis Testing in Multivariate GARCH Models . . . . . . 2184.1 General misspecication tests . . . . . . . . . . . . . . . . . . . 2194.2 Tests for extensions of the CCCGARCH model . . 2215 An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2226 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226Part II Recent Developments in Stochastic Volatility ModelingStochastic Volatility: Origins and Overview . . . . . . . . . . . . . . . . . . 233Neil Shephard and Torben G. Andersen1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2332 The Origin of SV Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2353 Second Generation Model Building . . . . . . . . . . . . . . . . . . . . . 2403.1 Univariate models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2403.2 Multivariate models . . . . . . . . . . . . . . . . . . . . . . . . . . . 2414 Inference Based on Return Data . . . . . . . . . . . . . . . . . . . . . . . 2424.1 Momentbased inference . . . . . . . . . . . . . . . . . . . . . . . 2424.2 Simulationbased inference . . . . . . . . . . . . . . . . . . . . . 2435 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2465.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2466 Realized Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250Probabilistic Properties of Stochastic Volatility Models . . . . . . 255Richard A. Davis and Thomas Mikosch1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2552 Stationarity, Ergodicity and Strong Mixing . . . . . . . . . . . . . . 2562.1 Strict stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2562.2 Ergodicity and strong mixing . . . . . . . . . . . . . . . . . . . 2573 The Covariance Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2584 Moments and Tails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2615 Asymptotic Theory for the Sample ACVF and ACF . . . . . . 263xii ContentsReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266MomentBased Estimation of Stochastic Volatility Models . . . 269Eric Renault1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2702 The Use of a Regression Model to Analyze Fluctuations inVariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2722.1 The linear regression model for conditional variance 2722.2 The SRSARV(p) model . . . . . . . . . . . . . . . . . . . . . . . 2742.3 The Exponential SARV model . . . . . . . . . . . . . . . . . . 2772.4 Other parametric SARV models. . . . . . . . . . . . . . . . . 2793 Implications of SV Model Specication for Higher OrderMoments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2813.1 Fat tails and variance of the variance . . . . . . . . . . . . 2813.2 Skewness, feedback and leverage eects . . . . . . . . . . 2844 Continuous Time Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2864.1 Measuring volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . 2874.2 Moment-based estimation with realized volatility . . 2884.3 Reduced form models of volatility . . . . . . . . . . . . . . . 2924.4 High frequency data with random times separatingsuccessive observations . . . . . . . . . . . . . . . . . . . . . . . . 2935 SimulationBased Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 2955.1 Simulation-based bias correction . . . . . . . . . . . . . . . . 2965.2 Simulation-based indirect inference . . . . . . . . . . . . . . 2985.3 Simulated method of moments . . . . . . . . . . . . . . . . . . 3005.4 Indirect inference in presence of misspecication . . 3046 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307Parameter Estimation and Practical Aspects of ModelingStochastic Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313Borus Jungbacker and Siem Jan Koopman1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3132 A Quasi-Likelihood Analysis Based on Kalman FilterMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3162.1 Kalman lter for prediction and likelihoodevaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3192.2 Smoothing methods for the conditional mean,variance and mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3202.3 Practical considerations for analyzing thelinearized SV model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3213 A Monte Carlo Likelihood Analysis . . . . . . . . . . . . . . . . . . . . . 3223.1 Construction of a proposal density . . . . . . . . . . . . . . 3233.2 Sampling from the importance density and MonteCarlo likelihood. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3254 Some Generalizations of SV Models . . . . . . . . . . . . . . . . . . . . . 327Contents xiii4.1 Basic SV model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3274.2 Multiple volatility factors . . . . . . . . . . . . . . . . . . . . . . 3284.3 Regression and xed eects . . . . . . . . . . . . . . . . . . . . 3294.4 Heavy-tailed innovations . . . . . . . . . . . . . . . . . . . . . . . 3304.5 Additive noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3314.6 Leverage eects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3314.7 Stochastic volatility in mean . . . . . . . . . . . . . . . . . . . 3335 Empirical Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3335.1 Standard & Poors 500 stock index: volatilityestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3345.2 Standard & Poors 500 stock index: regressioneects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3355.3 Daily changes in exchange rates: dollarpound anddollaryen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3376 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342Stochastic Volatility Models with Long Memory . . . . . . . . . . . . . 345Cliord M. Hurvich and Philippe Soulier1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3452 Basic Properties of the LMSV Model . . . . . . . . . . . . . . . . . . . 3463 Parametric Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3474 Semiparametric Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3495 Generalizations of the LMSV Model . . . . . . . . . . . . . . . . . . . . 3526 Applications of the LMSV Model . . . . . . . . . . . . . . . . . . . . . . . 352References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353Extremes of Stochastic Volatility Models . . . . . . . . . . . . . . . . . . . . 355Richard A. Davis and Thomas Mikosch1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3552 The Tail Behavior of the Marginal Distribution . . . . . . . . . . 3562.1 The light-tailed case . . . . . . . . . . . . . . . . . . . . . . . . . . . 3562.2 The heavy-tailed case . . . . . . . . . . . . . . . . . . . . . . . . . . 3573 Point Process Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3583.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3583.2 Application to stochastic volatility models . . . . . . . 360References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364Multivariate Stochastic Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365Siddhartha Chib, Yasuhiro Omori and Manabu Asai1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3662 Basic MSV Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3692.1 No-leverage model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3692.2 Leverage eects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3732.3 Heavy-tailed measurement error models . . . . . . . . . . 377xiv Contents3 Factor MSV Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3793.1 Volatility factor model . . . . . . . . . . . . . . . . . . . . . . . . . 3793.2 Mean factor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3823.3 Bayesian analysis of mean factor MSV model . . . . . 3844 Dynamic Correlation MSV Model . . . . . . . . . . . . . . . . . . . . . . 3884.1 Modeling by reparameterization . . . . . . . . . . . . . . . . 3884.2 Matrix exponential transformation . . . . . . . . . . . . . . 3904.3 Wishart process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3915 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397Part III Topics in Continuous Time ProcessesAn Overview of AssetPrice Models . . . . . . . . . . . . . . . . . . . . . . . . . 403Peter J. Brockwell1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4042 Shortcomings of the BSM Model . . . . . . . . . . . . . . . . . . . . . . . 4093 A General Framework for Option Pricing . . . . . . . . . . . . . . . . 4104 Some Non-Gaussian Models for Asset Prices . . . . . . . . . . . . . 4115 Further Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416OrnsteinUhlenbeck Processes and Extensions . . . . . . . . . . . . . . . 421Ross A. Maller, Gernot Mller and Alex Szimayer1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4222 OU Process Driven by Brownian Motion . . . . . . . . . . . . . . . . 4223 Generalised OU Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4243.1 Background on bivariate Lvy processes . . . . . . . . . 4243.2 Lvy OU processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4263.3 Self-decomposability, self-similarity, class L,Lamperti transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 4294 Discretisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4304.1 Autoregressive representation, and perpetuities . . . 4304.2 Statistical issues: Estimation and hypothesis testing 4314.3 Discretely sampled process . . . . . . . . . . . . . . . . . . . . . 4314.4 Approximating the COGARCH . . . . . . . . . . . . . . . . . 4325 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435JumpType Lvy Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439Ernst Eberlein1 Probabilistic Structure of Lvy Processes . . . . . . . . . . . . . . . . 4392 Distributional Description of Lvy Processes . . . . . . . . . . . . . 4433 Financial Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4464 Examples of Lvy Processes with Jumps . . . . . . . . . . . . . . . . 4494.1 Poisson and compound Poisson processes . . . . . . . . 449Contents xv4.2 Lvy jump diusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 4504.3 Hyperbolic Lvy processes . . . . . . . . . . . . . . . . . . . . . 4504.4 Generalized hyperbolic Lvy processes . . . . . . . . . . . 4514.5 CGMY and variance gamma Lvy processes . . . . . . 4524.6 -Stable Lvy processes . . . . . . . . . . . . . . . . . . . . . . . . 4534.7 Meixner Lvy processes . . . . . . . . . . . . . . . . . . . . . . . . 453References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454LvyDriven ContinuousTime ARMA Processes . . . . . . . . . . . . 457Peter J. Brockwell1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4582 SecondOrder LvyDriven CARMA Processes . . . . . . . . . . 4603 Connections with DiscreteTime ARMA Processes . . . . . . . 4704 An Application to Stochastic Volatility Modelling . . . . . . . . 4745 ContinuousTime GARCH Processes . . . . . . . . . . . . . . . . . . . 4766 Inference for CARMA Processes . . . . . . . . . . . . . . . . . . . . . . . . 478References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479Continuous Time Approximations to GARCH and StochasticVolatility Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481Alexander M. Lindner1 Stochastic Volatility Models and Discrete GARCH . . . . . . . 4812 Continuous Time GARCH Approximations . . . . . . . . . . . . . . 4822.1 Preserving the random recurrence equation property 4832.2 The diusion limit of Nelson . . . . . . . . . . . . . . . . . . . 4842.3 The COGARCH model . . . . . . . . . . . . . . . . . . . . . . . . 4862.4 Weak GARCH processes . . . . . . . . . . . . . . . . . . . . . . . 4882.5 Stochastic delay equations . . . . . . . . . . . . . . . . . . . . . 4892.6 A continuous time GARCH model designed foroption pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4903 Continuous Time Stochastic Volatility Approximations . . . . 4913.1 Sampling a continuous time SV model atequidistant times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.2 Approximating a continuous time SV model . . . . . . 493References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495Maximum Likelihood and Gaussian Estimation of ContinuousTime Models in Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497Peter C. B. Phillips and Jun Yu1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4982 Exact ML Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4992.1 ML based on the transition density . . . . . . . . . . . . . . 4992.2 ML based on the continuous record likelihood . . . . 5023 Approximate ML Methods Based on Transition Densities . . 5033.1 The Euler approximation and renements . . . . . . . . 5043.2 Closedform approximations . . . . . . . . . . . . . . . . . . . . 509xvi Contents3.3 Simulated inll ML methods . . . . . . . . . . . . . . . . . . . 5123.4 Other approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5144 Approximate ML Methods Based on the ContinuousRecord Likelihood and Realized Volatility . . . . . . . . . . . . . . . 5165 Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5196 Estimation Bias Reduction Techniques . . . . . . . . . . . . . . . . . . 5206.1 Jackknife estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . 5216.2 Indirect inference estimation . . . . . . . . . . . . . . . . . . . 5227 Multivariate Continuous Time Models . . . . . . . . . . . . . . . . . . 5248 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527Parametric Inference for Discretely Sampled StochasticDierential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531Michael Srensen1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5312 Asymptotics: Fixed Frequency . . . . . . . . . . . . . . . . . . . . . . . . . 5323 Likelihood Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5364 Martingale Estimating Functions . . . . . . . . . . . . . . . . . . . . . . . 5385 Explicit Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5436 High Frequency Asymptotics and Ecient Estimation . . . . 548References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551Realized Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555Torben G. Andersen and Luca Benzoni1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5562 Measuring Mean Return versus Return Volatility . . . . . . . . . 5573 Quadratic Return Variation and Realized Volatility . . . . . . . 5594 Conditional Return Variance and Realized Volatility . . . . . . 5615 Jumps and Bipower Variation . . . . . . . . . . . . . . . . . . . . . . . . . . 5636 Ecient Sampling versus Microstructure Noise . . . . . . . . . . . 5647 Empirical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5667.1 Early work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5667.2 Volatility forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . 5677.3 The distributional implications of the no-arbitragecondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5687.4 Multivariate quadratic variation measures . . . . . . . . 5687.5 Realized volatility, model specication andestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5698 Possible Directions for Future Research . . . . . . . . . . . . . . . . . 569References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570Contents xviiEstimating Volatility in the Presence of MarketMicrostructure Noise: A Review of the Theory and PracticalConsiderations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577Yacine At-Sahalia and Per A. Mykland1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5772 Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5792.1 The parametric volatility case . . . . . . . . . . . . . . . . . . 5792.2 The nonparametric stochastic volatility case . . . . . . 5823 Renements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5853.1 Multi-scale realized volatility . . . . . . . . . . . . . . . . . . . 5853.2 Non-equally spaced observations . . . . . . . . . . . . . . . . 5863.3 Serially-correlated noise . . . . . . . . . . . . . . . . . . . . . . . . 5873.4 Noise correlated with the price signal . . . . . . . . . . . . 5893.5 Small sample edgeworth expansions . . . . . . . . . . . . . 5913.6 Robustness to departures from the data generatingprocess assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5914 Computational and Practical ImplementationConsiderations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5924.1 Calendar, tick and transaction time sampling . . . . . 5924.2 Transactions or quotes . . . . . . . . . . . . . . . . . . . . . . . . . 5924.3 Selecting the number of subsamples in practice . . . 5934.4 High versus low liquidity assets . . . . . . . . . . . . . . . . . 5944.5 Robustness to data cleaning procedures . . . . . . . . . . 5944.6 Smoothing by averaging . . . . . . . . . . . . . . . . . . . . . . . 5955 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596Option Pricing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599Jan Kallsen1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5992 Arbitrage Theory from a Market Perspective. . . . . . . . . . . . . 6003 Martingale Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6034 Arbitrage Theory from an Individual Perspective . . . . . . . . . 6055 Quadratic Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6066 Utility Indierence Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611An Overview of Interest Rate Theory . . . . . . . . . . . . . . . . . . . . . . . 615Tomas Bjrk1 General Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6152 Interest Rates and the Bond Market . . . . . . . . . . . . . . . . . . . . 6183 Factor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6204 Modeling under the Objective Measure P . . . . . . . . . . . . . . . 6214.1 The market price of risk . . . . . . . . . . . . . . . . . . . . . . . 6225 Martingale Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6235.1 Ane term structures . . . . . . . . . . . . . . . . . . . . . . . . . 624xviii Contents5.2 Short rate models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6255.3 Inverting the yield curve . . . . . . . . . . . . . . . . . . . . . . . 6276 Forward Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6296.1 The HJM drift condition . . . . . . . . . . . . . . . . . . . . . . . 6296.2 The Musiela parameterization . . . . . . . . . . . . . . . . . . 6317 Change of Numeraire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6327.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6327.2 Forward measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6357.3 Option pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6358 LIBOR Market Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6388.1 Caps: denition and market practice . . . . . . . . . . . . 6388.2 The LIBOR market model . . . . . . . . . . . . . . . . . . . . . 6408.3 Pricing caps in the LIBOR model . . . . . . . . . . . . . . . 6418.4 Terminal measure dynamics and existence . . . . . . . . 6419 Potentials and Positive Interest . . . . . . . . . . . . . . . . . . . . . . . . 6429.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6429.2 The FlesakerHughston fractional model . . . . . . . . . 6449.3 Connections to the Riesz decomposition . . . . . . . . . 6469.4 Conditional variance potentials . . . . . . . . . . . . . . . . . 6479.5 The Rogers Markov potential approach . . . . . . . . . . 64810 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651Extremes of ContinuousTime Processes. . . . . . . . . . . . . . . . . . . . . 653Vicky Fasen1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6532 Extreme Value Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6542.1 Extremes of discretetime processes . . . . . . . . . . . . . 6552.2 Extremes of continuoustime processes . . . . . . . . . . 6562.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6563 The Generalized Ornstein-Uhlenbeck (GOU)Model . . . . . . 6573.1 The OrnsteinUhlenbeck process . . . . . . . . . . . . . . . . 6583.2 The nonOrnsteinUhlenbeck process . . . . . . . . . . . . 6593.3 Comparison of the models . . . . . . . . . . . . . . . . . . . . . . 6614 Tail Behavior of the Sample Maximum . . . . . . . . . . . . . . . . . . 6615 Running sample Maxima and Extremal Index Function . . . 6636 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665Part IV Topics in Cointegration and Unit RootsCointegration: Overview and Development . . . . . . . . . . . . . . . . . . 671Sren Johansen1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6711.1 Two examples of cointegration . . . . . . . . . . . . . . . . . . 672Contents xix1.2 Three ways of modeling cointegration. . . . . . . . . . . . 6731.3 The model analyzed in this article . . . . . . . . . . . . . . 6742 Integration, Cointegration and Grangers RepresentationTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6752.1 Denition of integration and cointegration . . . . . . . 6752.2 The Granger Representation Theorem . . . . . . . . . . . 6772.3 Interpretation of cointegrating coecients . . . . . . . . 6783 Interpretation of the I(1) Model for Cointegration . . . . . . . . 6803.1 The models H(r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6803.2 Normalization of parameters of the I(1) model . . . . 6813.3 Hypotheses on long-run coecients . . . . . . . . . . . . . . 6813.4 Hypotheses on adjustment coecients . . . . . . . . . . . 6824 Likelihood Analysis of the I(1) Model . . . . . . . . . . . . . . . . . . . 6834.1 Checking the specications of the model . . . . . . . . . 6834.2 Reduced rank regression . . . . . . . . . . . . . . . . . . . . . . . 6834.3 Maximum likelihood estimation in the I(1) modeland derivation of the rank test . . . . . . . . . . . . . . . . . . 6845 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6865.1 Asymptotic distribution of the rank test . . . . . . . . . 6865.2 Asymptotic distribution of the estimators . . . . . . . . 6876 Further Topics in the Area of Cointegration . . . . . . . . . . . . . 6896.1 Rational expectations . . . . . . . . . . . . . . . . . . . . . . . . . 6896.2 The I(2) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6907 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692Time Series with Roots on or Near the Unit Circle . . . . . . . . . . 695Ngai Hang Chan1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6952 Unit Root Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6962.1 First order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6972.2 AR(p) models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6992.3 Model selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7023 Miscellaneous Developments and Conclusion . . . . . . . . . . . . . 704References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705Fractional Cointegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709Willa W. Chen and Cliord M. Hurvich1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7092 Type I and Type II Denitions of I(d) . . . . . . . . . . . . . . . . . . 7102.1 Univariate series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7102.2 Multivariate series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7133 Models for Fractional Cointegration . . . . . . . . . . . . . . . . . . . . 7153.1 Parametric models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7164 Tapering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7175 Semiparametric Estimation of the Cointegrating Vectors . . 718xx Contents6 Testing for Cointegration; Determination of CointegratingRank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724Part V Special Topics RiskDierent Kinds of Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729Paul Embrechts, Hansjrg Furrer and Roger Kaufmann1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7292 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7322.1 Risk measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7322.2 Risk factor mapping and loss portfolios . . . . . . . . . . 7353 Credit Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7363.1 Structural models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7373.2 Reduced form models . . . . . . . . . . . . . . . . . . . . . . . . . . 7373.3 Credit risk for regulatory reporting . . . . . . . . . . . . . . 7384 Market Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7384.1 Market risk models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7394.2 Conditional versus unconditional modeling . . . . . . . 7404.3 Scaling of market risks . . . . . . . . . . . . . . . . . . . . . . . . . 7405 Operational Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7426 Insurance Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7446.1 Life insurance risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7446.2 Modeling parametric life insurance risk . . . . . . . . . . 7456.3 Non-life insurance risk . . . . . . . . . . . . . . . . . . . . . . . . . 7477 Aggregation of Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7488 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750ValueatRisk Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753Peter Christoersen1 Introduction and Stylized Facts . . . . . . . . . . . . . . . . . . . . . . . . 7532 A Univariate Portfolio Risk Model . . . . . . . . . . . . . . . . . . . . . . 7552.1 The dynamic conditional variance model . . . . . . . . . 7562.2 Univariate ltered historical simulation . . . . . . . . . . 7572.3 Univariate extensions and alternatives . . . . . . . . . . . 7593 Multivariate, BaseAsset Return Methods . . . . . . . . . . . . . . . 7603.1 The dynamic conditional correlation model . . . . . . . 7613.2 Multivariate ltered historical simulation. . . . . . . . . 7613.3 Multivariate extensions and alternatives . . . . . . . . . 7634 Summary and Further Issues. . . . . . . . . . . . . . . . . . . . . . . . . . . 764References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764Contents xxiCopulaBased Models for Financial Time Series . . . . . . . . . . . . . 767Andrew J. Patton1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7672 CopulaBased Models for Time Series. . . . . . . . . . . . . . . . . . . 7712.1 Copulabased models for multivariate time series . 7722.2 Copulabased models for univariate time series . . . 7732.3 Estimation and evaluation of copulabased modelsfor time series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7753 Applications of Copulas in Finance and Economics . . . . . . . 7784 Conclusions and Areas for Future Research . . . . . . . . . . . . . . 780References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781Credit Risk Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787David Lando1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7872 Modeling the Probability of Default and Recovery . . . . . . . . 7883 Two Modeling Frameworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7894 Credit Default Swap Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . 7925 Corporate Bond Spreads and Bond Returns . . . . . . . . . . . . . 7956 Credit Risk Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797Part V Special Topics Time Series MethodsEvaluating Volatility and Correlation Forecasts . . . . . . . . . . . . . . 801Andrew J. Patton and Kevin Sheppard1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8011.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8032 Direct Evaluation of Volatility Forecasts . . . . . . . . . . . . . . . . 8042.1 Forecast optimality tests for univariate volatilityforecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8052.2 MZ regressions on transformations of 2t . . . . . . . . . 8062.3 Forecast optimality tests for multivariate volatilityforecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8072.4 Improved MZ regressions using generalised leastsquares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8082.5 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8103 Direct Comparison of Volatility Forecasts . . . . . . . . . . . . . . . 8153.1 Pairwise comparison of volatility forecasts . . . . . . . 8163.2 Comparison of many volatility forecasts . . . . . . . . . . 8173.3 Robust loss functions for forecast comparison . . . . 8183.4 Problems arising from nonrobust loss functions . 8193.5 Choosing a robust loss function . . . . . . . . . . . . . . . 8233.6 Robust loss functions for multivariate volatilitycomparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825xxii Contents3.7 Direct comparison via encompassing tests . . . . . . . . 8284 Indirect Evaluation of Volatility Forecasts . . . . . . . . . . . . . . . 8304.1 Portfolio optimisation . . . . . . . . . . . . . . . . . . . . . . . . . 8314.2 Tracking error minimisation . . . . . . . . . . . . . . . . . . . . 8324.3 Other methods of indirect evaluation . . . . . . . . . . . . 8335 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835Structural Breaks in Financial Time Series . . . . . . . . . . . . . . . . . . 839Elena Andreou and Eric Ghysels1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8392 Consequences of Structural Breaks in Financial Time Series 8403 Methods for Detecting Structural Breaks . . . . . . . . . . . . . . . . 8433.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8443.2 Historical and sequential partialsumschangepoint statistics . . . . . . . . . . . . . . . . . . . . . . . . . 8453.3 Multiple breaks tests . . . . . . . . . . . . . . . . . . . . . . . . . . 8484 ChangePoint Tests in Returns and Volatility . . . . . . . . . . . . 8514.1 Tests based on empirical volatility processes . . . . . . 8514.2 Empirical processes and the SV class of models . . . 8544.3 Tests based on parametric volatility models . . . . . . 8584.4 Changepoint tests in long memory . . . . . . . . . . . . . 8614.5 Changepoint in the distribution . . . . . . . . . . . . . . . . 8635 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866An Introduction to Regime Switching Time Series Models . . . 871Theis Lange and Anders Rahbek1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8711.1 Markov and observation switching. . . . . . . . . . . . . . . 8722 Switching ARCH and CVAR. . . . . . . . . . . . . . . . . . . . . . . . . . . 8742.1 Switching ARCH and GARCH . . . . . . . . . . . . . . . . . 8752.2 Switching CVAR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8773 LikelihoodBased Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 8794 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8815 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 889Hannes Leeb and Benedikt M. Ptscher1 The Model Selection Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 8891.1 A general formulation . . . . . . . . . . . . . . . . . . . . . . . . . 8891.2 Model selection procedures . . . . . . . . . . . . . . . . . . . . . 8922 Properties of Model Selection Procedures and ofPostModelSelection Estimators . . . . . . . . . . . . . . . . . . . . . . . 9002.1 Selection probabilities and consistency . . . . . . . . . . . 900Contents xxiii2.2 Risk properties of post-model-selection estimators 9032.3 Distributional properties of post-model-selectionestimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9063 Model Selection in Large- or InniteDimensional Models . 9084 Related Procedures Based on Shrinkage and ModelAveraging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9155 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916Nonparametric Modeling in Financial Time Series . . . . . . . . . . . 927Jrgen Franke, Jens-Peter Kreiss and Enno Mammen1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9272 Nonparametric Smoothing for Time Series . . . . . . . . . . . . . . . 9292.1 Density estimation via kernel smoothing . . . . . . . . . 9292.2 Kernel smoothing regression . . . . . . . . . . . . . . . . . . . . 9322.3 Diusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9353 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9374 Nonparametric Quantile Estimation . . . . . . . . . . . . . . . . . . . . 9405 Advanced Nonparametric Modeling . . . . . . . . . . . . . . . . . . . . . 9426 Sieve Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947Modelling Financial High Frequency Data Using PointProcesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953Luc Bauwens and Nikolaus Hautsch1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9532 Fundamental Concepts of Point Process Theory . . . . . . . . . . 9542.1 Notation and denitions . . . . . . . . . . . . . . . . . . . . . . . 9552.2 Compensators, intensities, and hazard rates . . . . . . 9552.3 Types and representations of point processes . . . . . 9562.4 The random time change theorem . . . . . . . . . . . . . . . 9593 Dynamic Duration Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9603.1 ACD models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9603.2 Statistical inference . . . . . . . . . . . . . . . . . . . . . . . . . . . 9633.3 Other models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9643.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9654 Dynamic Intensity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9674.1 Hawkes processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9674.2 Autoregressive intensity processes . . . . . . . . . . . . . . . 9694.3 Statistical inference . . . . . . . . . . . . . . . . . . . . . . . . . . . 9734.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976xxiv ContentsPart V Special Topics Simulation Based MethodsResampling and Subsampling for Financial Time Series . . . . . . 983Efstathios Paparoditis and Dimitris N. Politis1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9832 Resampling the Time Series of LogReturns . . . . . . . . . . . . . 9862.1 Parametric methods based on i.i.d. resampling ofresiduals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9862.2 Nonparametric methods based on i.i.d. resamplingof residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9882.3 Markovian bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . 9903 Resampling Statistics Based on the Time Series ofLogReturns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9923.1 Regression bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . 9923.2 Wild bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9933.3 Local bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9944 Subsampling and SelfNormalization. . . . . . . . . . . . . . . . . . . . 995References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997Markov Chain Monte Carlo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1001Michael Johannes and Nicholas Polson1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10012 Overview of MCMC Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 10022.1 CliordHammersley theorem . . . . . . . . . . . . . . . . . . 10022.2 Constructing Markov chains . . . . . . . . . . . . . . . . . . . . 10032.3 Convergence theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 10073 Financial Time Series Examples . . . . . . . . . . . . . . . . . . . . . . . . 10083.1 Geometric Brownian motion . . . . . . . . . . . . . . . . . . . . 10083.2 Time-varying expected returns . . . . . . . . . . . . . . . . . . 10093.3 Stochastic volatility models . . . . . . . . . . . . . . . . . . . . 10104 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012Particle Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015Michael Johannes and Nicholas Polson1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10152 A Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10173 Particle Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10193.1 Exact particle ltering . . . . . . . . . . . . . . . . . . . . . . . . . 10213.2 SIR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10243.3 Auxiliary particle ltering algorithms . . . . . . . . . . . . 10264 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031List of ContributorsYacine At-SahaliaPrinceton University and NBER, Bendheim Center for Finance, PrincetonUniversity, U.S.A..Torben G. AndersenKellogg School of Management and NBER, Northwestern University, U.S.A.and CREATES, Aarhus, Denmark.Elena AndreouDepartment of Economics, University of Cyprus, Nicosia, Cyprus.Manabu AsaiFaculty of Economics, Soka University, Tokyo, Japan.Luc BauwensCORE, Universit Catholique de Louvain, Belgium.Luca BenzoniFederal Reserve Bank of Chicago, U.S.A..Thomas BjrkDepartment of Finance, Stockholm School of Economics, Sweden.Peter J. BrockwellDepartment of Statistics, Colorado State University, Fort Collins, U.S.A..Ngai Hang ChanDepartment of Statistics, Chinese University of Hong Kong, Shatin, NT,Hong Kong.Willa W. ChenDepartment of Statistics, Texas A&M University, College Station, U.S.A..xxvi List of ContributorsSiddhartha ChibOlin Business School, Washington University in St. Louis, U.S.A..Peter ChristoersenDesautels Faculty of Management, McGill University, Quebec, Canada.Pavel ekDepartment of Econometrics & OR, Tilburg University, The Netherlands.Richard A. DavisDepartment of Statistics, Columbia University, New York, U.S.A..Ernst EberleinDepartment of Mathematical Stochastics, University of Freiburg, Germany.Paul EmbrechtsDepartment of Mathematics, ETH Zrich, Switzerland.Vicky FasenZentrum Mathematik, Technische Universitt Mnchen, Germany.Christian FrancqUniversity Lille III, EQUIPPE-GREMARS, France.Jrgen FrankeDepartment of Mathematics, Universitt Kaiserslautern, Germany.Hansjrg FurrerSwiss Life, Zrich, Switzerland.Eric GhyselsDepartment of Economics, University of North Carolina at Chapel Hill,U.S.A..Liudas GiraitisDepartment of Economics, Queen Mary University of London, UnitedKingdom.Nikolaus HautschInstitute for Statistics and Econometrics, HumboldtUniversitt zu Berlin,Germany.Cliord M. HurvichLeonard N. Stern School of Business, New York University, U.S.A..Michael JohannesGraduate School of Business, Columbia University, New York, U.S.A..Sren JohansenDepartment of Applied Mathematics and Statistics, University ofCopenhagen, Denmark.List of Contributors xxviiBorus JungbackerDepartment of Econometrics, Vrije Universiteit Amsterdam, TheNetherlands.Jan KallsenMathematisches Seminar, Christian-Albrechts-Universitt zu Kiel, Germany.Roger KaufmannAXA Winterthur, Winterthur, Switzerland.Siem Jan KoopmanDepartment of Econometrics, Vrije Universiteit Amsterdam, TheNetherlands.Jens-Peter KreissInstitut fr Mathematische Stochastik, Technische Universitt Braunschweig,Germany.David LandoCopenhagen Business School, Department of Finance, Denmark.Theis LangeDepartment of Economics, University of Copenhagen, Denmark.Hannes LeebDepartment of Statistics, Yale University, U.S.A..Remigijus LeipusVilnius University and Institute of Mathematics and Informatics, Vilnius,Lithuania.Alexander M. LindnerTechnische Universitt Braunschweig, Institut fr Mathematische Stochastik,Germany.Oliver B. LintonDepartment of Economics, London School of Economics and PoliticalScience, United Kingdom.Ross A. MallerSchool of Finance & Applied Statistics and Centre for Mathematics & itsApplications, Australian National University, Canberra, Australia.Enno MammenAbteilung Volkswirtschaftslehre, Universitt Mannheim, Germany.Thomas MikoschLaboratory of Actuarial Mathematics, University of Copenhagen, Denmark.Gernot MllerZentrum Mathematik, Technische Universitt Mnchen, Germany.xxviii List of ContributorsPer A. MyklandDepartment of Statistics, The University of Chicago, U.S.A..Yasuhiro OmoriFaculty of Economics, University of Tokyo, Japan.Efstathios PaparoditisDepartment of Mathematics and Statistics, University of Cyprus, Nicosia,Cyprus.Andrew J. PattonDepartment of Economics and Oxford-Man Institute of QuantitativeFinance, University of Oxford, United Kingdom.Peter C. B. PhillipsCowles Foundation for Research in Economics, Yale University, U.S.A.;University of Auckland; University of York; and Singapore ManagementUniversity.Benedikt M. PtscherDepartment of Statistics, University of Vienna, Austria.Dimitris N. PolitisDepartment of Mathematics, University of California, San Diego, U.S.A..Nicholas PolsonGraduate School of Business, University of Chicago, U.S.A..Anders RahbekDepartment of Economics, University of Copenhagen, Denmark.Eric RenaultDepartment of Economics, University of North Carolina, Chapel Hill,U.S.A..Neil ShephardOxford-Man Institute and Department of Economics, University of Oxford,United Kingdom.Kevin SheppardDepartment of Economics and Oxford-Man Institute of QuantitativeFinance, University of Oxford, United Kingdom.Annastiina SilvennoinenSchool of Finance and Economics, University of Technology Sydney,Australia.Michael SrensenDepartment of Mathematical Sciences, University of Copenhagen, Denmark.Philippe SoulierDepartment of Mathematics, University Paris X, France.List of Contributors xxixVladimir SpokoinyWeierstrassInstitut, Berlin, Germany.Donatas SurgailisVilnius University and Institute of Mathematics and Informatics, Vilnius,Lithuania.Alex SzimayerFraunhofer-Institut fr Techno-und Wirtschaftsmathematik, Kaiserslautern,Germany.Timo TersvirtaCREATES, School of Economics and Management, University of Aarhus,Denmark and Department of Economic Statistics, Stockholm School ofEconomics, Sweden.Jun YuSchool of Economics, Singapore Management University, Singapore.Jean-Michel ZakoanUniversity Lille III, EQUIPPE-GREMARS, and CREST, France.Eric ZivotDepartment of Economics, University of Washington, Seattle, U.S.A..An Introduction to Univariate GARCHModelsTimo TersvirtaAbstract This paper contains a survey of univariate models of conditionalheteroskedasticity. The classical ARCH model is mentioned, and various ex-tensions of the standard Generalized ARCH model are highlighted. This in-cludes the Exponential GARCH model. Stochastic volatility models remainoutside this review.1 IntroductionFinancial economists are concerned with modelling volatility in asset returns.This is important as volatility is considered a measure of risk, and investorswant a premium for investing in risky assets. Banks and other nancial insti-tutions apply so-called value-at-risk models to assess their risks. Modellingand forecasting volatility or, in other words, the covariance structure of assetreturns, is therefore important.The fact that volatility in returns uctuates over time has been known fora long time. Originally, the emphasis was on another aspect of return series:their marginal distributions were found to be leptokurtic. Returns were mod-elled as independent and identically distributed over time. In a classic work,Mandelbrot (1963) and Mandelbrot and Taylor (1967) applied so-called sta-ble Paretian distributions to characterize the distribution of returns. Rachevand Mittnik (2000) contains an informative discussion of stable Paretian dis-tributions and their use in nance and econometrics.Observations in return series of nancial assets observed at weekly andhigher frequencies are in fact not independent. While observations in theseTimo TersvirtaCREATES, School of Economics and Management, University of Aarhus, DK-8000 AarhusC, and Department of Economic Statistics, Stockholm School of Economics, Box 6501, SE-113 83 Stockholm, e-mail: tterasvirta@econ.au.dk T.G. Anderson et al., Handbook of Financial Time Series, 17 DOI: 10.1007/978-3-540-71297-8_1, Springer-Verlag Berlin Heidelberg 2009 18 T. Tersvirtaseries are uncorrelated or nearly uncorrelated, the series contain higher-order dependence. Models of Autoregressive Conditional Heteroskedastic-ity (ARCH) form the most popular way of parameterizing this dependence.There are several articles in this Handbook devoted to dierent aspects ofARCH models. This article provides an overview of dierent parameteriza-tions of these models and thus serves as an introduction to autoregressiveconditional heteroskedasticity. The article is organized as follows. Section2 introduces the classic ARCH model. Its generalization, the GeneralizedARCH (GARCH) model is presented in Section 3. This section also describesa number of extensions to the standard GARCH models. Section 4 considersthe Exponential GARCH model whose structure is rather dierent from thatof the standard GARCH model, and Section 5 discusses ways of comparingEGARCH models with GARCH ones. Suggestions for further reading can befound at the end.2 The ARCH ModelThe autoregressive conditional heteroskedasticity (ARCH) model is the rstmodel of conditional heteroskedasticity. According to Engle (2004), the orig-inal idea was to nd a model that could assess the validity of the conjectureof Friedman (1977) that the unpredictability of ination was a primary causeof business cycles. Uncertainty due to this unpredictability would aect theinvestors behaviour. Pursuing this idea required a model in which this un-certainty could change over time. Engle (1982) applied his resulting ARCHmodel to parameterizing conditional heteroskedasticity in a wage-price equa-tion for the United Kingdom. Let t be a random variable that has a meanand a variance conditionally on the information set Ft1 (the -eld gener-ated by tj, j 1). The ARCH model of t has the following properties.First, E{t|Ft1} = 0 and, second, the conditional variance ht = E{2t|Ft1}is a nontrivial positive-valued parametric function of Ft1. The sequence {t}may be observed directly, or it may be an error or innovation sequence of aneconometric model. In the latter case,t = ytt(yt) (1)where yt is an observable random variable and t(yt) = E{yt|Ft1}, theconditional mean of yt given Ft1. Engles (1982) application was of thistype. In what follows, the focus will be on parametric forms of ht, and forsimplicity it is assumed that t(yt) = 0.Engle assumed that t can be decomposed as follows:t = zth1/2t (2)An Introduction to Univariate GARCH Models 19where {zt} is a sequence of independent, identically distributed (iid) randomvariables with zero mean and unit variance. This implies t|Ft1 D(0, ht)where D stands for the distribution (typically assumed to be a normal or aleptokurtic one). The following conditional variance denes an ARCH modelof order q:ht = 0 +q

j=1j2tj (3)where 0 > 0, j 0, j = 1, . . . , q1, and q > 0. The parameter restrictionsin (3) form a necessary and sucient condition for positivity of the conditionalvariance. Suppose the unconditional variance E2t = 2< . The denitionof t through the decomposition (2) involving zt then guarantees the whitenoise property of the sequence {t}, since {zt} is a sequence of iid variables.Although the application in Engle (1982) was not a nancial one, Engle andothers soon realized the potential of the ARCH model in nancial applicationsthat required forecasting volatility.The ARCH model and its generalizations are thus applied to modelling,among other things, interest rates, exchange rates and stock and stock indexreturns. Bollerslev et al. (1992) already listed a variety of applications in theirsurvey of these models. Forecasting volatility of these series is dierent fromforecasting the conditional mean of a process because volatility, the objectto be forecast, is not observed. The question then is how volatility should bemeasured. Using 2t is an obvious but not necessarily a very good solutionif data of higher frequency are available; see Andersen and Bollerslev (1998)and Andersen and Benzoni (2008) for discussion.3 The Generalized ARCH Model3.1 Why Generalized ARCH?In applications, the ARCH model has been replaced by the so-called gen-eralized ARCH (GARCH) model that Bollerslev (1986) and Taylor (1986)proposed independently of each other. In this model, the conditional vari-ance is also a linear function of its own lags and has the formht = 0 +q

j=1j2tj +p

j=1jhtj. (4)The conditional variance dened by (4) has the property that the uncondi-tional autocorrelation function of 2t, if it exists, can decay slowly, albeit stillexponentially. For the ARCH family, the decay rate is too rapid comparedto what is typically observed in nancial time series, unless the maximum20 T. Tersvirtalag q in (3) is long. As (4) is a more parsimonious model of the conditionalvariance than a high-order ARCH model, most users prefer it to the simplerARCH alternative.The overwhelmingly most popular GARCH model in applications has beenthe GARCH(1,1) model, that is, p = q = 1 in (4). A sucient condition forthe conditional variance to be positive is 0 > 0, j 0, j = 1, . . . , q; j 0,j = 1, . . . , p. The necessary and sucient conditions for positivity of theconditional variance in higher-order GARCH models are more complicatedthan the sucient conditions just mentioned and have been given in Nelsonand Cao (1992). The GARCH(2,2) case has been studied in detail by Heand Tersvirta (1999). Note that for the GARCH model to be identied ifat least one j > 0 (the model is a genuine GARCH model) one has torequire that also at least one j > 0. If 1 = . . . = q = 0, the conditionaland unconditional variances of t are equal and 1, . . . , p are unidentiednuisance parameters. The GARCH(p,q) process is weakly stationary if andonly if qj=1 j + pj=1 j < 1.The stationary GARCH model has been slightly simplied by variancetargeting, see Engle and Mezrich (1996). This implies replacing the inter-cept 0 in (4) by (1

qj=1 j

pj=1 j)2where 2= E2t. The estimate 2= T1

Tt=1 2t is substituted for 2before estimating the other param-eters. As a result, the conditional variance converges towards the long-rununconditional variance, and the model contains one parameter less than thestandard GARCH(p,q) model.It may be pointed out that the GARCH model is a special case of aninnite-order (ARCH()) model (2) withht = 0 +

j=1j2tj. (5)The ARCH() representation is useful in considering properties of ARCHand GARCH models such as the existence of moments and long memory; seeGiraitis et al. (2000). The moment structure of GARCH models is consideredin detail in Lindner (2008).3.2 Families of univariate GARCH modelsSince its introduction the GARCH model has been generalized and extendedin various directions. This has been done to increase the exibility of theoriginal model. For example, the original GARCH specication assumes theresponse of the variance to a shock to be independent of the sign of theshock and just be a function of the size of the shock. Several extensions ofthe GARCH model aim at accommodating the asymmetry in the response.An Introduction to Univariate GARCH Models 21These include the GJR-GARCH model of Glosten et al. (1993), the asym-metric GARCH models of Engle and Ng (1993) and the quadratic GARCHof Sentana (1995). The GJR-GARCH model has the form (2), whereht = 0 +q

j=1{j + jI(tj > 0)}2tj +p

j=1jhtj. (6)In 6), I(tj > 0) is an indicator function obtaining value one when theargument is true and zero otherwise.In the asymmetric models of both Engle and Ng, and Sentana, the centreof symmetry of the response to shocks is shifted away from zero. For example,ht = 0 + 1(t1)2+ 1ht1 (7)with = 0 in Engle and Ng (1993). The conditional variance in SentanasQuadratic ARCH (QARCH) model (the model is presented in the ARCHform) is dened as follows:ht = 0 +

t1 +

t1At1 (8)where t = (t, . . . , tq+1)

is a q 1 vector, = (1, . . . , q)

is a q 1parameter vector and A a q q symmetric parameter matrix. In (8), notonly squares of ti but also cross-products titj, i = j, contribute tothe conditional variance. When = 0, the QARCH generates asymmetricresponses. The ARCH equivalent of (7) is a special case of Sentanas model.Constraints on parameters required for positivity of ht in (8) become clearby rewriting (8) as follows:ht = _t1 1

_ A /2

/2 0_ _t11_. (9)The conditional variance ht is positive if and only if the matrix in thequadratic form on the right-hand side of (9) is positive denite.Some authors have suggested modelling the conditional standard deviationinstead of the conditional variance: see Taylor (1986), Schwert (1990), and foran asymmetric version, Zakoan (1994). A further generalization of this ideaappeared in Ding et al. (1993). These authors proposed a GARCH model forhkt where k > 0 is also a parameter to be estimated. Their GARCH modelis (2) withhkt = 0 +q

j=1j|tj|2k+p

j=1jhktj, k > 0. (10)The authors argued that this extension provides exibility lacking in theoriginal GARCH specication of Bollerslev (1986) and Taylor (1986).The proliferation of GARCH models has inspired some authors to denefamilies of GARCH models that would accommodate as many individual22 T. Tersvirtamodels as possible. Hentschel (1995) dened one such family . The rst-orderGARCH model has the general formh/2t 1 = + h/2t1f(zt1) + h/2t11 (11)where > 0 andf(zt) = |ztb| c(ztb).Family (11) contains a large number of well-known GARCH models. The Box-Cox type transformation of the conditional standard deviation h1/2t makes itpossible, by allowing 0, to accommodate models in which the logarithmof the conditional variance is parameterized, such as the exponential GARCHmodel to be considered in Section 4. Parameters b and c in fv(zt) allow theinclusion of dierent asymmetric GARCH models such as the GJR-GARCHor threshold GARCH models in (11).Another family of GARCH models that is of interest is the one He andTersvirta (1999) dened as follows:hkt =q

j=1g(ztj) +p

j=1cj(ztj)hktj, k > 0 (12)where {g(zt)} and {c(zt)} are sequences of independent and identically dis-tributed random variables. In fact, the family was originally dened for q = 1,but the denition can be generalized to higher-order models. For example,the standard GARCH(p, q) model is obtained by setting g(zt) = 0/q andcj(ztj) = jz2tj + j, j = 1, . . . , q, in (12). Many other GARCH mod-els such as the GJR-GARCH, the absolute-value GARCH, the QuadraticGARCH and the power GARCH model belong to this family.Note that the power GARCH model itself nests several well-known GARCHmodels; see Ding et al. (1993) for details. Denition (12) has been usedfor deriving expressions of fourth moments, kurtosis and the autocorrelationfunction of 2t for a number of rst-order GARCH models and the standardGARCH(p, q) model.The family of augmented GARCH models, dened by Duan (1997), is arather general family. The rst-order augmented GARCH model is denedas follows. Consider (2) and assume thatht =_|t 1| if = 0exp{t 1} if = 0 (13)wheret = 0 + 1,t1t1 + 2,t1. (14)In (14), (1t, 2t) is a strictly stationary sequence of random vectors with acontinuous distribution, measurable with respect to the available informationAn Introduction to Univariate GARCH Models 23until t. Duan dened an augmented GARCH(1,1) process as (2) with (13)and (14), such that1t = 1 + 2|tc|+ 3 max(0, c t)2t = 4|tc|1 + 5max(0, c t)1 .This process contains as special cases all the GARCH models previously men-tioned, as well as the Exponential GARCH model to be considered in Section4. Duan (1997) generalized this family to the GARCH(p, q) case and derivedsucient conditions for strict stationarity for this general family as well asconditions for the existence of the unconditional variance of t. Furthermore,he suggested misspecication tests for the augmented GARCH model.3.3 Nonlinear GARCH3.3.1 Smooth transition GARCHAs mentioned above, the GARCH model has been extended to characterizeasymmetric responses to shocks. The GJR-GARCH model, obtained as set-ting qj=1 g(ztj) = 0 and cj(ztj) = (j + jI(ztj > 0))z2tj + j, j =1, . . . , q, in (12), is an example of that. A nonlinear version of the GJR-GARCH model is obtained by making the transition between regimes smooth.Hagerud (1997), Gonzales-Rivera (1998) and Anderson et al. (1999) proposedthis extension. A smooth transition GARCH (STGARCH) model may bedened as equation (2) withht = 10 +q

j=11j2tj + (20 +q

j=12j2tj)G(, c; tj) +p

j=1jhtj (15)where the transition functionG(, c; tj) = (1 + exp{K

k=1(tj ck)})1, > 0. (16)When K = 1, (16) is a simple logistic function that controls the changeof the coecient of 2tj from 1j to 1j + 2j as a function of tj, andsimilarly for the intercept. In that case, as , the transition functionbecomes a step function and represents an abrupt switch from one regimeto the other. Furthermore, at the same time setting c1 = 0 yields the GJR-GARCH model because t and zt have the same sign. When K = 2 and, inaddition, c1 = c2 in (16), the resulting smooth transition GARCH model isstill symmetric about zero, but the response of the conditional variance to a24 T. Tersvirtashock is a nonlinear function of lags of 2t. Smooth transition GARCH modelsare useful in situations where the assumption of two distinct regimes is notan adequate approximation to the asymmetric behaviour of the conditionalvariance. Hagerud (1997) also discussed a specication strategy that allowsthe investigator to choose between K = 1 and K = 2 in (16). Values of K > 2may also be considered, but they are likely to be less common in applicationsthan the two simplest cases.The smooth transition GARCH model (15) with K = 1 in (16) is designedfor modelling asymmetric responses to shocks. On the other hand, the stan-dard GARCH model has the un