essays on forecasting with linear state-space...

Essays on Forecasting with Linear

State-Space Systems

2016-2

Lorenzo Boldrini

PhD Thesis

DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS

AARHUS UNIVERSITY DENMARK

ESSAYS ON FORECASTING WITH LINEAR

STATE-SPACE SYSTEMS

By Lorenzo Boldrini

A PhD thesis submitted to

School of Business and Social Sciences, Aarhus University,

in partial fulfilment of the PhD degree

in Economics and Business Economics

February 2016

CREATESCenter for Research in Econometric Analysis of Time Series

PREFACE

This dissertation is the result of my PhD studies at the Department of Economics

and Business Economics at Aarhus University and was written in the period from

September 2012 to August 2015. I am grateful to the Department of Economics and

Business Economics as well as CREATES - Center for Research in Econometric Analy-

sis of Time Series-DNRF78 -, funded by the Danish National Research Foundation,

for providing both a unique research environment and financial support.

Several people deserve my gratitude. In the first place, I wish to thank my main

advisor, Prof. Eric T. Hillebrand for the support, advice and encouragement he has

given me during these three years. Working with Eric has never failed to be challeng-

ing and thought provoking. Thank you to my co-advisor and Center Director, Prof.

Niels Haldrup and the Center Administrator Solveig Nygaard Sørensen, for creating

a great research environment and for the many interesting PhD courses that were

organized by CREATES during my studies. I also wish to thank Prof. Eduardo Rossi

and Prof. Maria E. De Giuli for encouraging me to apply for a PhD position.

From January 2015 to April 2015 I had the great pleasure of visiting Prof. Siem

Jan Koopman at the Department of Econometrics and Operations Research at VU

Amsterdam. I had the opportunity to have frequent, helpful meetings with Siem Jan,

who never lacked enthusiasm for the subject at hand and always offered a diverse

perspective on the topic.

At Aarhus University I would like to thank the faculty and staff at the Depart-

ment of Economics and Business Economics. In particular, I would like to thank

Ulrich, Johannes, Nima, Matt, Ulises, Anders Kock and Henning B. for their help

on some computational and analytical aspects involved in this work and Solveig,

Mikkel, Jonas and Niels S. for their editorial help involved in the writing of this thesis.

I am very grateful to my fellow PhD students for the many interesting conversations

and enjoyable activities we shared and in particular to Camilla, Silvia, Alice, Andrea,

Silvana, Vladimir and Eduardo. Special thanks go to Federico, Orimar and Hassan. I

am indebted to them for the copious fruitful discussions we had and fun moments

we shared. Lastly, I am thankful for the support my family and friends have shown

me during the last three years.

Lorenzo Boldrini

Aarhus, August 2015

i

UPDATED PREFACE

The pre-defence meeting was held on the 25th November 2015, in Aarhus. I am grate-

ful to the members of the assessment committee consisting of Siem Jan Koopman

- VU University Amsterdam -, Tommaso Proietti - University of Rome Tor Vergata

- and Asger Lunde - Aarhus University and CREATES - for their careful reading of

the dissertation and their many insightful comments and suggestions. Some of the

suggestions have been incorporated into the present version of the dissertation while

others remain for future research.

Lorenzo Boldrini

Aarhus, February 2016

iii

CONTENTS

Summary vii

Danish summary xi

1 Supervision in Factor Models Using a Large Number of Predictors 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Forecasting with dynamic factor models . . . . . . . . . . . . . . . . . 5

1.3 Quantifying supervision . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Computational aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Empirical application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.8 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.10 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2 The Forecasting Power of the Yield Curve 41

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2 Dynamic factor models and supervision . . . . . . . . . . . . . . . . . 45


2.4 Empirical application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3 Forecasting the Global Mean Sea Level 75

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.2 Model specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.4 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90


v

vi CONTENTS

3.6 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

SUMMARY

This dissertation is composed of three self-contained, independent chapters. The

statistical framework in all three chapters is that of linear, Gaussian, state-space

systems. In the first two chapters, we study a method to forecast macroeconomic

time series whereas the third one is focused on the forecasting of the global mean sea

level, conditional on the global mean temperature.

More in detail, in Chapter 1 we investigate the forecasting performance of a partic-

ular factor model in which the factors are extracted from a large number of predictors.

We use a semi-parametric state-space representation of the factor model in which the

forecast objective, as well as the factors, is included in the state vector. The factors are

informed of the forecast target (supervised) through the state equation dynamics. We

propose a way to assess the contribution of the forecast objective on the extracted fac-

tors that exploits the Kalman filter recursions. We forecast one target at a time based

on the filtered states and estimated parameters of the state-space system. We assess

the out-of-sample forecast performance of the proposed method in a simulation

study and in an empirical application to US macroeconomic data. We compare our

specification to other multivariate and univariate approaches and use the Giacomini

and White (2006) test to assess the relative conditional predictive ability of the mod-

els. In particular, we choose as competing models: principal components regression,

partial least squares regression, ARMA(p,q) processes, an unsupervised factor model,

and a standard dynamic factor model with separate forecast and state equations.

We find that, variables which contribute more to the variance of the filtered states,

are the ones which benefit more from the supervised framework and vice versa. The

proposed specification performs particularly well in forecasting the federal funds

rate, the unemployment rate, and real disposable income.

In Chapter 2, we study the forecast power of the yield curve for macroeconomic

time series. We employ a state-space system in which the forecasting objective is

included in the state vector. This amounts to an augmented dynamic factor model in

which the factors (level, slope, and curvature of the yield curve) are supervised for

the forecast target. In other words, the factors are informed about the dynamics of

the forecast objective. The factor loadings have the Nelson and Siegel (1987) struc-

ture and we consider one forecast target at a time. We forecast the consumer price

index, personal consumption expenditures, the producer price index, real disposable

vii

viii CONTENTS

income, unemployment rate, and industrial production. All time series are relative to

the US economy. We compare the forecasting performance of our specification to

benchmark models such as principal components regression, partial least squares

regression, ARMA(p,q) processes, and two-step forecasting procedures based on

factor models. We use the yield curve data from Gürkaynak, Sack, and Wright (2007)

and Diebold and Li (2006), and macroeconomic data from FRED. We compare the

different models by means of the conditional predictive ability test of Giacomini and

White (2006). We find that the yield curve has more forecast power for real variables

compared to inflation measures and that supervising the factor extraction for the

forecast target can improve the forecast performance of the factor model. We also

compare direct and indirect forecasts for the different models and find that the in-

direct forecasts perform better for our data and specification. We find that the yield

curve has forecast power for unemployment rate, real disposable income, and indus-

trial production. Similarly to Giacomini and Rossi (2006), Rudebusch and Williams

(2009), and Stock and Watson (1999) we find that the predictive ability of the yield

curve is somewhat unstable and has changed through the years. In particular, we

find the yield curve has more predictive power for the periods 1987-1994 (the early

Greenspan monetary regime) and 2006-2012 (the early Bernanke monetary regime)

as compared to the period 1994-2006 (the late Greenspan monetary regime).

In Chapter 3, we propose a continuous-time, Gaussian, linear state-space system

to model the relation between global mean sea level (GMSL) and the global mean

temperature (GMT) with the aim of making long term projections for the GMSL. We

provide a justification for the model specification based on popular semi-empirical

methods present in the literature and on zero-dimensional energy balance models.

We show that some of the models developed in the literature on semi-empirical mod-

els can be analysed within this framework. We use the sea-level data reconstruction

developed in Church and White (2011) and the temperature reconstruction from

Hansen, Ruedy, Sato, and Lo (2010). We compare the forecasting performance of

the proposed specification to the procedures developed in Rahmstorf (2007) and

Vermeer and Rahmstorf (2009). Finally, we compute projections for the sea level, con-

ditional on 21st century temperature scenarios, corresponding to the Special Report

on Emissions Scenarios (SRES) of the Intergovernmental Panel on Climate Change

(IPCC) fourth assessment report. Furthermore, we propose a bootstrap procedure to

compute confidence intervals for the projections, based on the method introduced

in Rodriguez and Ruiz (2009). We make projections of the sea-level rise from 2010

up to 2099. Across state-space specifications and temperature scenarios, we find as

best-case scenario an increase of the sea level of 0.09[m] and as worst-case scenario

an increase of 0.37[m] for the year 2099, with respect to the (smoothed) sea-level

value in 2009. This corresponds to an increase of roughly 0.14[m] (best case) and

0.42[m] (worst case) relative to the mean, smoothed 1990 sea-level value. The choice

of the trend component influences somewhat the forecast performance of the model.

CONTENTS ix

References

Church, J. A., White, N. J., 2011. Sea-level rise from the late 19th to the early 21st

century. Surveys in Geophysics 32 (4-5), 585–602.

Diebold, F. X., Li, C., 2006. Forecasting the term structure of government bond yields.

Journal of econometrics 130 (2), 337–364.

Giacomini, R., Rossi, B., 2006. How stable is the forecasting performance of the yield

curve for output growth? Oxford Bulletin of Economics and Statistics 68 (s1), 783–

795.

Giacomini, R., White, H., 2006. Tests of conditional predictive ability. Econometrica

74 (6), 1545–1578.

Gürkaynak, R. S., Sack, B., Wright, J. H., 2007. The us treasury yield curve: 1961 to the

present. Journal of Monetary Economics 54 (8), 2291–2304.

Hansen, J., Ruedy, R., Sato, M., Lo, K., 2010. Global surface temperature change.

Reviews of Geophysics 48 (4).

Nelson, C. R., Siegel, A. F., 1987. Parsimonious modeling of yield curves. Journal of

business, 473–489.

Rahmstorf, S., 2007. A semi-empirical approach to projecting future sea-level rise.

Science 315 (5810), 368–370.

Rodriguez, A., Ruiz, E., 2009. Bootstrap prediction intervals in state–space models.

Journal of time series analysis 30 (2), 167–178.

Rudebusch, G. D., Williams, J. C., 2009. Forecasting recessions: the puzzle of the

enduring power of the yield curve. Journal of Business & Economic Statistics 27 (4).

Stock, J. H., Watson, M. W., 1999. Business cycle fluctuations in us macroeconomic

time series. Handbook of macroeconomics 1, 3–64.

Vermeer, M., Rahmstorf, S., 2009. Global sea level linked to global temperature. Pro-

ceedings of the National Academy of Sciences 106 (51), 21527–21532.

DANISH SUMMARY

Denne afhandling indeholder tre selvstændige, uafhængige kapitler. Det statistiske

grundlag i alle tre kapitler er lineære, Gaussiske og state-space-systemer. I de første

to kapitler studerer vi en metode til at forudsige makroøkonomiske tidsserier og det

tredje er fokuseret på forudsigelse af det globale gennemsnitlige havniveau, betinget

af den globale gennemsnitstemperatur.

Mere konkret, så undersøger vi i Kapitel 1 prædiktionsevnen af en bestemt fak-

tormodel, hvor faktorerne er trukket ud fra et stort antal prædiktorer. Vi anvender en

semiparametrisk state-space-repræsentation af faktormodellen i hvilket prædiktions-

målet såvel som faktorerne er inkluderet i state-vektoren. Faktorerne er informerede

om prædiktionsmålet (superviseret) igennem state-ligningens dynamik. Baseret på

Kalman-rekursionerne fremsætter vi en metode til at bedømme bidraget fra prædik-

tionsmålet på de udtrukne faktorer. Vi prædikterer et mål ad gangen på baggrund af

de filtrerede states og estimerede parametre i state-space-systemet. Vi vurderer præ-

diktionsevnen af de foreslåede metoder uden for stikprøven i et simulationsstudie og

i en empirisk anvendelse på amerikanske makroøkonomiske data. Vi sammenligner

vores specifikation med andre multi- og univariate metoder og bruger Giacomini og

White (2006)’s test til at vurdere den relative betingede prædiktionsevne af model-

lerne. Vi vælger primært følgende konkurrerende modeller: principal components,

partiel mindste-kvadraters regression, ARMA(p,q) processer, en ikke-superviseret

faktormodel og en standard dynamisk faktormodel med separat prædiktions- og

state-ligning. Vi finder, at variablene som bidrager mest til variansen af de filtrerede

states, er de som drager størst fordel af det superviserede set up og vice versa. Den

foreslåede specifikation klarer sig specielt godt i forudsigelse af den amerikanske

centralbanksrente, arbejdsløshedsraten og den reelle disponible indkomst.

I Kapitel 2 studerer vi prædiktionsevnen af rentekurven for makroøkonomiske

tidsserier. Vi anvender et state-space-system, i hvilket prædiktionsobjektivet er in-

kluderet i state-vektoren. Dette resulterer i en udvidet dynamisk faktormodel i hvil-

ken faktorerne (niveau, hældning og krumning af rentekurven) er superviserede for

prædiktionsmålet. Faktorvægtene har struktur fra Nelson and Siegel (1987), og vi be-

tragter et prædiktionsmål ad gangen. Vi forudsiger forbrugerprisindekset, personlige

forbrugsudgifter, producentprisindekset, reel disponibel indkomst, arbejdsløsheds-

raten og industriel produktion. Alle tidsserier vedrører den amerikanske økonomi.

xi

xii CONTENTS

Vi sammenligner prædiktionspræstationen af vores specifikation til benchmark-

modeller såsom principal component regression, partiel mindste-kvadraters regres-

sion, ARMA(p,q) processer og totrins-prædiktionsmetoder baseret på faktormodeller.

Vi anvender rentekurvedata fra Gürkaynak, Sack og Wright (2007) og Diebold og Li

(2006) og makroøkonomisk data fra FRED. Vi sammenligner de forskellige modeller

ved hjælp af den betingede prædiktionsevnetest fra Giacomini og White (2006). Vi

finder, at rentekurven bedre prædikterer reelle variable, sammenlignet med infla-

tionsmål og at supervision af faktorudledningen kan forbedre prædiktionsevnen

af faktormodellen. Vi sammenligner også direkte og indirekte prædiktioner fra de

forskellige modeller og finder, at indirekte prædiktioner klarer sig bedre for vores data

og specifikationer. Vi finder, at rentekurven kan hjælpe med at prædiktere arbejdsløs-

hedsraten, reel disponibel indkomst og industriel produktion. Ligesom Giacomini

og Rossi (2006), Rudebusch og Williams (2009) og Stock og Watson (1999) finder vi,

at rentekurvens prædiktionsevne er noget ustabil og har ændret sig igennem årene.

I særdeleshed finder vi, at rentekurven har større prædiktionsevne for perioderne

1987-1994 (det tidlige Greenspan monetære regime) og 2006-2012 (det tidlige Bernan-

ke monetære regime), sammenlignet med 1994-2006 (det sene Greenspan monetære

regime).

I Kapitel 3 fremsætter vi et Gaussisk, lineært state-space-system i kontinuert

tid som model for relationen mellem det globale gennemsnitlige havniveau (GGH)

og den globale gennemsnitlige temperatur (GGT) med det formål at lave langsig-

tede forudsigelser af GGH. Vi giver en begrundelse for modelspecifikationen ba-

seret på populære semiempiriske metoder fra litteraturen samt nul-dimensionale

energibalance-modeller. Vi viser, at nogle af de semiempiriske metoder fra litteratu-

ren kan analyseres i dette set up. Vi bruger havniveaudata-rekronstruktion udviklet i

Church og White (2011) og temperatur-rekonstruktionen fra Hansen, Ruedy, Sato og

Lo (2010). Vi sammenligner prædiktionsevnerne af de foreslåede specifikationer til

procedurerne udviklet i Rahmstorf (2007) og Vermeer og Rahmstorf (2009). Endelig

udregner vi forudsigelserne for havniveauet betinget på det 21. århundredes SRES

temperaturscenarier fra IPCC’s fjerde vurderingsrapport. Vi fremsætter yderligere

en bootstrap-metode til at udregne konfidensintervaller for forudsigelserne base-

ret på metoden introduceret i Rodriguez og Ruiz (2009). Vi foretager forudsigelser

af havniveauet fra 2010 til og med 2099. På tværs af state-space-specifikationer og

temperaturscenearier finder vi, at i det bedste tilfælde vil der være en stigning i havni-

veauet på 0,09[m] og i det værste tilfælde vil stigningen være på 0,37[m] for året 2099,

mht. havniveauet i 2009. Dette svarer groft sagt til en stigning på mellem 0,14[m]

(bedste scenario) og 0,42[m] (værste scenario) relativt til det gennemsnitlige, udglat-

tede 1990-havniveau. Valget af trend-specifikationen påvirker prædiktionsevnen af

modellen i nogen grad.

CH

AP

TE

R

1SUPERVISION IN FACTOR MODELS USING A

LARGE NUMBER OF PREDICTORS

A STATE-SPACE APPROACH

Lorenzo Boldrini

Aarhus University and CREATES

Eric Hillebrand


1

2 CHAPTER 1. SUPERVISION IN FACTOR MODELS USING A LARGE NUMBER OF PREDICTORS

Abstract

In this paper we investigate the forecasting performance of a particular factor model

(FM) in which the factors are extracted from a large number of predictors. We use a

semi-parametric state-space representation of the FM in which the forecast objective,

as well as the factors, is included in the state vector. The factors are informed of

the forecast target (supervised) through the state equation dynamics. We propose

a way to assess the contribution of the forecast objective on the extracted factors

that exploits the Kalman filter recursions. We forecast one target at a time based on

the filtered states and estimated parameters of the state-space system. We assess the

out-of-sample forecast performance of the proposed method in a simulation study

and in an empirical application, comparing its forecasts to the ones delivered by

other popular multivariate and univariate approaches, e.g. a standard dynamic factor

model with separate forecast and state equations.

1.1. INTRODUCTION 3

1.1 Introduction

The availability of large datasets, the increase in computational power, and the ease

of implementation have made factor models an appealing tool in forecasting. Factor

models offer several advantages over other forecasting methods. For example, they do

not require the choice of the variables to include in the forecasting scheme (as struc-

tural models do), they make use of a large information set, they allow to concentrate

the information in all the candidate predictors in a relatively small number of factors,

and they can be estimated with simple and fast methods. Using many predictors

also allows to avoid the structural instability typical of low-dimensional systems. As

argued for instance in Stock and Watson (2006) and Stock and Watson (2002a), also

practitioners typically examine a large number of variables when making forecasts.

Forecasting using factor models is usually carried out in a two-step procedure, as

suggested for instance by Stock and Watson (2002b). In the first step the factors are

estimated using a set of predictors (that may include the lags of the forecast target)

and in a second step the estimated factors are used to forecast the target by means of

a forecast equation. In the two-step forecasting procedure suggested in Stock and

Watson (2002b) however, the same factors are used to forecast different targets. That

is, the selection of the factors is not supervised by the forecast target. In this paper we

study a method to supervise the factor extraction for the forecast objective in order to

improve on the predictive power of factor models. In the supervised framework, the

factors are informed of the forecast target (supervised) through the state equation

dynamics. Furthermore, we propose a way to assess the contribution of the forecast

objective on the extracted factors that exploits the Kalman filter recursions.

The forecasting properties of static, restricted, and general dynamic factor models

have been widely studied in the literature. Some examples are Boivin and Ng (2005)

and d’Agostino and Giannone (2012), who study the predictive power of different

approaches belonging to the class of general dynamic factor models. Alessi, Barigozzi,

and Capasso (2007), Stock and Watson (2002b), Stock and Watson (2002a), and Stock

and Watson (2006) compare the forecasting performance of factor models to dif-

ferent univariate and multivariate approaches. The evidence regarding the relative

merits of factor models in forecasting, compared to other methods, differs between

works. Stock and Watson (1999) and Stock and Watson (2002b) find a better forecast

performance of factor models compared to univariate methods for inflation and

industrial production, whereas Schumacher and Dreger (2002), Banerjee, Marcellino,

and Masten (2005), and Engel, Mark, and West (2012) find mixed evidence.

The latent factors in a FM can be estimated using principal components analysis

(PCA), as in Stock and Watson (2002a), by dynamic principal components analysis,

using frequency domain methods, as proposed by Forni, Hallin, Lippi, and Reichlin

(2000), or by Kalman filtering techniques. Comprehensive surveys on factor models

can be found in Bai and Ng (2008b), Breitung and Eickmeier (2006), and Stock and

Watson (2011).


In the standard approach to factor models, the extracted factors are the same for

all the forecast targets. One of the directions the literature has taken for improving

on this approach is to select factors based on their ability to forecast a specific target.

Different methods have been proposed in the literature that address this problem.

The method of partial least squares regression (PLSR), for instance, constructs a

set of linear combinations of the inputs (predictors and forecast target) for regres-

sion, for more details see for instance Friedman, Hastie, and Tibshirani (2001). Bai

and Ng (2008a) proposed performing PCA on a subset of the original predictors, se-

lected using thresholding rules. This approach is close to the supervised PCA method

proposed in Bair, Hastie, Paul, and Tibshirani (2006), that aims at finding linear com-

binations of the predictors that have high correlation with the target. In particular,

first a subset of the predictors is selected, based on the correlation with the target

(i.e. the regression coefficient exceeds a given threshold), then PCA is applied on the

resulting subset of variables. Bai and Ng (2009) consider ‘boosting’ (a procedure that

performs subset variable selection and coefficient shrinkage) as a methodology for

selecting the predictors in factor-augmented autoregressions. Finally, Giovannelli

and Proietti (2014) propose an operational supervised method that selects factors

based on their significance in the regression of the forecast target on the predictors.

The supervised dynamic factor model we study in this paper is based on a Gaus-

sian, factor-augmented, approximate, dynamic factor model in which the forecast

objective is modelled jointly with the factors. In this paper, by dynamic factor model

we mean a factor model in which the factors follow a dynamic equation. The system

has a linear state-space representation and we estimate it using maximum likelihood.

The likelihood function is delivered by the Kalman filter. Under this setup, we propose

a way to measure the contribution of the forecast objective on the extracted factors

that exploits the Kalman filter recursions. In particular, we compute the contribution

of the forecast target to the variance of the filtered factors and find a positive corre-

spondence between this quantity and the forecast performance of the supervised

scheme.

We assess the out-of-sample forecast performance of the supervised scheme by

means of a simulation study and in an empirical application. In the simulation study,

we vary the degree of correlation between the factors and forecast objective. We com-

pare the forecasts from the supervised model to two unsupervised FM specifications.

We find that the higher the correlation between factors and forecast target, the better

the forecasts of the supervised scheme. In the empirical application, we forecast

selected macroeconomic time series and compare the forecast performance of the

supervised FM to two unsupervised FM specifications and other multivariate and uni-

variate methods. We use the dataset from Jurado, Ludvigson, and Ng (2015), adding

two more variables: real disposable personal income and personal consumption

expenditure, excluding food and energy, and removing the index of aggregate weekly

Hours (BLS), because this series starts later than the others. The resulting dataset

1.2. FORECASTING WITH DYNAMIC FACTOR MODELS 5

comprises 132 variables. We forecast consumer price index (CPI), federal funds rate

(FFR), personal consumption expenditures deflator (PCEd), producer price index

(PPI), personal income (PEI), unemployment rate (UR), industrial production (IP),

real disposable income (RDI), and personal consumption expenditures (PCE). The

observations range from January 1960 to December 2011 and all variables refer to the

US economy.

The paper is organized as follows: in Section 1.2 we introduce the supervised

factor model and compare with other forecasting methods based on factor models;

in Section 1.3 we show how supervision can be measured using the Kalman filter

recursions; in Section 1.4 we provide some details on the computational aspects of

the analysis; in Sections 1.5 and 1.6 we describe the empirical application and the

simulation setup, respectively; finally, Section 1.7 concludes.

1.2 Forecasting with dynamic factor models

Let yt be the forecast objective, xt an N -dimensional vector of predictors (that may

or not include lags of the forecast objective), h the forecast horizon and T the last

available time-point in the estimation window.

Supervised factor model

We propose the following forecasting model. Consider the state-space system:[xt

yt

]=

[Λ 00 1

][ft

yt

]+

[εt

0

], εt ∼ N (0,H),[

ft+1

yt+1

]= c+T

[ft

yt

]+ηt , ηt ∼ N (0,Q), (1.1)

where ft ∈ Rk are latent factors, Λ is a matrix of factor loadings, T and c are a ma-

trix and a vector of coefficients, respectively, of suitable dimensions, εt ∈ RN and

ηt ∈Rk+1 are uncorrelated vectors of disturbances and H and Q are their respective

variance-covariance matrices. The forecast objective is placed in the state equation

together with the latent factors and the predictors are modelled in the measurement

equation. We consider joint estimation of the factors using the Kalman filter recur-

sions and maximum likelihood estimation for the parameters. The intuition behind

the model is that if the forecast objective is correlated with the factors, modelling

factors and forecast objective jointly should deliver a better estimate of the factors.

We define supervision to be the contribution of the forecast target to the estimation

of the latent factors. In the next section we derive the analytical expression of this

contribution and present a measure of supervision based on it.

The state equation can be understood as a factor augmented VAR (FAVAR), intro-

duced in Bernanke, Boivin, and Eliasz (2005), in which factors are included together


with observables in a VAR model. A specification similar to this one was used also in

Diebold, Rudebusch, and Boragan Aruoba (2006) to analyse the correlation between

the Nelson-Siegel factors and some macroeconomic variables.

We wish to extract factors from a large number of predictors and model them

jointly with the forecast objective. In order to find a parsimonious specification of

the factor model we select as factor loadings basis functions of RN . This corresponds

to taking a low order approximation of the vector of predictors at each point in

time. Virtually any basis of RN can be used. We choose discrete cosine basis for their

ease of implementation. Mallat (1999, Theorem 8.12) shows that a random vector in

CN can be decomposed into discrete cosine basis. In particular, any g ∈CN can be

decomposed into

gn = 2

N

N−1∑k=0

fnλk cos

[kπ

N

(n + 1

2

)],

for 0 ≤ n < N , where gn is the n − th component of g,

λk =2−1/2 if k = 0 and

1 otherwise

and

fn =⟨

gn ,λk cos

[kπ

N

(n + 1

2

)]⟩=λk

N−1∑n=0

gncos

[kπ

N

(n + 1

2

)],

are the discrete cosine transform of type I.

In our specification xt = gt + εt for each t = 1, ...,T and xt ,n = g t ,n + εt ,n with

n = 1, ..., N , where with xt we denote a vector of predictors. For each point in time we

then have g t ,n = 2N

∑N−1k=0 ft ,kλk cos

[kπN

(n + 1

2

)]. The weights ft ,k are estimated via

Kalman filter/smoother recursions. The cosine basis functions are then contained in

the factor loading matrix

Λ =

p2

N2N cos

[πN

(1+ 1

2

)]· · · 2

N cos

[(k−1)π

N

(1+ 1

2

)]...

......

p2

N2N cos

[πN

(N + 1

2

)]· · · 2

N cos

[(k−1)π

N

(N + 1

2

)] . (1.2)

The supervised factor model is then comprised of equations (1.1) and (1.2). The

forecasting scheme for this model is:

(i) estimation of the system parameters using maximum likelihood;

(ii) extraction of the factors using the Kalman filter;

1.2. FORECASTING WITH DYNAMIC FACTOR MODELS 7

(iii) the forecast yT+h is obtained as the last element of the vector[fT+h|TyT+h|T

]= T

h

[fT |TyT

]+

h−1∑i=0

Tic, (1.3)

where fT |T is the vector of filtered factors, h is the forecast lead, and T and c are

estimated parameters.

Note that the filtered and smoothed estimates for fT are the same.

Two-step procedure

Forecasting using dynamic factor models (DFM hereafter) is often carried out in a

two-step procedure as in Stock and Watson (2002a). Consider the model

yt+h = β(L)′ft +γ(L)yt +εt+h , (1.4)

xt ,i = λi (L)ft +ηt ,i , (1.5)

with i = 1, . . . , N and where ft = ( ft ,1, . . . , ft ,k ) are k latent factors, ηt = [ηt ,i , . . . ,ηt ,N ]′

and εt are idiosyncratic disturbances, β(L) = ∑qj=0β j+1L j , λi (L) = ∑p

j=0λi ( j+1)L j ,

and γ(L) = ∑sj=0γ j+1L j are finite lag polynomials in the lag operator L; β j ∈ Rk ,

γ j ∈ R, and λi j ∈ R are parameters and q, p, s ∈N0 are indices. The assumption on

the finiteness of the lag polynomials allows us to rewrite (1.4)-(1.5) as a static factor

model, i.e. a factor model in which the factors do not appear in lags:

yt+h = c +β′Ft +γ(L)yt +εt+h ,

xt = ΛFt +ηt , (1.6)

with Ft = [f′t , . . . , f′t−r ]′, r = max(q, p), the i -th row of Λ is [λi ,1, . . . ,λi ,r+1], and β =[β′

1, . . . ,β′r+1]′. The forecasting scheme is the following:

(i) extraction of the factors ft from the predictors xt , modelled in equation (1.5),

using either principal components analysis or the Kalman filter;

(ii) regression of the forecast objective on the lagged estimated factors and on its

lags, according to the forecasting equation (1.4);

(iii) the forecast is obtained from the estimated factors and regression coefficients

as

yT+h = c + β′Ft + γ(L)yT .

Stock and Watson (2002a) developed theoretical results for this two-step procedure,

in the case of principal components estimation. In particular, they show the asymp-

totic efficiency of the feasible forecasts and the consistency of the factor estimates.


The difference between the supervised DFM and the two-step forecasting proce-

dure is that in the former model the factors are extracted conditionally on the forecast

target. In the supervised framework the filtered/smoothed factors are tailored to the

forecast objective. Note that for a linear state-space system the Kalman filter delivers

the best linear predictions of the state vector, conditionally on the observations, for

a correctly specified model. Moreover, if the innovations are Gaussian, the filtered

states coincide with conditional expectations, for more details on the optimality

properties of the Kalman filter see Brockwell and Davis (2009).

1.3 Quantifying supervision

In this section we propose a statistic to quantify supervision. We are interested in

quantifying the influence of the forecast target on the filtered factors. To accomplish

this, we develop some results that hold for a general linear, state-space system with

non-random coefficient matrices. Consider the following state-space system:

yt = Ztαt +εt ,

αt+1 = Ttαt +Rtηt , (1.7)

where εt ∼W N (0,Ht ) and ηt ∼W N (0,Qt ) are uncorrelated random vectors, yt ∈RN ,

αt ∈Rk , and ηt ∈Rq , and the matrices Zt , Tt , and Rt are of suitable dimensions. Note

that in this context we assume white noise innovations.

In model (1.1), we include the observable forecast target as last element, both in

the measurement and in the state equation. In the notation of model (1.7), we are

therefore ultimately interested in the influence of the forecast target, the last element

in yt , denoted yt ,N (or yt in the notation of model (1.1)), on the filtered factors ft . To

be more precise, the objective is to quantify the influence of the sequence yi ,N i=1,...,t

(or yi i=1,...,t in the notation of model (1.1)) on ft , the filtered factors at time t .

The standard Kalman filter recursions (see for instance Durbin and Koopman

(2012)) for system (1.7) are:

vt = yt −Zt at ,

Ft = Zt Pt Z′t +Ht ,

Mt = Pt Z′t , (1.8)

at |t = at +Mt F−1t vt , Pt |t = Pt −Mt F−1

t M′t ,

at+1 = Tt at |t , Pt+1 = Tt Pt |t T′t +Rt Qt R′

t , (1.9)

for t = 1, . . . ,T , where Pt = E [(αt −at )(αt −at )], Pt |t = E[[αt −at |t ][αt −at |t ]′

], Ft =

E [vt v′t ], and at |t = Pt (αt ) = P (αt |y0, . . . ,yt ) and at = Pt−1(αt ) = P (αt |y0, . . . ,yt−1) are

the filtered state and one-step-ahead prediction of the state vector, respectively, and

Pt (·) is the best linear predictor operator, see Brockwell and Davis (2002) for more

1.3. QUANTIFYING SUPERVISION 9

details on the definition and properties of the best linear predictor operator. In the

particular case of Gaussian innovations in both the state and measurement equations,

the best linear predictor coincides with the conditional expectation. The forecasting

step in the Kalman recursions (1.9) can be written as

at+1 = Tt at +Kt vt

= Tt at +Kt (yt −Zt at )

= St at +Kt yt , (1.10)

with Kt = Tt Pt Z′t F−1

t and St = Tt −Kt Zt . Iterating backwards on the one-step-ahead

prediction of the state, the filtered state can be written as

at |t = at + Kt vt

= Nt at + Kt yt

= Nt(St−1at−1 +Kt−1yt−1

)+ Kt yt

= Nt

[St−1

(St−2at−2 +Kt−2yt−2

)+Kt−1yt−1

]+ Kt yt

= Nt[St−1St−2at−2 +St−1Kt−2yt−2 +Kt−1yt−1

]+ Kt yt

= . . .

= Nt

t−1∏i=1

St−i a1 +t−1∑i=1

(i−1∏`=1

St−`

)Kt−i yt−i

+ Kt yt , (1.11)

with Nt = (I− Kt Zt ), Kt = Pt Z′t F−1

t and the convention∏i−1`=1 St−` = Ik if i −1 < 1. The

contribution of the n-th observable on the filtered state at time t can be isolated from

the previous expression in the following way

at |t = Nt

t−1∏i=1

St−i a1 +t−1∑i=1

(i−1∏`=1

St−`

)Kt−i

N∑j=1

e j e′j yt−i

+ Kt

N∑j=1

e j e′j yt

= Nt

t−1∏i=1

St−i a1 +t−1∑i=1

(i−1∏`=1

St−`

)Kt−i

N∑j=1, j 6=n

e j e′j yt−i

+ Kt

N∑j=1, j 6=n

e j e′j yt

+ Nt

t−1∑i=1

(i−1∏`=1

St−`

)kt−i ,·n yt−i ,n

+ kt ,·n yt ,n , (1.12)

where with bt ,·n and bt ,n· we denote the n-th column and row of the matrix Bt , respec-

tively, and yt ,n is the n-th component of yt ; e j with j = 1, . . . , N are the canonical basis

vectors of RN . In (1.12) we made use of the identity∑N

j=1 e j e′j = IN . The contribution

of the n-th observable yi ,N i=1,...,t on the filtered state at |t is given by

snt = Nt

t−1∑i=1

(i−1∏`=1

St−`

)kt−i ,·n yt−i ,n + kt ,·n yt ,n . (1.13)


The first moment and the variance of snt are

E [snt ] = Nt

t−1∑i=1

(i−1∏`=1

St−`

)kt−i ,·nE [yt−i ,n]+ kt ,·nE [yt ,n],

var [snt ] = Nt

t−1∑i=1,i ′=1

(i−1∏`=1

St−`

)kt−i ,·ncov[yt−i ,n , yt−i ′,n]k′

t−i ′,·n

i ′−1∏`=1

S′t−`

N′t

+ kt ,·n var [yt ,n]k′t ,·n

+ 2Nt

t−1∑i=1

(i−1∏`=1

St−`

)kt−i ,·ncov[yt−i ,n , yt ,n]k

′t ,·n . (1.14)

Note that Ft = E [vt v′t ], Pt = E [(αt −at )(αt −at )′], and Kt = Pt Z′t F−1

t are non-random

matrices. If yt ,n is stationary with mean E [yt ,n] =µy.,n and autocovariance function

cov[yt ,n , yt−h,n] = γn(h), we can rewrite (1.14) as

E [snt ] =

Nt

t−1∑i=1

(i−1∏`=1

St−`

)kt−i ,·n + kt ,·n

µy.,n ,

var [snt ] = Nt

t−1∑i=1,i ′=1

(i−1∏`=1

St−`

)kt−i ,·nγn(i − i ′)k′

t−i ′,·n

i ′−1∏`=1

S′t−`

N′t

+ kt ,·nγn(0)k′t ,·n

+ 2Nt

t−1∑i=1

(i−1∏`=1

St−`

)kt−i ,·nγn(i )k

′t ,·n ,

(1.15)

or, in a more compact form

E [snt ] =

(Wt−1ιt−1 + kt ,·n

)µy.,n ,

var [snt ] = Wt−1Γ

nt−2W′

t−1

+ kt ,·nγn(0)k′t ,·n

+ 2Wt−1γnt−1k

′t ,·n ,

where Wt−1 =[

Nt kt−1,·n ,Nt

(∏1`=1 St−`

)kt−2,·n , . . . ,Nt

(∏t−2`=1 St−`

)k1,·n

], ιt−1 is a vec-

tor of ones of length t −1, γnt−1 =

[γn(1), . . . ,γn(t −1)

]′, and

Γnt−2 =

γn(0) γn(1) · · · γn(t −2)

γn(1) γn(0) · · · γn(t −3)...

.... . .

...

γn(t −2) γn(t −3) · · · γn(0)

. (1.16)

1.3. QUANTIFYING SUPERVISION 11

The distribution of the contribution of the n-th observable on the filtered state at

time t is

snt ∼ p

(E

[sn

t

], var

[sn

t

]), (1.17)

where p(µ,Σ) denotes the distribution function of yt ,n with mean µ and covariance

matrix Σ. The contribution of observable n on the j -th filtered state at time t is given

by snt , j = e′j sn

t with e j the j -th canonical basis vector of Rk .

Variance of filtered states explained by forecast objective

In this section we derive the variance ratio used as a measure of supervision. In

particular, we compute the fraction of the total variance of the filtered factors that is

explained by snt , the contribution of the forecast target. According to eqn. (1.11) and

assuming a1 to be a constant vector (typically a1 =µ= E [αt ] for a stationary system),

the variance of at |t can be written as

var [at |t ] = var

[Nt

t−1∑i=1

Bit yt−i

]+ var

[Kt yt

]+2cov

[Nt

t−1∑i=1

Bit yt−i , Kt yt

]

= Nt

t−1∑i=1, j=1

Bit cov

[yt−i ,yt− j

](B j

t )′N′t

+ Kt var[yt

]K′

t +2Nt

t−1∑i=1

Bit cov

[yt−i ,yt

]K′

t ,

where Bit =

(∏i−1`=1 St−`

)Kt−i and as in the previous section St = Tt −Kt Zt . If yt is

stationary, we can write

var[at |t

] = Nt

t−1∑i=1, j=1

BitΣ(i − j )(B j

t )′N′t

+ KtΣ(0)K′t +2Nt

t−1∑i=1

BitΣ(i )K′

t , (1.18)

where Σ(i − j ) = cov(yt−i ,yt− j ).

Notice that, since the sequence of filtered states depends on the initial values of

the filter, so does the sequence sni i=1,...,t . As a consequence, it is not a stationary

and ergodic sequence. Its variance changes in time and in order to estimate it, we

first need to estimate the autocovariance function of the sequence of observations

yi i=1,...,t and the parameters of the system and then evaluate expressions (1.16) and

(1.18).

The variance of the j -th filtered factor explained by the n-th variable can then be


assessed by means of the ratio

r j ,nt =

e′j var[

snt

]e j

e′j var[at |t

]e j

, (1.19)

where e j is the j -th canonical basis vector of Rk , as before. This quantity can be

estimated by consistently replacing the data-generating parameters with consistent

estimates and the autocovariances of yt by their sample counterparts (under the con-

dition of ergodic stationarity). Note that the variance ratio has the same expression

also when adding a constant ct to the state equation.

1.4 Computational aspects

The objective of this study is to determine the forecasting power of the supervised fac-

tor model (1.1)-(1.2). The forecast performance is based on out-of-sample forecasts

for which a rolling window of fixed size is used for the estimation of the parameters.

The log-likelihood is maximized for each estimation window.

Estimation method

The parameters of the state-space model are estimated by maximum likelihood. The

likelihood is delivered by the Kalman filter. We employ the univariate Kalman filter

derived in Koopman and Durbin (2000) as we assume a diagonal covariance matrix

for the innovations in the measurement equation. The maximum of the likelihood

function has no explicit form solution and numerical methods have to be employed.

We make use of the following two algorithms.

• CMA-ES. Covariance Matrix Adaptation Evolution Strategy, see Hansen and

Ostermeier (1996)1. This is a genetic algorithm that samples the parameter

space according to a Gaussian search distribution which changes according to

where the best solutions are found in the parameter space;

• BFGS. Broyden-Fletcher-Goldfarb-Shanno, see for instance Press, Teukolsky,

Vetterling, and Flannery (2002). This algorithm belongs to the class of quasi-

Newton methods and requires the computation of the gradient of the function

to be minimized.

The CMA-ES algorithm performs very well when no good initial values are available

but it is slower to converge than the BFGS routine. The BFGS algorithm, on the

other hand, requires good initial values but converges considerably faster than the

CMA-ES algorithm (once good initial values have been obtained). Hence, we use

1See https://www.lri.fr/~hansen/cmaesintro.html for references and source codes. The au-thors provide C source code for the algorithm which can be easily converted into C++ code.

https://www.lri.fr/~hansen/cmaesintro.html

1.5. EMPIRICAL APPLICATION 13

the CMA-ES algorithm to find good initial values and then the BFGS one to perform

the minimizations with the different rolling windows of data. We use algorithmic (or

automatic) differentiation2 to compute gradients. We make use of the ADEPT C++

library, see Hogan (2013)3. The advantage of using algorithmic differentiation over

finite differences is twofold: increased speed and elimination of approximation errors

in the computation of the gradient.

Speed improvements

To gain speed we chose C++ as the programming language, using routines from the

Numerical Recipes, Press et al. (2002) 4. We compile and run the executables on a

Linux 64-bit operating system using the GCC compiler 5. We use Open MPI 1.6.4

(Message Passing Interface) with the Open MPI C++ wrapper compiler mpic++ to

parallelise the maximum likelihood estimations 6. We compute gradients using the

ADEPT library for algorithmic differentiation, see Hogan (2013).

1.5 Empirical application

We wish to assess the forecasting performance of model (1.1)-(1.2). We fix the number

of latent factors at 1, 2, and 3 for the models involving factors7. The complete sample

size is T = 617, the rolling window for the parameter estimation has size R = 306, and

the number of forecasts is S = 300.

Data

We use the Jurado, Ludvigson and Ng dataset as used in Jurado et al. (2015) adding

two more variables, namely real disposable income (RDI) and personal consumption

expenditure excluding food and energy (PCE) and removing the Index of Aggregate

Weekly Hours (BLS). The resulting dataset comprises 132 variables. We have applied

the same transformations as in Jurado et al. (2015) to achieve stationarity for the

series in common with this dataset. For RDI and PCE we used the same transforma-

tions used for the personal income (PI) and for the personal consumption deflator

(PCEd), respectively. Details on the Jurado, Ludvigson and Ng dataset used in Ju-

rado et al. (2015) are provided by the authors at http://www.econ.nyu.edu/user/

2See for instance Verma (2000) for an introduction to algorithmic differentiation.3For a user guide see http://www.cloud-net.org/~clouds/adept/adept_documentation.

pdf.4See Aruoba and Fernández-Villaverde (2014) for a comparison of different programming languages

in economics and Fog (2006) for many suggestions on how to optimize software in C++.5See http://gcc.gnu.org/onlinedocs/ for more information on the Gnu Compiler Collection,

GCC.6See http://www.open-mpi.org/ for more details on Open MPI and Karniadakis (2003) for a review

of parallel scientific computing in C++ and MPI.7See below for more details on the choice of the number of factors.

http://www.econ.nyu.edu/user/ludvigsons/data.htm



http://www.cloud-net.org/~clouds/adept/adept_documentation.pdf






http://gcc.gnu.org/onlinedocs/



http://www.open-mpi.org/





ludvigsons/data.htm.

The details for the time series added to the Jurado, Ludvigson and Ng dataset are

the following:

• PCE. Series ID: DPCCRC1M027SBEA, Title: Personal consumption expendi-

tures excluding food and energy, Source: U.S. Department of Commerce: Bu-

reau of Economic Analysis, Release: Personal Income and Outlays, Units: Bil-

lions of Dollars, Frequency: Monthly, Seasonal Adjustment: Seasonally Adjusted

Annual Rate, Notes: BEA Account Code: DPCCRC1, For more information about

this series see http://www.bea.gov/national/.

• RDI. Series ID: DSPIC96, Title: Real Disposable Personal Income, Source: U.S.

Department of Commerce: Bureau of Economic Analysis, Release: Personal In-

come and Outlays, Units: Billions of Chained 2009 Dollars, Frequency: Monthly,

Seasonal Adjustment: Seasonally Adjusted Annual Rate, Notes: BEA Account,

Code: A067RX1, A Guide to the National Income and Product Accounts of

the United States (NIPA) - (http://www.bea.gov/national/pdf/nipaguid.

pdf).

The RDI and PCE series have been taken from the FRED (Federal Reserve Economic

Data) database and can be downloaded from the website of the Federal Reserve Bank

of St. Louis: http://research.stlouisfed.org/fred2, Help: http://research.

stlouisfed.org/fred2/help-faq.

The macroeconomic variables selected as forecast objectives are: consumer price

index (CPI), federal funds rate (FFR), personal consumption expenditures deflator

(PCEd), producer price index (PPI), personal income (PEI), unemployment rate (UR),

industrial production (IP), real disposable income (RDI), and personal consumer

expenditures (PCE).

The observations in levels range from January 1960 to December 2011 for a total

of 624 observations, and from March 1960 to December 2011 after being transformed

to stationarity, for a total of 622 data points. The data refers to the US economy.

Selection of number of factors

The selection of the number of factors is a key aspect in dynamic/static factor mod-

els. A widely used information-criterion-based method for static factor models was

derived by Bai and Ng (2002). Under appropriate assumptions, they show that their

method can consistently identify the number of factors as both the cross-section and

the sample-size tend to infinity. The method was extended to the case of restricted

dynamic models by Bai and Ng (2007) and Amengual and Watson (2007). The Bai

and Ng (2002) criterion was found to overestimate the true number of factors in

simulation studies by e.g. Hallin and Liška (2007), who propose a new method, valid

under more general assumptions, that exploits the properties of the eigenvalues of



http://www.bea.gov/national/

http://www.bea.gov/national/pdf/nipaguid.pdf


http://research.stlouisfed.org/fred2

http://research.stlouisfed.org/fred2/help-faq



sample spectral density matrices. Alessi, Barigozzi, and Capasso (2010) follow the idea

of Hallin and Liška (2007) to improve on Bai and Ng (2002) in the less general case of

static factor models. They show using simulations that their method performs well, in

particular under large idiosyncratic disturbances. Another method for selecting the

number of factors in static approximate factor models and based on the eigenvalues

of the variance-covariance matrix of the panel of data, was proposed by Ahn and

Horenstein (2013).

We find that the Alessi et al. (2010) test is somewhat dependent on the number

and sizes of the subsamples required by the test. Similarly, the number of factors

selected using the Ahn and Horenstein (2013) eigenvalue ratio test, is somewhat

sensitive to the choice of the maximum allowed number of factors. Motivated by

this and by the empirical finding that models using a low number of factors tend to

forecast better (see e.g. Stock and Watson (2002b) for the case of output and inflation)

in this work we consider models with a fixed, low number of factors. In particular, we

consider factor models with 1, 2, and 3 factors. Increasing the number of factors was

seen not to further improve the forecasts.

Competing models

We choose different competing models widely used in the forecasting literature in

order to assess the relative forecasting performance of the supervised DFM. We divide

these models into direct multi-step and indirect (recursive) forecasting models. In

the factor models considered as well as in the principal components regressions

and partial least squares regressions we extract 1, 2, and 3 factors. In the following

we denote with h the forecast horizon, yt the forecast objective, xt = [x1t , . . . , xN

t ] an

(N ×1) vector of predictors, εt a Gaussian white noise innovation, ft = [ f 1t , . . . , f k

t ] a

(k ×1) vector of factors andΛ a matrix of factor loadings.

Direct forecasting models

The first model is the following restricted AR(p) process

yt+h = c +φ1 yt + . . .φp yt−p +εt+h . (1.20)

The second model is a restricted M A(q) process

yt+h = c +θ1εt + . . .θqεt−q +εt+h . (1.21)

Both models are estimated by maximum likelihood. The lags p and q are selected

for each estimation sample as the values that minimize the Bayesian information

criterion. In particular, we consider p, q ∈ 1,2,3.

The third model is principal component regression (PCR). In the first step, prin-

cipal components are extracted from the regressors Xt = [x1t , . . . , xN

t , yt ]; yt+h is then

regressed on them to obtain βPC R for time indexes 1 ≤ t ≤ Ti −h. In the second step,


the principal components are projected at time Ti and then multiplied by βPC R to

obtain the h-period ahead forecast.

The fourth model considered is partial least squares regression (PLSR). In the

first step, the partial least squares components ymt are computed using the forecast

target yt : h ≤ t ≤ Ti and the predictors Xt = [x1t , . . . , xN

t , yt ] with 1 ≤ t ≤ Ti −h where

M ≤ (N +1) is the number of partial least squares components and N +1 is the num-

ber of predictors, including the lagged value of the forecast objective. In the second

step, the partial least squares components ymt are regressed on the predictors Xt to

recover the coefficient vector βPLSR . Note that as the partial least squares compo-

nents are a linear combination of the regressors, the relation is exact, i.e. the residuals

from this regression are (algebraically) null. In the third step, the partial least squares

components are projected at time Ti by multiplying YTi by βPLSR . The projected

PLSR components at time Ti are then summed to obtain the h-period ahead forecast

yTi+h =∑Mm=1 ym

Ti.

The fifth direct forecasting method considered is a two-step procedure as de-

scribed in equations (1.4).

The sixth direct forecasting method is the one based on the principal components

estimation approach in Stock and Watson (2002a) to a specific version of model (1.6).

In particular we take as forecasting equation

yt+h = c +β′ft + yt +εt+h ,

xt = Λft +ηt , (1.22)

with ft = [ f 1t , . . . , f k

t ]. The predictors are in xt and are standardized for each forecasting

window. As described in Stock and Watson (2002a) the factor loadings Λ and the

factors ft for t = 1, . . . ,T can be estimated using principal components. Denote with

F = [f1, . . . , fT ] and X′ = [x1, . . . ,xT ]; the estimator of the loadings Λ is the matrix made

of the eigenvectors corresponding to the largest eigenvalues of the matrix X′X and the

factors are estimated by F = XΛ. In the present paper, we are focusing on forecasting

and hence any estimated rotation of the factors will suffice for the analysis.

Indirect forecasting models

The first model is the following AR(p) process

yt+h = c +φ1 yt+h−1 + . . .φp yt+h−p +εt+h . (1.23)

The second model is a M A(q) process

yt+h = c +θ1εt+h−1 + . . .θqεt+h−q +εt+h . (1.24)





The third indirect forecasting method is an alternative two-step procedure. We

specify a dynamic equation for the factors and a static one between the factors and

the predictors/forecast target. In particular, we allow the factors Ft ∈Rk to follow the

autoregressive dynamics:

Ft+1 = c+TFt +νt , (1.25)

where T and c are a matrix and a vector of coefficients, respectively, and νt is a vector

of disturbances with E [νtν′t ] =Σ, and a static equation is specified for the mapping

between the factors and the predictors/forecast target, such as[xt

yt

]= ZFt +εt , (1.26)

where Z is a matrix of factor loadings and εt is an innovation vector, with E [εtε′t ] =Ω.

In particular, we use the factor loading matrix (1.2). Forecasts can be constructed by

estimating the system, iterating on the factor equation and then mapping the factors

to the forecast objective using the estimated factor loadings. Assuming Gaussianity

of the idiosyncratic errors, for instance, the system (1.25-1.26) can be estimated

maximizing the likelihood delivered by the Kalman filter. In this case a forecasting

scheme would be of this type:

(i) estimation of the system parameters by maximum likelihood;

(ii) extraction of the factors using the Kalman filter;

(iii) forecasting of factors using the state equation

fT+h =[

Th

fT +h−1∑i=0

Tic

], (1.27)

where ft represent estimated factors;

(iv) the forecast is then the last element of the vector[xT+h

yT+h

]= ZfT+h . (1.28)

Finally, we compare the forecast performance of the supervised model (1.1-1.2) to its

unsupervised counterpart. Namely, in this specification the factors are first extracted

using the Kalman filter and the forecast are then obtained using the forecast equation

yt+h = c + f′tβ+γyt +ut , (1.29)

where c, β, and γ are parameters to be estimated, ut is the error term, and ft is the

vector of filtered factors.


Forecasting

Forecasting scheme

The aim is to compute the forecast of the objective variable yt at time t +h, i.e. yt+h ,

where h is the forecast lead. We consider a rolling windows scheme. The reason is

that one of the requirements for the application of the Giacomini and White (2006)

test, in case of nested models, is to use rolling windows. The variables, including the

forecast target, are made stationary according to the transformations used in Jurado

et al. (2015). We standardize the variables in the estimation windows by subtracting

the time average and dividing by the standard deviation.

We build series of forecast errors of length S for all forecast targets. The com-

plete time series is indexed Yt : t ∈ N>0, t ≤ T where T is the sample length of

the complete dataset and Yt = x1t , ..., xN

t , yt . The estimation sample takes into ac-

count observations indexed Yt : t ∈ N>0,Ti −R +1 ≤ t ≤ Ti for i ∈ N>0, i ≤ S with

T1 = R = T ∗−S −hmax +1 the index of the last observation of the first estimation

sample, which coincides with the size of the rolling window, and Ti = T1 + i for

i ∈N>0, i ≤ S and hmax is the maximum forecast lead. The forecasting strategy for

h-step ahead forecasts for the supervised factor model (1.1)-(1.2) is the following (for

the competing models the forecasting scheme is analogous), for i = 1, . . . ,S:

(i) estimation of the system parameters using information from time Ti −R +1 up

to time Ti by maximizing the log-likelihood function delivered by the Kalman

filter;

(ii) computation of the filtered state vector at time Ti , i.e. αTi |Ti (note that the last

element of αTi |Ti is yTi );

(iii) the forecast is then:

yTi+h|Ti = [01×L : 1]

[T

hαTi |Ti +

h−1∑i=0

Tic

], (1.30)

where the parameter matrices are relative to equation (1.1).

The forecasting scheme for the competing methods is analogous.

In particular, the complete sample size is T = 622, the rolling window has size

R = 311, and the number of forecasts is S = 300. The 1-step ahead forecasts range

from February 1986 to January 2011. The 12-step ahead forecasts range from January

1987 to December 2011.


Test of forecast performance

We make use of the conditional predictive ability test proposed in Giacomini and

White (2006)8 to assess the forecasting performance of the supervised factor model

(1.1)-(1.2), relative to the other forecasting methods. In particular, we use a quadratic

loss function. This test is valid also when comparing nested models, provided a rolling

scheme for parameter estimation is used. The autocorrelations of the loss differentials

are taken into account by computing Newey and West (1987) standard errors. We

follow the “rule of thumb” in Clark and McCracken (2011) and take a sample split

ratio π= SR approximately equal to one.

Empirical application results

In this subsection we present results corresponding to the empirical application. The

mean square prediction error ratios between forecasts from the supervised model

and the competing models can be found in tables 1.1-1.9 in Appendix 1.10. The

supervised factor model corresponds to equations (1.1) with discrete cosine basis as

loadings, equation (1.2). In the tables, three, two, and one stars refer to significance

levels 0.01, 0.05, and 0.10 for the null hypothesis of equal conditional predictive ability

for the Giacomini and White (2006) test. The different forecasting models are labelled

according to the following convention:

• model 1. Principal component regression (PCR);

• model 2. Partial least squares regression (PLSR);

• model 3. AR(p) direct, eqn. (1.20);

• model 4. MA(q) direct, eqn. (1.21);

• model 5. AR(p) indirect, eqn. (1.23);

• model 6. MA(q) indirect, eqn. (1.24);

• model 7. Stock and Watson two-step procedure, eqn. (1.22);

• model 8. Unsupervised factor model (1.25), and (1.26) with discrete cosine

basis factor loadings, eqn. (1.2);

• model 9. Unsupervised factor model as in Section 1.2 with discrete cosine basis

factor loadings, eqn. (1.2);

• model 10. Supervised factor model as in eqn. (1.1) with discrete cosine basis

factor loadings (1.2).

8The authors provide MATLAB codes at http://www.runshare.org/CompanionSite/site.do?siteId=116 for the test.

http://www.runshare.org/CompanionSite/site.do?siteId=116



For reasons explained in Section 1.5, we estimate the supervised and unsupervised

factor models using 1, 2, and 3 factors.

Looking at the tables 1.1-1.9, we can make the following remarks (divided with

respect to the different number of factors used):

(i) 1 factor. The supervised factor model, eqn. (1.1), in general delivers forecasts

better than or similar to the other forecasting methods. In more than 56% of the

cases the model performs better than the competing ones, in 23% equally well

and in roughly 20% of the cases it performs worse. However, of the 56% cases

in which the model performs better, 37% of them are statistically significant at

theα= 10% significance level, whereas of the 17% of cases in which it performs

worse, only 9% are statistically significant at the α = 10% significance level.

The supervised factor model offers better forecasts relative to unsupervised

ones for most targets. The model forecasts particularly well the federal funds

rate (FFR). The improvements over unsupervised factor models 7, 8, and 9, are

particularly marked for this variable;

(ii) 2 factors. In most cases the supervised factor model delivers forecasts similar

to or better than the other methods. In more than 51% of the cases the model

performs better than the competing ones, in 36% of which the differences are

statistically significant at α= 10% significance level, in 23% equally well and in

roughly 26% of the cases it performs worse, in 15% of which the differences are

statistically significant at the α= 10% significance level. The supervised factor

model forecasts particularly well the federal funds rate (FFR);

(iii) 3 factors. In most cases the supervised factor model delivers forecasts similar

to or better than the other methods. In more than 60% of the cases the model

performs better than the competing ones, in 33% of which the differences

are statistically significant at the α = 10% significance level, in 20% equally

well and in roughly 20% of the cases it performs worse, in 17% of which the

differences are statistically significant at the α = 10% significance level. The

supervised factor model forecasts particularly well the unemployment rate

(UR), personal income (PEI), and real disposable income (RDI). Improvements

over the unsupervised models 7, 8, and 9 are particularly clear for UR and RDI.

The indirect MA(q) process is hard to beat in forecasting inflation measures and

the federal funds rate at lead h = 1. For the rest of variables/leads the supervised

factor model performs well. The supervised factor model (model 10) performs well

in forecasting unemployment rate, real disposable income, and the federal funds

rate. These findings are somewhat similar to the ones in Stock and Watson (2002b),

in which it was found that the factors have more predictive power for real variables

rather than for inflation measures. The results are robust to the choice of sample split

as can be seen from tables 1.1-1.9.

1.6. SIMULATIONS 21

In table 1.10, in Appendix 1.10, are reported the ratios between the variance of the

contribution to the filtered factors of the forecast target and the total variance of the

filtered factors, equation (1.19), for all variables. We notice a positive relation between

the value of this ratio and the forecast performance of the supervised factor model.

For example CPI has a much lower impact on the filtered factors compared to FFR

and UR. A possible interpretation is that the forecast objectives which influence more

the extraction of the unobserved factors benefit more from the supervised framework.

This suggests that the supervised factors may contain additional information with

respect to unsupervised ones.

From tables 1.1-1.9 it remains unclear what the best number of factors is, in terms

of forecasting performance. The best number of factors seems to change with the

forecast target and sample split.

1.6 Simulations

We perform two simulation experiments according to two different data generating

processes (hereafter DGPs). We simulate a state-space system according to equations

(1.1) with different loading coefficients:

case 1 discrete cosine basis as loadings, equation (1.2);

case 2 random loadings, generated as independent draws from a normal distri-

bution N (0,1).

In both cases, the state vector follows a three dimensional, stable VAR(1). The first two

components of the state vector ft ,1 and ft ,2, are treated as latent factors whereas the

third component ft ,3, is regarded as the forecast objective. We simulate the system

under different correlations between the factors and the forecast objective, namely

ρ f1,y and ρ f2,y , by restricting the unconditional variance-covariance matrix of the

state vector. For pairs of indexes i ; j = 1;2 and i ; j = 2;1 we fix the correlations

ρ fi ,y = 0.5 and ρ fi , f j = 0.1 and let ρ f j ,y ∈ 0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9. We then

compute forecasts using models 8, 9, and 10 previously defined and reported again

here for convenience:

model 8. Unsupervised factor model (1.25) and (1.26) with discrete cosine basis


model 9. Unsupervised factor model as in Section 1.2 with discrete cosine basis


model 10. Supervised factor model as in eqn. (1.1) with discrete cosine basis

factor loadings (1.2).

The complete sample size in the simulations is T = 600, the rolling window has size

R = 289, and the number of forecasts is S = 300.


Simulations results

In this subsection we present results relative to the simulation exercise. The mean

square prediction error ratios between forecasts from the supervised model and the

competing models can be found in tables 1.11-1.12 in Appendix 1.10. The supervised

factor model corresponds to equations (1.1) with discrete cosine basis as loadings,

equation (1.2). Looking at tables 1.11 and 1.12, we can make the following remarks

divided according to the DGP and correlations.

I discrete cosine basis loadings

(i) varying ρ f1,y and fixed ρ f2,y . In around 50% of the cases the supervised

factor model performs better than the unsupervised counterparts (in

34% of which the difference is statistically significant at the α = 0.10

significance level), in 30% of the cases it delivers the same forecasting

performance as the other two methods and in the remaining 20% of

cases it delivers slightly worse forecasts (in 11% of which the difference is

statistically significant at the α= 0.10 significance level);

(ii) varying ρ f2,y and fixed ρ f1,y . In around 60% of the cases the supervised

factor model performs better than the unsupervised counterparts (in 30%

of which the difference is statistically significant at the α = 0.10 signifi-

cance level), in 25% of cases it delivers the same forecast performance

as the other two methods and in the remaining 15% of cases it delivers

slightly worse forecasts (in 21% of which the difference is statistically

significant at the α= 0.10 significance level).

II random loadings

(i) varying ρ f1,y and fixed ρ f2,y . In around 58% of the cases the supervised



cance level), in 19% of cases it delivers the same forecast performance

as the other two methods and in the remaining 23% of cases it delivers

slightly worse forecasts (in 42% of which the difference is statistically

significant at the α= 0.10 significance level);

(ii) varying ρ f2,y and fixed ρ f1,y . In around 75% of the cases the supervised



cance level), in 15% of cases it delivers the same forecast performance as

the other two methods and in the remaining 10% of cases it delivers slighly

worse forecasts (in 11% of which the difference is statistically significant

at the α= 0.10 significance level).

1.7. CONCLUSIONS 23

Furthermore, we notice that even with moderate levels of correlation between the

forecast objective and the factors, the supervised specification delivers on average

better forecasts with respect to the unsupervised counterparts.

1.7 Conclusions

In this paper we study the forecasting properties of a supervised factor model. In this

framework the factors are extracted conditionally on the forecast target. The model

has a linear state-space representation and standard Kalman filtering techniques

can be used. Under this setup, we propose a way to measure the contribution of the

forecast objective on the extracted factors that exploits the Kalman filter recursions.

In particular, we compute the contribution of the forecast target to the variance of

the filtered factors and find a positive correspondence between this quantity and the

forecast performance of the supervised scheme.

We assess the forecast performance of the supervised factor model with a simula-

tion study and an empirical application. The simulated data are generated according

to different levels of correlation between the forecast objective and the factors. In

the simulations experiment, we find that if the forecast objective is correlated with

the factors the supervised factor model improves, on average, forecast performance

compared to unsupervised schemes.

In the empirical application the supervised FM is used to forecast macroeconomic

variables using factors extracted from a large number of predictors. The macroeco-

nomic data are taken from the Jurado, Ludvigson and Ng dataset and FRED. We

estimate the model considering one, two, and three factors. We forecast consumer

price index (CPI), the federal funds rate (FFR), personal consumption expenditures

deflator (PCEd), the producer price index (PPI), personal income (PEI), the unem-

ployment rate (UR), industrial production (IP), the real disposable income (RDI), and

personal consumption expenditures (PCE) relative to the US economy.

We find that supervising the factor extraction can improve forecasting perfor-

mance compared to unsupervised factor models and other popular multivariate and

univariate forecasting models. For this dataset and specification the supervised factor

model outperforms partial least squares regressions and principal components re-

gressions on most targets. In forecasting inflation, both measured by consumer price

index and producer price index, M A(q) processes are difficult to beat whereas the

supervised factor model performs particularly well in forecasting the federal funds

rate, the unemployment rate, and real disposable income. These findings are similar

to the ones in Stock and Watson (2002b), in which it was found that the factors have

more predictive power for real variables rather than for inflation measures.

We find that variables which contribute more to the variance of the filtered states,

i.e. a higher r j ,nt , equation (1.19), are the ones which benefit more from the supervised

framework and vice versa. Furthermore, supervising the factor extraction leads in


most cases to improved forecasts, compared to unsupervised two-step forecasting

schemes.

1.8 Acknowledgements

This research was supported by the European Research Executive Agency in the

Marie-Sklodowska-Curie program under grant number 333701-SFM.

1.9. REFERENCES 25

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1.10 Appendix

Tables

In this section we report mean square forecast errors (MSFE) ratios corresponding to

the empirical application (see Section 1.5) and the simulation exercise (see Section

1.6). The results relative to the empirical application correspond to MSFE ratios

between model 10 and the competing models (see Section 1.5 for the description

of the different models involved) and are contained in tables 1.1-1.9. We consider

different subsamples of the dataset and estimate the factor models using 1, 2, and

3 factors. The results relative to the simulation exercise correspond to MSFE ratios

between model 10 and models 8 and 9 and are contained in tables 1.11-1.12. We

estimate the factor models using 2 factors. In the tables below, three, two, and one

stars refer to significance levels 0.01, 0.05, and 0.10 for the null hypothesis of equal

conditional predictive ability for the Giacomini and White (2006) test. In table 1.10

are reported the ratios between the variance of the contribution to the filtered factors

of the forecast target and the total variance of the filtered factors, equation (1.19), for

all variables.

1.10. APPENDIX 29

Table 1.1. MSFE ratios for whole forecast sample (1 factor).

h mod 1 mod 2 mod 3 mod 4 mod 5 mod 6 mod 7 mod 8 mod 9

1 0,98 1,04 1,11 1,34* 1,11 1,34* 0,99 0,94 0,993 0,98 0,9 0,99 0,99 0,96 1,01 0,99 0,99 0,99

CPI 6 0,99 0,82** 0,99 0,95 1 1 0,99 1 0,999 0,98 0,79** 1,01 1,01 1 1 0,98 1 0,9812 1,02 0,83*** 1,01 1,01 1 1 1,02 1 1,02

1 0,78*** 0,47*** 0,95 1 0,95 1 1,11 1,03 0,98***3 0,75*** 0,32*** 0,68*** 0,65*** 0,73*** 0,79** 0,82 0,94 0,85**

FFR 6 0,72*** 0,26*** 0,89** 0,87*** 0,94* 0,92** 0,83 0,97 0,85***9 0,86 0,37*** 1,02 1,03 0,97*** 0,98*** 0,98 0,99 112 0,75*** 0,37*** 0,91** 0,92* 0,99 0,99 0,87 1 0,89***

1 0,94** 0,99 1,15 1,31* 1,15 1,31* 0,99 0,91* 13 1 0,91 0,99 0,99 0,98 1 0,99 0,98 0,99

PCEd 6 0,99 0,83*** 0,99 0,99 0,99* 1 0,99 1 0,999 0,99 0,87 1,01 1 1 1 1 1 112 1 0,85*** 0,97 0,97** 1 1 0,98 1 0,97

1 0,91** 0,88 1,1 1,27** 1,1 1,27** 1 0,79*** 13 0,98 0,85*** 0,97 0,97 0,98 1 0,98 0,98 0,98

PPI 6 0,99 0,82 1 1 1,01 1 1 1 19 0,99 0,79 0,97 0,97 1 1 0,98 1 0,9812 1,04 0,9* 1,05 1,03 1 1 1 1 1

1 1,03 0,95 0,94* 0,94** 0,94* 0,94** 1,04 1,03 13 1,02 0,92 0,96** 0,95** 0,97* 0,97* 1,03 1,01 1,02

PEI 6 1,01 0,89* 0,99*** 0,99*** 0,99** 0,99** 1,02 1 1,029 1 0,91** 0,97 0,96 1 1 1,01 1 1,0112 0,99 0,84 0,97 0,98 1 1 0,98 1 0,97

1 0,97 1,02 0,9 0,85* 0,9 0,85* 1,06 0,66*** 13 1,04 1,01 0,99 0,94** 0,99 0,92** 1,05 0,77*** 1,06

UR 6 1,05 0,83 0,96 0,99 0,99 0,95** 1,05 0,92** 1,059 1,03 0,89 0,95 1 0,99 0,98* 0,99 0,97* 1,0112 0,98*** 0,82 0,96** 0,95** 0,99 1** 0,94 0,99** 0,95**

1 0,92* 0,99 0,9 0,87 0,9 0,87 0,97 0,94 0,99***3 1,05 0,98 1,08 1,07 1,06 0,97 1,04 0,96 1,02

IP 6 1,02 0,93 1,01 1,02 1 0,98 0,98 0,99 0,999 0,96 0,89* 0,99 0,98 0,99 0,99 0,95 0,99 0,95**12 0,97 0,82** 0,97** 0,98*** 0,99 1 0,95 1 0,95

1 0,96 0,8*** 0,99 1,01 0,99 1,01 1,01 0,95 13 0,99 0,82*** 0,99 0,99 0,98 0,99 1 1 1*

RDI 6 0,99 0,78** 0,99 0,99 1 1 1 1 19 0,99 0,87*** 1 1 1 1 1,01 1 1,0112 1 0,85*** 1 1 1 1 1 1 1

1 0,63*** 1,01 1,38** 1,72*** 1,38** 1,72*** 0,99 0,58*** 13 1,05 0,84** 1,04 1,04 0,99 1,06 1,03 1,05 1,03

PCE 6 0,99 0,75*** 0,99 0,99 0,99 1 0,99 1 0,999 0,98 0,68*** 1,01 1 1 1 1 1 112 0,97 0,75*** 1,01 1,01 1 1 1,01 1 1,01

Jurado et al. (2013) dataset. MSFE ratios between model 10 and competing models for CPI, FFR, PCEd,PPI, PEI, UR, IP, RDI, and PCE for forecasting leads h. A value lower than one indicates a lower MSFEof model 10 w.r.t. the competing models. One, two, and three stars mean .10, .05, and .01 statisticalsignificance, respectively, for the Giacomini and White (2006) test with quadratic loss function. Number offorecasts is S = 300. The number of factors in the methods involving factor models is 1. The 1-step aheadforecasts range from February 1986 to January 2011. The 12-step ahead forecasts range from January 1987to December 2011.


Table 1.2. MSFE ratios for first half of forecast sample (1 factor).


1 0,96* 0,92 1,05 1,14 1,05 1,14 0,99 0,92 13 0,99* 0,74*** 1 0,99 0,98 1,01* 0,99 1 1

CPI 6 1,01 0,63** 1 1 1 1 1 1 19 0,99 0,66*** 1 1,01 1 1 1,01 1 1,0112 1,01 0,62*** 1,01 1 1 1 1 1 1

1 0,84 0,47*** 0,9 0,92 0,9 0,92 1,09* 1,06 13 0,79** 0,32*** 0,74** 0,73** 0,77* 0,81 0,91 0,94 0,94

FFR 6 0,74 0,32*** 0,89* 0,88* 0,94* 0,94* 0,94 0,97 0,959 0,91 0,51** 1,02 1,02 0,98*** 0,98** 1,07 0,99*** 1,0712 0,77 0,39*** 0,93 0,93 0,99 0,99 0,91 0,99 0,93

1 0,92* 1 1,15** 1,3*** 1,15** 1,3*** 0,99 0,89*** 13 1,01** 0,93 0,98** 0,99** 0,98 1,01 0,98 1,01 0,98*

PCEd 6 1 0,74*** 0,96 0,96 1 1* 0,96 1 0,969 1 0,84 1,01 1,01 1 1 1,01 1 1,0112 1 0,75*** 1 1 1 1 1 1 0,99

1 0,99 0,85 1 1,2 1 1,2 1 0,9 13 0,99 0,7*** 0,92* 0,92 0,94 0,99 0,97 0,99* 0,97

PPI 6 1,01 0,79** 0,96 0,95 1 1 0,96 1 0,969 1 0,75*** 0,94 0,95 1 1 0,99 1 0,9912 0,97 0,7 0,99** 0,99** 1* 1 0,99 1 0,99*

1 1 0,91 0,96 0,97 0,96 0,97 1,01 1,01 13 1 0,98 0,98 0,98 0,99 0,99 1 1 1

PEI 6 0,99 0,94*** 0,99*** 0,99*** 0,99*** 0,99*** 1 1** 19 0,98 0,94 0,96 0,95 1 1 0,98 1 0,9912 1 0,96*** 0,99 1,02 1 1 1,04 1 1,03

1 0,95 0,97 0,91 0,92* 0,91 0,92* 1,02 0,85 13 0,98 1 1,01 1 1,03 0,99* 1 0,92 1

UR 6 1,01* 0,91 1 0,99 1,01 0,98* 1,02** 0,97* 1,01**9 1,06 0,93 1 1 1,01 1 1,01 1 112 1,01** 0,83 0,95*** 0,92** 1 1* 0,95*** 1** 0,94**

1 0,96 0,95 0,89 0,93 0,89 0,93 0,96 1,02 0,97**3 1,07* 0,79*** 1,04 1,02 1,01 1 1 0,99 0,97

IP 6 1,03 0,82* 1 1 1 0,99 1 1 0,96*9 0,98 0,8** 1,02** 1,01*** 1 1 1,02* 1 0,9812 1,05 0,78*** 0,99** 0,98*** 1 1 1,01 1 1

1 0,93 0,86*** 0,99 0,99 0,99 0,99 1 0,93 13 1 0,91 0,99 0,99 1 0,99 0,99 1 0,99

RDI 6 0,99 0,82*** 1 1 1 1 1 1 1,019 1 0,92* 1 1 1* 1** 1 1* 1**12 0,99 0,88*** 1 1 1 1 1 1 1

1 0,67 1,15 1,32 1,72*** 1,32 1,72*** 0,99* 0,59* 13 1,08 0,97 1,03 1,03 0,96 1,07 1,03 1,08 1,03

PCE 6 1 0,85*** 0,99 0,98 1* 1 0,99 1 0,999 0,99 0,83** 1,02 1** 1 1 1** 1 112 0,99 0,84*** 1,02 1,03 1* 1 1,02 1* 1,02

Jurado et al. (2013) dataset. MSFE ratios between model 10 and competing models for CPI, FFR, PCEd, PPI,PEI, UR, IP, RDI, and PCE for forecasting leads h. A value lower than one indicates a lower MSFE of model10 w.r.t. the competing models. One, two, and three stars mean .10, .05, and .01 statistical significance,respectively, for the Giacomini and White (2006) test with quadratic loss function. Number of forecastsis S′ = 150 (the first half of the S = 300 out-of-sample forecasts). The number of factors in the methodsinvolving factor models is 1. The 1-step ahead forecasts range from February 1986 to July 1998. The 12-stepahead forecasts range from January 1987 to June 1999.

1.10. APPENDIX 31

Table 1.3. MSFE ratios for second half of forecast sample (1 factor).


1 0,99 1,07 1,13 1,4 1,13 1,4 0,99 0,95 0,993 0,98 0,95 0,99 0,99 0,96 1,01 0,98 0,98 0,99

CPI 6 0,98 0,89* 0,99 0,94 1 1 0,99 1 0,999 0,98 0,83 1,01 1,01 1 1 0,98*** 1 0,98**12 1,02 0,91*** 1,01 1,02 1 1 1,03 1 1,03

1 0,72** 0,47*** 1,03 1,14 1,03 1,14 1,14 0,99 0,97***3 0,69*** 0,32*** 0,62** 0,57** 0,69* 0,77 0,72** 0,94 0,77**

FFR 6 0,69*** 0,21*** 0,9 0,85** 0,94 0,9 0,71** 0,97 0,74**9 0,81 0,27* 1,01 1,05 0,97 0,98 0,9 1 0,9212 0,73*** 0,36*** 0,89 0,91 1 1 0,83** 1,01* 0,84**

1 0,95 0,99 1,15 1,31 1,15 1,31 0,99* 0,92 0,993 0,99 0,9 1 1 0,98 0,99 0,99 0,97 0,99

PCEd 6 0,99 0,87* 1,01 1,01 0,98* 1 1 1 19 0,98* 0,88 1,01 1 1 1 1 1 112 1,01 0,91 0,95 0,95** 1* 1 0,97 1 0,97*

1 0,9** 0,89 1,13 1,29* 1,13 1,29* 0,99 0,76*** 0,993 0,98 0,89* 0,98 0,98 0,99 1 0,99 0,98 0,98

PPI 6 0,98 0,82 1,01 1,01 1,01 1 1 1 19 0,99 0,8 0,97 0,98 1 1 0,98 1 0,9712 1,06 0,95 1,06 1,04 1 1 1 1 1

1 1,08 1,01 0,91 0,9 0,91 0,9 1,08 1,06 1,013 1,04 0,85 0,92* 0,92* 0,94* 0,94* 1,08 1,03 1,05

PEI 6 1,04 0,82 0,98** 0,98** 0,98 0,98 1,05 1,01 1,049 1,03 0,87** 0,98 0,98 0,99 0,99 1,05* 1 1,04**12 0,99 0,72 0,93 0,92 1 1 0,91 1 0,91

1 1 1,07 0,88 0,79* 0,88 0,79* 1,11 0,53*** 13 1,1 1,02 0,98 0,9*** 0,95** 0,87** 1,1 0,68** 1,11

UR 6 1,07 0,78 0,94 0,99 0,98 0,93** 1,07 0,88** 1,099 1,01 0,86 0,92 0,99 0,98** 0,97* 0,98 0,95* 1,0212 0,95* 0,81 0,97 0,97 0,99 0,99* 0,94 0,99* 0,95

1 0,9* 1,01 0,9 0,84 0,9 0,84 0,98 0,9 13 1,04 1,13 1,11 1,09 1,09 0,96 1,06 0,95 1,05

IP 6 1,01 1 1,02 1,03 1,01 0,97 0,97 0,99 19 0,95 0,94 0,98* 0,97* 0,98 0,99 0,92*** 0,99 0,94**12 0,94* 0,85 0,97* 0,97 0,98 1 0,93 1 0,93

1 0,98 0,75** 1 1,03 1 1,03 1,01 0,96 13 0,99 0,76** 1 1 0,97 0,99 1,01 1 1,01*

RDI 6 0,99 0,75 0,99 0,99 0,99 1 1 1,01 19 0,98 0,83*** 1,01 1,01 1 1 1,02 1 1,0212 1 0,83*** 1 0,99 1 1 0,99 1 1

1 0,56** 0,85 1,48** 1,71** 1,48** 1,71** 1 0,56* 1,013 1,01 0,69** 1,05 1,05 1,03 1,04 1,04 1 1,04

PCE 6 0,97 0,63*** 0,99 0,99 0,99 0,99 0,99 0,99 0,999 0,97 0,53*** 1 1 1* 1 1 1 112 0,95 0,67*** 1 1 1 1 1 1 0,99

Jurado et al. (2013) dataset. MSFE ratios between model 10 and competing models for CPI, FFR, PCEd, PPI,PEI, UR, IP, RDI, and PCE for forecasting leads h. A value lower than one indicates a lower MSFE of model10 w.r.t. the competing models. One, two, and three stars mean .10, .05, and .01 statistical significance,respectively, for the Giacomini and White (2006) test with quadratic loss function. Number of forecasts isS′ = 150 (the second half of the S = 300 out-of-sample forecasts). The number of factors in the methodsinvolving factor models is 1. The 1-step ahead forecasts range from August 1998 to January 2011. The12-step ahead forecasts range from July 1999 to December 2011.


Table 1.4. MSFE ratios for whole forecast sample (2 factors).


1 0,98 1,04 1,12 1,34* 1,12 1,34* 0,99 0,99 13 0,98 0,9 0,99* 0,99* 0,96 1 0,97 1 0,98*

CPI 6 0,99 0,82** 0,99 0,95 1 1 0,98 1 0,999 0,98 0,79** 1,01 1,01 1 1 0,98 1 0,9712 1,02 0,83*** 1,01 1,01 1 1 1,03 1 1,02

1 1,37*** 0,83 1,66*** 1,75*** 1,66*** 1,75*** 1,88 1,68*** 13 0,78*** 0,33*** 0,71*** 0,67*** 0,76*** 0,82** 0,81 0,91 0,84***

FFR 6 0,74*** 0,27*** 0,92* 0,89*** 0,97* 0,95** 0,83 0,97* 0,84***9 0,87 0,37*** 1,03 1,04 0,98*** 0,99*** 0,96 0,99*** 0,9312 0,75*** 0,37*** 0,91** 0,92* 1 1 0,85 1 0,86***

1 0,94 0,99 1,15 1,31* 1,15 1,31* 1 0,95 13 1 0,91 0,99 1 0,98 1 0,99 1 0,98

PCEd 6 0,99 0,83*** 0,99 0,99 0,99* 1 0,99 1 19 0,99 0,87 1,01 1 1 1 1 1 0,9912 1 0,85*** 0,97 0,97** 1 1 0,98 1 0,97

1 0,92* 0,89 1,11 1,28** 1,11 1,28** 1,01 0,84*** 0,993 0,98 0,85*** 0,98 0,98 0,98 1 0,97 1 0,99

PPI 6 0,99 0,82 1 1 1,01 1 0,99 1 0,999 0,99 0,79 0,97 0,97 1 1 0,98 1 0,9812 1,04 0,9* 1,05 1,03 1 1 1,02 1 1,02

1 1 0,92 0,91 0,91 0,91 0,91 1,01 0,96 13 1,02 0,93 0,96* 0,96** 0,97 0,97 1,02 1,03 1,02

PEI 6 1,02 0,89 0,99*** 0,99*** 0,99* 0,99* 1,01 1,01 1,019 1 0,91** 0,97 0,97 1 1 1,01 1 1,0112 1 0,84 0,97 0,98 1 1 0,97 1 0,98

1 0,98 1,02 0,9 0,85* 0,9 0,85* 1,04 0,56*** 0,993 1,06 1,03 1,02 0,97** 1,01 0,94* 1,09 0,75*** 1,07

UR 6 1,06 0,84 0,98 1,01 1,01 0,97** 1,05 0,93** 1,069 1,04 0,9 0,96 1,01 1 0,99** 1,02 0,98** 0,9912 0,98*** 0,82 0,96** 0,95** 1 1** 0,95 0,99** 0,95**

1 0,94 1,01 0,92 0,89 0,92 0,89 0,99 1 0,95***3 1,05 0,98 1,08 1,06 1,06 0,97 1,04 0,98 1,01

IP 6 1,04 0,95 1,03 1,04 1,02 1 1,01 1,01 0,989 0,99 0,93* 1,03 1,02 1,03 1,03 0,98 1,03 112 1,01 0,86** 1,01 1,02 1,03 1,04 0,99 1,04 0,99

1 0,95 0,79*** 0,99 1,01 0,99 1,01 1 0,97 13 0,99 0,82*** 0,99 0,99 0,98 0,99 1 0,99 0,98

RDI 6 0,99 0,78** 0,99 0,99 1 1 0,98 1 0,999 0,99 0,87*** 1 1 1* 1 1 1 112 1 0,85*** 1 1 1 1 0,99 1 1

1 0,63*** 1,02 1,39** 1,74*** 1,39** 1,74*** 1 0,63** 13 1,05 0,83** 1,03 1,03 0,98 1,05 1,03 1,05 1,02

PCE 6 0,99 0,75*** 0,99 0,99 0,99* 1 0,99 1 0,979 0,98 0,68*** 1,01 1 1 1 0,98 1 112 0,97 0,75*** 1,01 1,02 1 1 1,01 1 0,99


1.10. APPENDIX 33

Table 1.5. MSFE ratios for first half of forecast sample (2 factors).


1 0,94 0,9 1,03 1,12 1,03 1,12 0,97 0,94 13 0,99* 0,74*** 1 0,98 0,98 1 0,99** 1,01 0,98

CPI 6 1,01 0,63** 1 1 1 1 1 1 0,999 0,99 0,66*** 1 1,01 1 1 1 1 0,9812 1,01 0,62*** 1,01 1 1 1 1 1 1,01

1 1,34** 0,75* 1,43** 1,47*** 1,43** 1,47*** 1,75*** 1,59*** 1,013 0,81** 0,33*** 0,76** 0,74** 0,79** 0,83* 0,89 0,91 0,93

FFR 6 0,76 0,33*** 0,91 0,9* 0,96* 0,96* 0,92 0,98* 0,939 0,92 0,51** 1,03 1,03 0,99*** 0,99** 1,08 0,99*** 1,0112 0,77 0,39*** 0,94 0,93 0,99 0,99 0,87 0,99 0,91

1 0,92* 1 1,14** 1,29*** 1,14** 1,29*** 0,99 0,9*** 13 1,01** 0,93 0,98** 0,99* 0,98 1,01 0,98 1,01** 0,98

PCEd 6 1 0,74*** 0,96 0,96 1 1* 0,96 1 0,969 1 0,84 1,01 1,01 1 1 1,01 1 1,0112 1 0,75*** 1 1 1 1 1 1** 0,99

1 0,99 0,85* 0,99 1,19 0,99 1,19 1,01 0,94 0,993 1 0,71*** 0,93 0,93 0,96 1 0,98 1 0,97

PPI 6 1,01 0,79** 0,96 0,95 1 1 0,96 1 0,969 1 0,75*** 0,94 0,95 1 1 0,99 1 0,9912 0,97 0,7 0,99** 0,99** 1 1 0,99 1 0,94

1 1 0,91 0,95 0,96 0,95 0,96 1 0,98 13 1 0,99 0,99 0,99 1 1 1,01 1 0,99

PEI 6 0,99 0,94*** 0,99*** 0,99*** 1*** 1*** 0,98 1* 19 0,98 0,94 0,96 0,95 1 1 0,99 1 0,9812 1 0,96*** 0,99 1,02 1 1 1 1 1,03

1 0,97 0,99 0,93 0,94 0,93 0,94 1,03 0,78* 13 0,98 1 1,02 1 1,04 0,99* 1 0,9* 1,01

UR 6 1,02* 0,91 1,01 0,99 1,01 0,98** 1,01** 0,97** 1,01*9 1,06 0,93 1,01 1,01 1,01 1 1,09 1 0,9512 1,01** 0,83 0,95*** 0,92** 1 1 0,98*** 1* 0,94***

1 0,95 0,94 0,88 0,92 0,88 0,92 0,95 1,03 0,96**3 1,08* 0,8*** 1,05 1,03 1,02 1,02 1,06 1,02 0,98

IP 6 1,03 0,82* 1 1 1 0,99* 1,01 1 0,979 0,97 0,8** 1,02* 1,01** 1 1 1,01 1 0,9812 1,05 0,78*** 0,99* 0,98** 1 1 1,05 1 1,02

1 0,93 0,86*** 0,99 0,99 0,99 0,99 0,99 0,94 13 1 0,91 0,99 0,99 1 0,99 0,99 1 0,96

RDI 6 0,99 0,82*** 1 1 1 1 1 1 19 1 0,92* 1 1 1* 1** 1 1* 1,0112 0,99 0,88*** 1 1 1 1 0,99 1 1,01

1 0,68 1,17 1,34 1,75*** 1,34 1,75*** 1,01** 0,65 0,99**3 1,07 0,97 1,02 1,03 0,96 1,06 1,02 1,07 1,03

PCE 6 1,01 0,85*** 0,99 0,98 1* 1 0,99 1 0,999 0,99 0,83** 1,02 1** 1 1 0,99*** 1 112 0,99 0,84*** 1,02 1,03 1 1 1,01 1 1

Jurado et al. (2013) dataset. MSFE ratios between model 10 and competing models for CPI, FFR, PCEd, PPI,PEI, UR, IP, RDI, and PCE for forecasting leads h. A value lower than one indicates a lower MSFE of model10 w.r.t. the competing models. One, two, and three stars mean .10, .05, and .01 statistical significance,respectively, for the Giacomini and White (2006) test with quadratic loss function. Number of forecasts isS′ = 150 (first half of the S = 300 out-of-sample forecasts). The number of factors in the methods involvingfactor models is 2. The 1-step ahead forecasts range from February 1986 to July 1998. The 12-step aheadforecasts range from January 1987 to June 1999.


Table 1.6. MSFE ratios for second half of forecast sample (2 factors).


1 1 1,08 1,14 1,41 1,14 1,41 1 1,01 0,993 0,97 0,95 0,99 0,99 0,96 1 0,96 1 0,98

CPI 6 0,98 0,89* 0,99 0,94 1 1 0,98 1 0,999 0,98 0,83 1,01 1,01 1 1 0,98* 1 0,97***12 1,02 0,91*** 1,01 1,02 1* 1 1,03 1 1,02

1 1,4** 0,92 2,01*** 2,23*** 2,01*** 2,23*** 2,06*** 1,8*** 0,99**3 0,73*** 0,33*** 0,65** 0,6** 0,73* 0,82 0,73** 0,92 0,76***

FFR 6 0,71** 0,22*** 0,93 0,88** 0,97 0,93 0,73** 0,97 0,74***9 0,82 0,27* 1,02 1,06 0,98* 0,98* 0,84 1 0,8512 0,73*** 0,36*** 0,89 0,91 1 1* 0,82*** 1,01 0,82***

1 0,95 0,99 1,15 1,31 1,15 1,31 1 0,97 13 1 0,9 1 1 0,98 1 0,99 1 0,98

PCEd 6 0,99 0,87* 1,01 1,01 0,98* 1 1,01 1 1,019 0,98* 0,88 1,01 1 1 1 0,99 1 0,99*12 1,01 0,91 0,95 0,95** 1* 1** 0,98 1 0,97

1 0,91* 0,9 1,15 1,31* 1,15 1,31* 1,01 0,81** 0,993 0,98 0,89* 0,98 0,99 0,99 1 0,97 1 0,99

PPI 6 0,98 0,82 1,01 1,01 1,01 1 1 1 19 0,99 0,8 0,97 0,98 1 1 0,98 1 0,9712 1,06 0,95 1,06 1,04 1 1 1,03 1 1,04

1 1 0,94 0,84 0,83 0,84 0,83 1,03 0,95 13 1,04 0,85 0,92* 0,92* 0,94* 0,94* 1,05 1,06 1,07

PEI 6 1,05 0,83 0,99** 0,99** 0,99 0,99 1,05 1,02 1,039 1,03 0,87* 0,98 0,98 1 1 1,04 1,01 1,05**12 0,99 0,72 0,93 0,92 1 1 0,92 1 0,93

1 0,99 1,06 0,87 0,78* 0,87 0,78* 1,06 0,43*** 0,993 1,15 1,06 1,02 0,94*** 0,99 0,91** 1,17 0,67** 1,12

UR 6 1,1 0,8 0,96 1,02 1 0,95** 1,08 0,9** 1,19 1,02 0,87 0,93 1,01 0,99 0,98* 0,97 0,96** 1,0312 0,96 0,82 0,97 0,97 0,99 1* 0,93** 0,99* 0,96

1 0,94 1,06 0,94 0,87 0,94 0,87 1,01** 0,98 0,94***3 1,03 1,12 1,1 1,08 1,08 0,95 1,03 0,96 1,03

IP 6 1,04 1,03* 1,05 1,06 1,04 1 1,01 1,02 0,999 1 1* 1,03 1,02 1,04 1,05 0,97 1,05 112 1 0,9 1,03 1,03 1,04 1,06 0,97 1,06 0,97

1 0,98 0,74** 0,99 1,02 0,99 1,02 1,01 0,99 13 0,99 0,76** 0,99 0,99 0,97 0,99 1 0,99 1,01

RDI 6 0,99 0,75 0,99 0,99 0,99 1 0,97 1 0,999 0,98 0,83*** 1,01 1,01 1* 1 1,01 1 0,9912 1 0,83** 1 1 1 1 1 1 0,99

1 0,56** 0,85 1,48** 1,71* 1,48** 1,71* 1 0,61 13 1 0,68** 1,04 1,04 1,03 1,04 1,04 1,04 1,01

PCE 6 0,96 0,63*** 0,99 0,99 0,98 0,99 0,99 0,99 0,959 0,97 0,53*** 1 1 1 1 0,97 1 0,9912 0,95 0,67*** 1 1 1 1 1,01 1 0,99

Jurado et al. (2013) dataset. MSFE ratios between model 10 and competing models for CPI, FFR, PCEd, PPI,PEI, UR, IP, RDI, and PCE for forecasting leads h. A value lower than one indicates a lower MSFE of model10 w.r.t. the competing models. One, two, and three stars mean .10, .05, and .01 statistical significance,respectively, for the Giacomini and White (2006) test with quadratic loss function. Number of forecastsis S′ = 150 (second half of the S = 300 out-of-sample forecasts). The number of factors in the methodsinvolving factor models is 2. The 1-step ahead forecasts range from August 1998 to January 2011. The12-step ahead forecasts range from July 1999 to December 2011.

1.10. APPENDIX 35

Table 1.7. MSFE ratios for whole forecast sample (3 factors).


1 1 1,06 1,14 1,37** 1,14 1,37** 1 1,02 1,013 0,98 0,9 0,99 0,99 0,96 1 0,97 1 0,97

CPI 6 0,99 0,83** 1 0,95 1 1 0,99 1 0,999 0,98 0,79** 1,01 1,01 1 1 0,96 1 0,9712 1,02 0,83*** 1,01 1,01 1 1 1,01 1 1,02

1 1,15 0,69** 1,39*** 1,47*** 1,39*** 1,47*** 1,32 1,61*** 0,98**3 0,79** 0,34*** 0,72** 0,69** 0,78 0,84 0,83 1,06 0,94

FFR 6 0,71** 0,26*** 0,89 0,86 0,93 0,91 0,72 1,02 0,8***9 0,85 0,36*** 1 1,02 0,96 0,96 0,92 1,02 0,92*12 0,76** 0,38*** 0,92 0,94 1,01 1,01 0,78 1,03 0,9

1 0,93 0,98 1,13 1,29* 1,13 1,29* 1 0,94 13 1 0,91 0,99 0,99 0,98 1 0,98 1 0,98

PCEd 6 0,99 0,83*** 0,99 0,99 0,99 1 1 1 1*9 0,99 0,87 1,01 1 1 1 0,99 1 112 1 0,85*** 0,97 0,97** 1* 1 0,99 1 0,98

1 0,93* 0,9 1,13 1,3** 1,13 1,3** 1,01 0,86** 13 0,98 0,85*** 0,98 0,98 0,98 1 0,97 1 0,98

PPI 6 0,99 0,82 1 1 1,01 1 0,99 1 0,999 0,99 0,79 0,97 0,97 1 1 0,98 1 0,9712 1,04 0,9* 1,05 1,03 1 1 1,06 1 1,03

1 0,97 0,9** 0,88* 0,88* 0,88* 0,88* 0,99 0,97 13 0,99 0,9*** 0,93** 0,93** 0,95* 0,95* 1 1,02 0,99

PEI 6 0,99 0,87* 0,96** 0,96* 0,97** 0,97** 0,98 1,01 0,99*9 0,99 0,9*** 0,96 0,96 0,99 0,99 0,99 1* 0,9912 0,99 0,84 0,97 0,98 1 1 0,97 0,99 0,97

1 0,89*** 0,93 0,82*** 0,78*** 0,82*** 0,78*** 0,94 0,49*** 1,013 0,98 0,95 0,94** 0,89** 0,93 0,87** 0,99 0,6*** 0,99

UR 6 0,99 0,78 0,91** 0,94 0,94 0,9* 0,99 0,76** 0,98**9 0,99 0,86 0,92* 0,96 0,95 0,94 1 0,85 0,91**12 0,97*** 0,82 0,95* 0,94** 0,99 0,99 0,97 0,94* 0,94**

1 0,92*** 0,99 0,9 0,87 0,9 0,87 0,96 1 0,97**3 1,06 0,99 1,09 1,07 1,07 0,98 1,07 1,01 1,02

IP 6 1,03 0,94 1,02 1,03 1,01 0,99 1,03 1,01 0,979 0,98 0,92 1,02 1,01 1,02 1,02 0,99 1,01 0,9812 1 0,85** 1 1 1,02 1,03 0,99 1,01 0,96

1 0,94 0,78*** 0,98 1 0,98 1 0,99 0,96 13 0,99 0,82*** 0,99 0,99 0,98 0,99* 0,99 0,99 0,98

RDI 6 0,98 0,77** 0,99 0,99 0,99 0,99 0,97 0,99 0,98*9 0,98 0,87*** 1 1 1 1 0,98 0,99 0,9912 1 0,85*** 1 1 1 1 0,99 1 0,99

1 0,64*** 1,03 1,4*** 1,75*** 1,4*** 1,75*** 1 0,65** 0,99*3 1,04 0,83** 1,03 1,03 0,98 1,05 1,02 1,05 1,01

PCE 6 0,99 0,75*** 0,99 0,99 0,99 1 0,97 1 0,979 0,98 0,68*** 1,01 1 1 1 0,98 1 0,9912 0,97 0,75*** 1,01 1,01 1 1 0,98 1 0,98



Table 1.8. MSFE ratios for first half of forecast sample (3 factors).


1 0,93 0,89 1,02 1,1 1,02 1,1 0,95* 0,92 0,993 0,99* 0,74*** 1 0,98 0,98 1 0,99** 1 0,98

CPI 6 1,01 0,63** 1* 1* 1 1 1,02 1 0,999 0,99 0,66*** 1 1,01 1 1 1 1 0,9912 1,01 0,62*** 1,01 1 1 1 1 1 1

1 1,15 0,65*** 1,23 1,27 1,23 1,27 1,24** 1,49*** 0,983 0,8 0,33*** 0,75* 0,73* 0,78 0,82 0,85 0,98 1

FFR 6 0,68* 0,3*** 0,82 0,81 0,87 0,87 0,69* 0,94 0,869 0,84 0,47** 0,94 0,94 0,9* 0,9* 0,92 0,94 0,9612 0,73* 0,37*** 0,88 0,88 0,94 0,94 0,75 0,96 0,88

1 0,9** 0,98 1,12** 1,27*** 1,12** 1,27*** 0,96* 0,88*** 0,97**3 1,01** 0,93 0,98* 0,99* 0,98 1,01 0,97 1,01** 0,98

PCEd 6 1 0,74*** 0,96 0,96 1 1 0,97 1* 0,969 1 0,84 1,01 1,01 1 1 1,01 1 1,0112 1 0,75*** 1 1 1 1 1 1 0,98

1 1 0,86 1 1,2 1 1,2 1,02 0,95 0,99**3 1,01 0,71*** 0,94 0,94 0,96 1,01 0,97 1,01 0,96

PPI 6 1,01 0,79** 0,96 0,95 1 1 0,98 1 0,979 1 0,75*** 0,94 0,95 1 1 0,99 1 0,9912 0,97 0,7 0,99** 0,99** 1 1 0,97* 1 0,93

1 0,99 0,9 0,94 0,95 0,94 0,95 1 0,98 13 1 0,98 0,98* 0,98 0,99 0,99 1,01 1 1

PEI 6 0,98 0,93*** 0,98*** 0,98*** 0,99** 0,99** 0,97* 1,01** 19 0,98 0,94 0,96 0,95 1 1 0,96 1 0,9812 0,99 0,96*** 0,99 1,02 1 1 1 1 1,02

1 0,91 0,93 0,87 0,88* 0,87 0,88* 0,97 0,7** 1,013 0,97 0,99 1,01 0,99 1,03* 0,98** 0,98 0,83** 0,98

UR 6 1 0,9 0,99 0,97 1 0,97* 1* 0,92** 0,999 1,04 0,92 0,99 0,99 1 0,99 1,04 0,97* 0,9***12 1** 0,83 0,94*** 0,92*** 0,99 0,99** 0,97*** 0,98** 0,93***

1 0,87* 0,86 0,81** 0,84* 0,81** 0,84* 0,89* 0,99 0,973 1,09 0,8** 1,06 1,04 1,03 1,02 1,05 1,02 0,98

IP 6 1,03 0,82** 1 1 1 0,99 1,01 1,01 0,979 0,97 0,8** 1,01 1,01 1 1 0,97 1 0,9712 1,05 0,78*** 0,99 0,98* 1 1 1,04 1 0,99

1 0,92 0,85*** 0,98 0,98 0,98 0,98 0,98 0,93 13 1 0,9 0,99 0,99 0,99 0,99 0,99 0,99 0,95

RDI 6 0,99 0,81*** 0,99 0,99 0,99** 0,99** 0,99 0,99** 19 0,99 0,92* 0,99 0,99 1* 1* 0,99 1* 1,0212 0,99 0,88*** 1 1 1 1 0,99 1 1

1 0,68 1,17 1,35 1,76*** 1,35 1,76*** 1,01** 0,67 13 1,07 0,97 1,02 1,03 0,96 1,06 1,03 1,06 1,02

PCE 6 1,01 0,85*** 0,99 0,98 1* 1 0,99 1 0,989 0,99 0,83** 1,02 1** 1 1 0,99 1 1,0112 0,99 0,84*** 1,02 1,03 1** 1* 0,99 1* 0,99

Jurado et al. (2013) dataset. MSFE ratios between model 10 and competing models for CPI, FFR, PCEd, PPI,PEI, UR, IP, RDI, and PCE for forecasting leads h. A value lower than one indicates a lower MSFE of model10 w.r.t. the competing models. One, two, and three stars mean .10, .05, and .01 statistical significance,respectively, for the Giacomini and White (2006) test with quadratic loss function. Number of forecasts isS′ = 150 (first half of S = 300 out-of-sample forecasts). The number of factors in the methods involvingfactor models is 3. The 1-step ahead forecasts range from February 1986 to July 1998. The 12-step aheadforecasts range from January 1987 to June 1999.

1.10. APPENDIX 37

Table 1.9. MSFE ratios for second half of forecast sample (3 factors).


1 1,02 1,11 1,17 1,45** 1,17 1,45** 1,01 1,04 1,013 0,97 0,95 0,99 0,99 0,96 1 0,96 1 0,97

CPI 6 0,98 0,89* 0,99 0,94 1 1 0,99 1 0,999 0,98 0,83 1,02 1,01 1 1 0,95 1 0,97***12 1,02 0,91*** 1,01 1,02 1 1 1,01 1 1,02

1 1,15 0,75 1,64** 1,82*** 1,64** 1,82*** 1,41*** 1,77*** 0,983 0,78 0,36*** 0,7 0,65 0,78 0,87 0,81 1,17 0,89

FFR 6 0,74 0,23** 0,97 0,92 1,01 0,97 0,76 1,13 0,74**9 0,87 0,29* 1,08 1,12 1,03 1,04 0,91 1,11 0,89*12 0,8 0,39** 0,97 1 1,09 1,09 0,82 1,11 0,92

1 0,94 0,98 1,13 1,29 1,13 1,29 1,01 0,96 13 1 0,9 1 1 0,98 1 0,99 1 0,99

PCEd 6 0,99 0,87** 1,01 1,01 0,99 1 1,01 1 1,02*9 0,98* 0,88 1,01 1 1 1 0,98 1 0,9912 1,01 0,91 0,95 0,95** 1** 1 0,99 1 0,98

1 0,92* 0,91 1,16 1,33* 1,16 1,33* 1 0,84** 13 0,98 0,89* 0,98 0,99 0,99 1 0,97 1 0,99

PPI 6 0,99 0,83 1,01 1,02 1,01 1 1 1 0,999 0,99 0,8 0,97 0,98 1 1 0,98 1 0,9712 1,06 0,95 1,06 1,04 1 1 1,08 1 1,05

1 0,95 0,9 0,8 0,79 0,8 0,79 0,98 0,96 1,013 0,99 0,81** 0,87* 0,87* 0,89* 0,89* 0,99 1,05 0,99

PEI 6 1 0,79 0,94 0,94 0,94 0,94 0,99 1,02 0,979 1,01 0,85** 0,96 0,96 0,98 0,98 1,02 1,01* 1,0112 0,99 0,73 0,93 0,92 1 1 0,94 0,99 0,91

1 0,86** 0,93 0,77** 0,68*** 0,77** 0,68*** 0,9** 0,36*** 1,013 0,98 0,91 0,87*** 0,81** 0,85*** 0,78** 0,99 0,47** 0,99

UR 6 0,98 0,71 0,85** 0,91 0,89 0,85* 0,99** 0,66** 0,97***9 0,95 0,81 0,86* 0,93 0,92 0,91 0,96 0,78 0,9212 0,95** 0,81 0,96 0,96 0,98 0,99 0,97 0,91* 0,94

1 0,94 1,06 0,95 0,88 0,95 0,88 0,99 1 0,973 1,04 1,13 1,1 1,09 1,09 0,95 1,08 1,01 1,04

IP 6 1,03 1,01 1,03 1,05 1,02 0,99 1,04 1,01 0,979 0,99 0,98 1,02 1,01 1,03 1,03 0,99 1,02 0,9812 0,98 0,88 1,01 1,01 1,02 1,04 0,97 1,01 0,95

1 0,96 0,73*** 0,98 1,01 0,98 1,01 0,99 0,98 13 0,99 0,76*** 0,99 0,99 0,97 0,99* 0,99 0,98 1

RDI 6 0,98 0,74 0,98 0,98 0,98 0,99 0,95 0,98 0,979 0,97 0,83*** 1 1 0,99 0,99 0,98 0,99 0,9712 1 0,83*** 1 0,99 1 1 0,99 1 0,98

1 0,57** 0,86 1,5** 1,73* 1,5** 1,73* 0,99 0,62 0,99*3 1 0,68** 1,04 1,04 1,03 1,04 1,01 1,04 1

PCE 6 0,96 0,63*** 0,99 0,99 0,98 0,99 0,95 0,99 0,949 0,97 0,53*** 1 1 1 1 0,96 1 0,9712 0,95 0,67*** 1 1 1 1 0,97 1* 0,96

Jurado et al. (2013) dataset. MSFE ratios between model 10 and competing models for CPI, FFR, PCEd, PPI,PEI, UR, IP, RDI, and PCE for forecasting leads h. A value lower than one indicates a lower MSFE of model10 w.r.t. the competing models. One, two, and three stars mean .10, .05, and .01 statistical significance,respectively, for the Giacomini and White (2006) test with quadratic loss function. Number of forecasts isS′ = 150 (second half of S = 300 out-of-sample forecasts. The number of factors in the methods involvingfactor models is 3. The 1-step ahead forecasts range from February 1986 to July 1998. The 12-step aheadforecasts range from January 1987 to June 1999. The 1-step ahead forecasts range from August 1998 toJanuary 2011. The 12-step ahead forecasts range from July 1999 to December 2011.


Table

1.10.Varian

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-070,0063

4,26E-08

3,75E-07

1,38E-07

1,71E-02

2,61E-05

4,16E-07

1,11E-06

2facto

rs1

stfacto

r8,23E

-060,0068

2,28E-06

1,30E-05

5,34E-07

2,89E-03

7,09E-06

3,24E-07

1,04E-07

2n

dfacto

r6,93E

-050,02

2,33E-05

0,000158,07E

-062,49E

-037,39E

-066,83E

-061,08E

-05

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rs1

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r5,63E

-060,04

1,28E-06

8,42E-06

3,93E-07

3,30E-03

8,09E-05

4,72E-08

5,55E-07

2n

dfacto

r0,00017

0,00975,37E

-050,00017

5,82E-06

2,56E-03

3,27E-03

5,53E-06

4,17E-06

3rd

factor

8,24E-06

0,093,56E

-064,14E

-062,17E

-072,68E

-038,80E

-041,10E

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1.10. APPENDIX 39

Table 1.11. MSFE ratios, cosine basis loadings.

ρ f2,y = 0.5 ρ f1,y = 0.5

ρ f1,y h mod 8 mod 9 ρ f2,y h mod 8 mod 9

1 1,01 1,02 1 0,98 0,943 0,99 1 3 0,97 1,01

0.1 6 1 1,01 0.1 6 1 1,019 1,01 1 9 0,96** 112 0,98 1 12 0,97 0,99

1 0,95** 0,73*** 1 0,98 0,64***3 1 0,99 3 0,99 0,95

0.2 6 0,98 1,01* 0.2 6 0,96*** 19 1 1 9 0,97 112 0,99 1 12 0,98 1

1 0,98 0,91* 1 0,96 13 0,98** 1 3 0,98 0,98

0.3 6 0,98 1 0.3 6 0,98 0,999 1 1 9 0,97*** 112 0,97** 1 12 0,98** 1

1 0,98 0,6*** 1 0,92** 0,85***3 0,98 0,96 3 0,99 0,98

0.4 6 0,97 1,01 0.4 6 0,98 0,999 0,96*** 1 9 0,98 112 0,98 1 12 0,98 1

1 1,01 0,99 1 0,94** 0,34***3 0,99* 1 3 0,93** 0,74**

0.5 6 0,98 1 0.5 6 0,94 0,979 0,98** 1 9 0,98 1,0312 0,99 1 12 1 1,03

1 0,97 0,75*** 1 0,92** 0,87***3 0,98 0,99 3 1 1,01

0.6 6 1 1,02 0.6 6 0,99 19 1,01 1,02 9 0,98 112 0,97** 1,02 12 0,99 1

1 0,92** 0,86*** 1 0,95** 0,993 0,97 1,01 3 0,97 1,01

0.7 6 0,98 1 0.7 6 0,99 19 0,99 1 9 0,99 1,0112 0,99 1 12 0,98** 1,01

1 0,96 1,03 1 0,98 0,86***3 1,01 1,03* 3 0,96 0,99

0.8 6 0,99 1 0.8 6 0,99 19 1,01 1 9 0,95 112 0,99** 1 12 0,98 1

1 0,96 0,81*** 1 1,03 1,033 0,98 0,97 3 1,02 1,01

0.9 6 0,98 1,03 0.9 6 1** 1,019 0,99 1,02 9 1 1,01*12 1 1 12 0,99 1*

Simulated data with cosine basis loadings. MSFE ratios between model 10 and models 8 and 9 for differentlevels of correlation between factors and forecast objective. Forecasting leads h = 1,3,6,9,12. A value lowerthan one indicates a lower MSFE of model 10 w.r.t. the competing models. One, two, and three stars mean.10, .05, and .01 statistical significance, respectively, for the Giacomini and White (2006) test with quadraticloss function. Number of forecasts is S = 300. The number of latent factors is 2. The correlation betweenthe two latent factors is ρ f1 , f2

= 0.10.


Table 1.12. MSFE ratios, random loadings.

ρ f2,y = 0.5 ρ f1,y = 0.5

ρ f1,y h mod 8 mod 9 ρ f2,y h mod 8 mod 9

1 1 0,44*** 1 1 0,963 0,98* 0,94 3 0,97 0,98

0.1 6 0,98 1,01* 0.1 6 0,98 19 0,97 1,01 9 0,96 112 0,98 1 12 0,96 1

1 1,04*** 0,9** 1 0,99 0,91***3 1,08** 1,1*** 3 0,97** 0,97

0.2 6 1,08* 1,08** 0.2 6 0,96* 0,989 1,04 1,07* 9 0,95 0,9812 1,03 1,05 12 0,96 0,98

1 0,93 0,9 1 1 0,85***3 0,97 1 3 0,99 1

0.3 6 0,98 1 0.3 6 1 19 0,98 1 9 0,98 112 0,97 1 12 0,98 1

1 1 0,28*** 1 0,99 0,89*3 0,99 0,66*** 3 0,97 0,98

0.4 6 0,97* 0,84* 0.4 6 0,97 0,999 0,97** 0,94 9 0,96* 0,9912 0,96 0,94 12 0,96* 0,99

1 0,98 0,34*** 1 0,97 0,5***3 0,98* 0,79*** 3 1 0,6***

0.5 6 0,97 0,99 0.5 6 1 0,78***9 1 1,02 9 0,99 0,88**12 1,01 1 12 0,98 0,95

1 0,97 0,87*** 1 1,01 0,32***3 1,01 1,01 3 0,99 0,72***

0.6 6 1,01 1,02 0.6 6 0,99 0,9**9 1,01 1,01 9 0,98 0,9612 0,99 0,99 12 0,98 1

1 1 0,65*** 1 1,01 0,77***3 0,99* 0,97* 3 0,98 0,96

0.7 6 0,99 1 0.7 6 0,99 0,989 0,99 1* 9 0,98 0,9712 0,98 1** 12 0,97 0,99

1 0,98 0,25*** 1 1,02 0,65***3 1,02* 0,77*** 3 1,04 0,86**

0.8 6 0,98 0,96 0.8 6 1,05 0,949 0,97 0,99 9 1,03 0,9912 0,97 0,99 12 1,03 1,04

1 0,99 0,95 1 1 0,29***3 0,99 1 3 1,01** 0,56***

0.9 6 1,01 1 0.9 6 0,99 0,839 0,98 1 9 0,97 0,9112 0,99** 1 12 0,99 0,98

Simulated data with random loadings. MSFE ratios between model 10 and models 8 and 9 for differentlevels of correlation between factors and forecast objective. Forecasting leads h = 1,3,6,9,12. A value lowerthan one indicates a lower MSFE of model 10 w.r.t. the competing models. One, two, and three stars mean.10, .05, and .01 statistical significance, respectively, for the Giacomini and White (2006) test with quadraticloss function. Number of forecasts is S = 300. The number of latent factors is 2. The correlation betweenthe two latent factors is ρ f1 , f2

= 0.10.

CH

AP

TE

R

2THE FORECASTING POWER OF THE YIELD CURVE

A SUPERVISED FACTOR MODEL APPROACH

Lorenzo Boldrini


Eric T. Hillebrand


41

42 CHAPTER 2. THE FORECASTING POWER OF THE YIELD CURVE

Abstract

We study the forecast power of the yield curve for macroeconomic time series, such

as consumer price index, personal consumption expenditures, producer price in-

dex, real disposable income, unemployment rate, and industrial production. We

employ a state-space model in which the forecasting objective is included in the state

vector. This amounts to an augmented dynamic factor model in which the factors

(level, slope, and curvature of the yield curve) are supervised for the macroeconomic

forecast target. In other words, the factors are informed about the dynamics of the

forecast objective. The factor loadings have the Nelson and Siegel (1987) structure and

we consider one forecast target at a time. We compare the forecasting performance

of our specification to benchmark models such as principal components regression,

partial least squares, and ARMA(p,q) processes. We use the yield curve data from

Gürkaynak, Sack, and Wright (2006) and Diebold and Li (2006) and macroeconomic

data from FRED. We compare the models by means of the conditional predictive

ability test of Giacomini and White (2006). We find that the yield curve has more

forecast power for real variables compared to inflation measures and that supervising

the factor extraction for the forecast target can improve forecast performance.

2.1. INTRODUCTION 43

2.1 Introduction

The forecasting power of the yield curve for macroeconomic variables has been docu-

mented in many papers, see among others Harvey (1988), Stock and Watson (1989),

Estrella and Hardouvelis (1991), and Chinn and Kucko (2010). However, the predictive

power of the yield curve has changed through the years, see for instance Giacomini

and Rossi (2006), Rudebusch and Williams (2009), and Stock and Watson (1999a)

raising doubt about its reliability as a predictor.

The aim of the present paper is twofold. The first one is to analyse the forecast

power of the yield curve for macroeconomic variables. This is carried out by com-

paring forecasting models that make use of the yield curve information to ones that

do not. We also study the stability of the yield curve as a predictor, by considering

different time spans. The second objective is to assess the forecast performance

of a supervised factor model, as proposed in Boldrini and Hillebrand (2015). This

model is a particular specification of a factor model in which the factors are extracted

conditionally on the forecast target.

There exists an extensive literature on the forecasting power of the yield curve.

One of the first works testifying the forecasting power of the yield curve for macroe-

conomic variables was Harvey (1988), who within the framework of the consumption

based asset pricing model found that the real term structure of interest rates is a

good predictor for consumption growth. Within the framework of dynamic factor

models, Stock and Watson (1989) find that two interest rate spreads, namely the

difference between the six-month commercial paper and the six-month Treasury bill

rates, and the difference between the ten-year and one-year Treasury bond rates, are

good predictors of real activity. In related papers, Bernanke (1990) and Friedman and

Kuttner (1993), using linear regressions, find that the spread between the commercial

paper rate and the Treasury bill rate is a particularly good predictor for real activity

indicators and inflation. In Estrella and Hardouvelis (1991) the authors conclude

that a positive slope of the yield curve is associated with a future increase in real

economic activity. They find that it outperforms both in-sample and out-of-sample

other variables, such as the index of leading indicators, real short-term interest rates,

lagged growth in economic activity, and lagged rates of inflation as well as survey

forecasts. Still in the framework of linear regressions Kozicki (1997) and Hamilton and

Kim (2000) confirm the predictive power of the spread for real growth and inflation.

Ang, Piazzesi, and Wei (2006) build a dynamic model for GDP growth and yields that

does not allow arbitrage and completely characterizes expectations of GDP. Contrary

to previous findings, they find that the short rate has more predictive power than any

term spread.

Diebold and Li (2006) focus on forecasting the yield curve by means of the Nelson

and Siegel (1987) (NS) model. They interpret the dynamically moving parameters

as level, slope, and curvature. Diebold et al. (2006) cast the NS model in state-space

form and analyse the correlations between the extracted factors and macroeconomic


variables. They find a strong correlation between the level factor and inflation and

between the slope factor and capacity utilization.

Giacomini and Rossi (2006) examine the stability of the forecasting power of the

yield curve for economic growth in the US economy, using forecast breakdown tests.

They find a forecast breakdown during the periods 1974-76 and in 1979-87, the Burns-

Miller and the Volcker monetary regimes respectively, whereas during 1987-2006,

corresponding to when Alan Greenspan was chairman of the FED, the yield curve

proved to be a more reliable forecaster for real growth. Similarly, Stock and Watson

(1999a) found some evidence of structural breaks in the relationship between the

slope of the yield curve and real activity during the past years.

The ability to extract information from large datasets has made factor models

an appealing tool in forecasting. Stock and Watson (1999b) and Stock and Watson

(2002a), for instance, investigate forecasts of output growth and inflation using a large

number of economic indicators, including many interest rates and yield spreads. The

advantage of factor models is that the information contained in a (potentially) large

number of predictors can be summarized in a few factors. Comprehensive surveys

on factor models can be found in Bai and Ng (2008b), Breitung and Eickmeier (2006),

and Stock and Watson (2011).

In the standard approach to factor models, the extracted factors are the same for

all the forecast targets. One of the directions the literature has taken for improving

on this approach is to select factors based on their ability to forecast a specific target.

Different methods have been proposed in the literature that address this problem.

The method of partial least squares regression (PLSR), for instance, constructs a set

of linear combinations of the inputs (predictors and forecast target) for regression,

for more details see for instance Friedman et al. (2001). Bai and Ng (2008a) proposed

performing PCA on a subset of the original predictors, selected using thresholding

rules. This approach is close to the supervised PCA method proposed in Bair et al.

(2006), that aims at finding linear combinations of the predictors that have high

correlation with the target. In particular, first a subset of the predictors is selected,

based on the correlation with the target (i.e. the regression coefficient exceeds a given

threshold), then PCA is applied on the resulting subset of variables. Bai and Ng (2009)

consider ‘boosting’ (a procedure that performs subset variable selection and coeffi-

cient shrinkage) as a methodology for selecting the predictors in factor-augmented

autoregressions. Finally, Giovannelli and Proietti (2014) propose an operational su-

pervised method that selects factors based on their significance in the regression of

the forecast target on the predictors.

Hillebrand, Huang, Lee, and Li (2012) propose a method to exploit the yield curve

information in forecasting macroeconomic variables. The model is a modified NS

factor model, where the new NS yield curve factors are supervised for a specific vari-

able to forecast. They show that it outperforms the conventional (non-supervised)

NS factor model in out-of-sample forecasting of monthly US output growth and

2.2. DYNAMIC FACTOR MODELS AND SUPERVISION 45

inflation.

In this paper we assess the forecast performance of the yield curve, using a super-

vised factor model, as presented in Boldrini and Hillebrand (2015). In the supervised

framework, the factors are informed of the forecast target (supervised) and the model

has a linear, state-space representation to which Kalman filtering techniques apply.

In particular, we select the Nelson and Siegel (1987) factor structure for the yield

curve. The latent factors in this specification are three and represent the level, slope,

and curvature of the yield curve. We consider also time variation in the factor loadings

using a specification similar to the one used in Koopman, Mallee, and Van der Wel

(2010). We include one forecast target at a time in the state vector, together with the

three NS factors. This allows us to estimate the latent factors using information also

in the forecast target, through the Kalman filter recursions. The factor extraction is

thus conditional on the forecast target.

We compare the forecasting performance of the proposed specification to that of

models that make use of the yield curve information (principal components regres-

sion, partial least squares, two-step forecasting procedures as in Stock and Watson

(2002a)) and models that do not (AR(p) and M A(q) processes). We use a rolling

windows scheme and consider both direct and indirect h-step ahead forecasts. We

compare the forecast performance of the different models by means of the Giacomini

and White (2006) test, considering different time spans. We use yield curve data from

Gürkaynak et al. (2007) and Diebold et al. (2006), and macroeconomic variables from

FRED. All the data is relative to the US economy. The selected forecast objectives are

consumer price index (CPI), personal consumption expenditures (PCE), producer

price index (PPI), real disposable income (RDI), unemployment rate (UR), and indus-

trial production (IP).

The paper is organized as follows: in Section 2.2 we introduce the supervised

factor model and relate it with other forecasting methods based on factor models; in

Section 2.3 we provide some details on the computational aspects of the analysis; in

Sections 2.4 we describe the empirical application; finally, Section 2.5 concludes.

2.2 Dynamic factor models and supervision

Let xt be the forecast objective, yt = [y1t , . . . , y N

t ] an N -dimensional vector of predic-

tors, h the forecast lead, and T the last available observation in the estimation window.

Throughout we indicate with a “caret”, estimates of scalars, vectors, or matrices.

Supervised factor model

In this section we present the supervised factor model as in Boldrini and Hillebrand

(2015), that we use to assess the forecasting power of the yield curve. Consider the


system

[yt

xt

]=

[Λ 00 1

][ft

xt

]+

[εt

0

], εt ∼ N (0,H),[

ft+1

xt+1

]= c+T

[ft

xt

]+ηt , ηt ∼ N (0,Q), (2.1)

where ft ∈Rk are latent factors,Λ is a matrix of factor loadings, T and c are a matrix

and a vector of coefficients, respectively, of suitable dimensions, εt ∈ RN and ηt ∈Rk+1 are vectors of disturbances and H and Q are their respective variance-covariance

matrices.

In this supervised framework, the forecast objective is placed in the state equation

together with the latent factors and the predictors are modelled in the measurement

equation. The intuition behind the model is that if the forecast objective is correlated

with the factors then their estimation by means of the Kalman filter will benefit

from the inclusion of the forecast target in the state vector. This follows because for a

general linear state-space system, the Kalman filter delivers the best linear predictions

of the latent states at time t , given the information from all the observables entering

the measurement equation up to and including time t . In the particular case of

Gaussian innovations, as in the system (2.1), the best linear prediction coincides

with the conditional expectation. For more details on the optimality properties of the

Kalman filter see for instance Brockwell and Davis (2009).

The forecasting scheme for this model is the following:

(i) the system parameters are estimated by maximizing the likelihood function,

delivered by the Kalman filter recursions;

(ii) given the parameter estimates, the Kalman filter is run on the data;

(iii) indicating the state vector with αt = [f′t , xt ], the forecast xT+h is then

xT+h = [0′,1]

[T

hαT +

h−1∑i=0

Tic

],

where αT represents the filtered αT .

Note that the last element of the filtered state vector at time t corresponds to xt .

To be able to compute direct h-period ahead forecasts, the state-space form (2.1)

can be modified extending the state vector by including h −1 lags of the factors. The

system (2.1) thus becomes


[yt

xt

]=

[Λ(N )×(K−1) 0N×1 0N×K (h−1)

01×(K−1) 1 01×K (h−1)

]

ft

xt...

ft−(h−1)

xt−(h−1)

+

[εt

0

],

ft+1

xt+1...

ft−h

xt−h

=

[c f

0K×(h−1)

]+

[0K×(h−1) T f

I(h−1)×(h−1) 0(h−1)×K

]

ft

xt...

ft−(h−1)

xt−(h−1)

+

[IK×K

0(h−1)×K

]ηt ,

(2.2)

where T f is a matrix of parameters relating the state vector at times t and t +h, c f

is a vector of parameters, and ηt is the same as in equation (2.1). The state vector

can be recognized to be a V AR(1) representation of a restricted V AR(h −1). For the

representation of V AR(p) processes in state-space form see for instance Aoki (1990).

The forecasting scheme for the direct approach is analogous to the previous one.

Two-step procedure

Forecasting using dynamic factor models (DFM hereafter) is often carried out in a

two-step procedure as in Stock and Watson (2002a). Consider the model

xt+h = β(L)′ft +γ(L)xt +εt+h , (2.3)

yt ,i = λi (L)ft +ηt ,i , (2.4)

with i = 1, . . . , N and where ft = ( ft ,1, . . . , ft ,k ) are k latent factors, ηt = [ηt ,i , . . . ,ηt ,N ]′

and εt are idiosyncratic disturbances, β(L) = ∑qj=0β j+1L j , λi (L) = ∑p

j=0λi ( j+1)L j ,

and γ(L) = ∑sj=0γ j+1L j are finite lag polynomials in the lag operator L; β j ∈ Rk ,

γ j ∈ R, and λi j ∈ R are parameters and q, p, s ∈N0 are indices. The assumption on

the finiteness of the lag polynomials allows us to rewrite (2.3)-(2.4) as a static factor

model, i.e. a factor model in which the factors do not appear in lags:

xt+h = c +β′Ft +γ(L)xt +εt+h ,

yt = ΛFt +ηt , (2.5)

with Ft = [f′t , . . . , f′t−r ]′, r = max(q, p), the i -th row of Λ is [λi ,1, . . . ,λi ,r+1], and β =[β′

1, . . . ,β′r+1]′. The forecasting scheme is the following:


(i) extraction of the factors ft from the predictors xt modelled in equation (2.4)

using either principal components, as in Stock and Watson (2002b), or the

Kalman filter;

(ii) regression of the forecast objective on the lagged estimated factors and on its

lags according to the forecasting equation (2.3) with t = 1, ...,T −h;

(iii) the forecast is obtained from the estimated factors and regression coefficients

as

xT+h = c + β′Ft + γ(L)xT . (2.6)

Stock and Watson (2002a) developed theoretical results for this two-step procedure, in

the case of principal components estimation. In particular, they show the asymptotic

efficiency of the feasible forecasts and the consistency of the factor estimates.

In our empirical application we estimate this models using the NS factor loadings

and extract the factors using the Kalman filter, we then use an auxiliary equation to

compute the forecasts.

The difference between the supervised model (2.1) and model (2.5) is that in the

latter the extraction of the factors is independent of the forecast objective whereas in

the former one the extracted factors are informed (supervised) of the specific forecast

target considered.

Nelson-Siegel factor model

We consider here the forecast of macroeconomic variables using the term structure

of interest rates as predictor. In the 1990s, factor models have gained popularity in

modelling the yield curve, for instance with the works of Litterman and Scheinkman

(1991) and Knez, Litterman, and Scheinkman (1994), who used factor analysis to

extract common features from yield curves in different countries and periods. They

concluded that three factors explained the greater part of the variation in the yield

curve. Nelson and Siegel (1987) proposed a way to model the yield curve based on

three functions describing level, slope, and curvature, of the term structure of interest

rates.

Following Diebold et al. (2006) we cast the Nelson and Siegel (1987) term struc-

ture model in state-space form and extract three factors whose interpretation is that

of level, slope, and curvature (in the following they are labelled LVt , SLt , and CVt

respectively). The latent factors corresponding to level, slope, and curvature are unre-

stricted whereas the factor loadings are restricted to have the Nelson-Siegel structure.

This guarantees positive forward rates at all horizons and a discount factor that ap-

proaches zero as maturity increases. The supervised factor model (2.1) becomes

then


yτ1

t...

yτNt

xt

= Λ

N×3(λ) 0

N×1

01×3

1

LVt

SLt

CVt

xt

+

ετ1t...

ετNt

0

,

LVt+1

SLt+1

CVt+1

xt+1

= c+T

LVt

SLt

CVt

xt

+ηt , (2.7)

with

Λ(λ) =

11−e−τ1λ

τ1λ

1−e−τ1λ

τ1λ−e−τ1λ

11−e−τ2λ

τ2λ

1−e−τ2λ

τ2λ−e−τ2λ

......

...

11−e−τNλ

τNλ

1−e−τNλ

τNλ−e−τNλ

, (2.8)

where yτ1t , . . . , yτN

t are the yields for maturities τ1, . . . ,τN , xt (h) is the forecast ob-

jective, and h the forecast lead, LVt ,SLt ,CVt are latent factors, Λ(λ) is a matrix

of factor loadings, T and c are a matrix and a vector of coefficients, respectively, of

suitable dimensions, εt and ηt are vectors of disturbances with H and Q as respective

variance-covariance matrices. The forecast objective xt (h) is a function of the forecast

horizon, see 2.4 for more details.

Koopman et al. (2010) suggested to make the parameter λ time varying in order

to get a better fit of the yield curve. The parameter λ, or rather its logarithm, is then

included in the state vector and follows joint dynamics with the slope, level, and cur-

vature factors. We also consider a variation of (2.8) in which λ is time-varying, but in

a more parsimonious way by letting λt follow an AR(1). The supervised factor model

is then modified to haveΛ=Λ(λt ). We parameterize the model in the following way

yτ1

t...

yτNt

xt (h)

= Λ

N×3(λt ) 0

N×1

01×3

1

LVt

SLt

CVt

xt (h)

+

ετ1t...

ετNt

0

, (2.9)


LVt+1

SLt+1

CVt+1

xt+1

log (λt+1)

= c+ T

LVt

SLt

CVt

xt

log (λt )

+ ηt , (2.10)

where T is now the block matrix

T =[

T 00 φ

], (2.11)

and ηt ∼ N (0,Q) with

Q =[

Q 00 σ2

λ

]. (2.12)

The system is now non-linear in the state vector and can be estimated via the extended

Kalman filter1, see for instance Durbin and Koopman (2012). Indicate with αt =[LVt ,SLt ,CVt , xt , l og (λt )]′ the state vector. The measurement equation has then the

form

[yt

xt

]=

[Λ(λt ) 0

0 1

]LVt

SLt

CVt

xt (h)

+

ετ1t...

ετNt

0

= Zt (αt )+[εt

0

], (2.13)

where the state vector αt follows the dynamics specified in equations (2.9)-(2.12).

The extended Kalman filter is based on a local linearization of Zt (αt ) at at |t−1, an

estimate of αt based on the past observations y1, . . . , yt−1 and x1(h), . . . , xt−1(h). The

linearized model is thus[yt

xt

]= Zt (at |t−1)+ Zt (at |t−1)(αt −at |t−1)+

[εt

0

]

= dt + Zt (at |t−1)αt +[εt

0

], (2.14)

where

dt = Zt (at |t−1)− Zt (at |t−1)at |t−1, (2.15)

and

Zt (at |t−1) = ∂Zt (αt )

∂α′t

∣∣∣∣αt=at |t−1

, (2.16)

1Exact estimation procedures for non-linear systems require a major computational effort as opposedto the extended Kalman filter.

2.3. COMPUTATIONAL ASPECTS 51

Zt (αt ) =Λ(λt ) 0 ∂Zt (αt )

∂log (λt )

0 1 0

, (2.17)

with

∂Zt (αt )

∂log (λt )=

e−λt τ1 (λtτ1−eλt τ1+1)

λ2t τ1

λt SLt + e−λt τ1 (τ21λ

2t +λtτ1−eλt τ1+1)

λ2t τ1

λt CVt

...e−λt τN (λtτN−eλt τN +1)

λ2t τN

λt SLt + e−λt τN (τ2Nλ

2t +λtτN−eλt τN +1)

λ2t τN

λt CVt

.

The state-space system is then made up of the measurement equation (2.14) and

state equation (2.10).


The objective of the study is to determine the forecasting power of the yield curve

and the supervised factor model (2.1). The forecast performance is based on out-of-

sample forecasts for which a rolling window of fixed size is used for the estimation of

the parameters. The log-likelihood is maximized for each estimation window.

Estimation method

The parameters of the state-space model are estimated by maximum likelihood. The

likelihood is delivered by the Kalman filter. We employ the univariate Kalman filter as

derived in Koopman and Durbin (2000) as we assume a diagonal covariance matrix

for the innovations in the measurement equation. The maximum of the likelihood

function has no explicit form solution and numerical methods have to be employed.

We make use of the following two algorithms.



space according to a Gaussian search distribution which changes according to


• BFGS. Broyden-Fletcher-Goldfarb-Shanno, see for instance Press et al. (2002).

This algorithm belongs to the class of quasi-Newton methods. The algorithm

needs the computation of the gradient of the function to be minimized.

The CMA-ES algorithm is used for the first estimation window for each forecast target.

This algorithm is particularly useful, in this context, if no good guess of initial values

is available. We then use the BFGS algorithm for the rest of the estimation windows




as this method is substantially faster than the CMA-ES but more dependent on initial

values. We use algorithmic (or automatic) differentiation3 to compute gradients. We

make use of the ADEPT library C++ library, see Hogan (2013)4. The advantage of using

algorithmic differentiation over finite differences is twofold: increased speed and

elimination of approximation errors.

Speed improvements

To gain speed we chose C++ as the programming language, using routines from

the Numerical Recipes, Press et al. (2002) 5. We compile and run the executables

on a Linux 64-bit operating system using GCC 6. We use Open MPI 1.6.4 (Message

Passing Interface) with the Open MPI C++ wrapper compiler mpic++ to parallelise the

maximum likelihood estimations 7. We compute gradients using the ADEPT library

for algorithmic differentiation, see Hogan (2013).

2.4 Empirical application

Data

The macroeconomic variables selected as forecast objectives are: consumer price

index (CPI), personal consumer expenditures (PCE), producer price index (PPI), real

disposable income (RDI), unemployment rate (UR), and industrial production (IP).

The macroeconomic data have been taken from FRED (Federal Reserve Economic

Data)8.

Yield curve

We use two datasets for the yield-curve data. The first one is from Gürkaynak et al.

(2007)9. We skip-sample the data in order to avoid inducing artificial persistence in

the time series. We take the first yield registered in the month as the yield for that

month. This is consistent with the macroeconomic variables whose values refer to

3See for instance Verma (2000) for an introduction to algorithmic differentiation.4For a user guide see http://www.cloud-net.org/~clouds/adept/adept_documentation.

pdf.5See Aruoba and Fernández-Villaverde (2014) for a comparison of different programming languages

in economics and Fog (2006) for many suggestions on how to optimize software in C++.6See http://gcc.gnu.org/onlinedocs/ for more information on the Gnu Compiler Collection,

GCC.7See http://www.open-mpi.org/ for more details on Open MPI and Karniadakis (2003) for a review

of parallel scientific computing in C++ and MPI.8The data can be downloaded from the website of the Federal Reserve Bank of St. Louis: http://

research.stlouisfed.org/fred2, Help: http://research.stlouisfed.org/fred2/help-faq.9The Gürkaynak, Sack and Wright dataset can be downloaded from http://www.federalreserve.

gov/pubs/feds/2006.




http://www.open-mpi.org/




http://www.federalreserve.gov/pubs/feds/2006

http://www.federalreserve.gov/pubs/feds/2006


the first days of the month. The second dataset is the yield-curve data from Diebold

and Li (2006). All data refer to the US economy.

Macroeconomic variables

• CPI. Series ID: CPIAUCSL, Title: Consumer Price Index for All Urban Con-

sumers: All Items, Source: U.S. Department of Labor: Bureau of Labor Statis-

tics, Release: Consumer Price Index, Units: Index 1982-84=100, Frequency:

Monthly, Seasonal Adjustment: Seasonally Adjusted, Notes: Handbook of Meth-

ods (http://www.bls.gov/opub/hom/pdf/homch17.pdf).

• PPI. Series ID: PPIFGS, Title: Producer Price Index: Finished Goods, Source: U.S.

Department of Labor: Bureau of Labor Statistics, Release: Producer Price Index,

Units: Index 1982=100, Frequency: Monthly, Seasonal Adjustment: Seasonally

Adjusted, Series ID: UNRATE, Title: Civilian Unemployment Rate, Source: U.S.

Department of Labor: Bureau of Labor Statistics, Release: Employment Situ-

ation, Units: Percent, Frequency: Monthly, Seasonal Adjustment: Seasonally

Adjusted.

• RDI. Series ID: DSPIC96, Title: Real Disposable Personal Income, Source: U.S.

Department of Commerce: Bureau of Economic Analysis, Release: Personal In-

come and Outlays, Units: Billions of Chained 2009 Dollars, Frequency: Monthly,

Seasonal Adjustment: Seasonally Adjusted Annual Rate, Notes: BEA Account,

Code: A067RX1, A Guide to the National Income and Product Accounts of

the United States (NIPA) - (http://www.bea.gov/national/pdf/nipaguid.

pdf).

• UR. Series ID: UNRATE, Title: Civilian Unemployment Rate, Source: U.S. De-

partment of Labor: Bureau of Labor Statistics, Release: Employment Situation,

Units: Percent, Frequency: Monthly, Seasonal Adjustment: Seasonally Adjusted.

• IP. Series ID: INDPRO, Title: Industrial Production Index, Source: Federal Re-

serve Economic Data, Units: Levels, Frequency: Monthly, Index: 2007=100, Sea-

sonal Adjustment: Seasonally Adjusted, Link: http://research.stlouisfed.

org/fred2.

Competing models

We choose different competing models diffusely used in the forecasting literature, in

order to assess the relative forecasting performance of the supervised factor model.

We divide these models into direct multi-step and indirect (recursive) forecasting

models.

http://www.bls.gov/opub/hom/pdf/homch17.pdf






Direct forecasting models

The first model is the following restricted AR(p) process

xt+h(h) = c +φ1xt (h)+ . . .φp xt−p (h)+εt+h . (2.18)

The second model is a restricted M A(q) process

xt+h(h) = c +θ1εt + . . .θqεt−q +εt+h . (2.19)




The third model is principal component regression (PCR). In the first step, princi-

pal components are extracted from the regressors Yt = [yτ1t , . . . , yτN

t , xt (h)]; xt+h(h) is

then regressed on them to obtain βPC R for time indexes 1 ≤ t ≤ Ti −h. In the second

step, the principal components are projected at time Ti and then multiplied by βPC R

to obtain the h-period ahead forecast. We estimate three factors.

The fourth model considered is partial least squares regression (PLSR). In the first

step, the partial least squares components xmt are computed using the forecast target

xt (h) : h ≤ t ≤ Ti and the predictors Yt = [yτ1t , . . . , yτN

t , xt (h)] with 1 ≤ t ≤ Ti −h

where M ≤ (N +1) is the number of partial least squares components and N +1 is

the number of predictors including the lagged value of the forecast objective. In the

second step, the partial least squares components xmt are regressed on the predic-

tors Yt to recover the coefficient vector βPLS . Note that as the partial least squares

components are a linear combination of the regressors, the relation is exact, i.e. the

residuals from this regression are (algebraically) null. In the third step, the partial

least squares components are projected at time Ti by multiplying YTi by βPLS . The

projected PLS components at time Ti are then summed to obtain the h-period ahead

forecast xTi+h(h) =∑Mm=1 xTi (h)m . We estimate three PLS directions.

The fifth direct forecasting method considered is a two-step procedure as de-

scribed in equations (2.5) in which the factors are extracted using the Kalman filter

and the factor loadings have the NS structure. We refer to these factor models as

unsupervised.

The sixth model is a two-step procedure where the factors are first extracted using

the semi-parametric factor model implemented in Härdle, Majer, and Schienle (2012)

and then used to forecast with an auxiliary equation as in eqn. (2.3).

The seventh competing method is the forecast combination (CF-NS) model de-

rived in Hillebrand et al. (2012).

Finally, we compare the forecast performance of the supervised models with fixed

λ (2.7), (2.8), and time varying λ (2.14), (2.10) to their unsupervised counterparts. In

these specifications the factors are first extracted using the Kalman filter, the forecasts

are then obtained using the forecast equation

xt+h(h) = c + ftβ+γxt (h)+ut , (2.20)


where c, β, and γ are parameters to be estimated, ut is the error term, and ft is the

vector of filtered factors.

Indirect forecasting models

The first model is the following AR(p) process

xt+h(h) = c +φ1xt+h−1(h)+ . . .φp xt+h−p (h)+εt+h . (2.21)

The second model is an M A(q) process

xt+h(h) = c +θ1εt+h−1 + . . .θqεt+h−q +εt+h . (2.22)

Both models are estimated using maximum likelihood. The lags p and q are selected



Forecasting

Forecast objective

We stationarize the forecast objectives by treating them as either I (2) or I (1) variables.

In particular we follow Stock and Watson (2002b) and treat price indexes as I (2)

processes and real variables as I (1) and take the following transformations

xt (h) =k(log (X t )− log (X t−h))/h if X t is I(1),

k(log (X t )− log (X t−h))/h −k(log (X t−h − log (X t−h−1)) if X t is I(2),(2.23)

where k is a scaling factor. In particular, we have k = 1200, h ∈ 1,3,6,9,12, X t ∈C PIt ,PC Et ,PPIt ,RD It ,U Rt .

Forecasting scheme

The aim is to compute the forecast of the macro variable xt (h) at time t +h, i.e.

xt+h(h) where h is the forecast lead. We consider a rolling windows scheme. The

reason is that one of the requirements for the application of the Giacomini and White

(2006) test, in case of nested models, is to use rolling windows. We build series of

forecast errors of length S for all forecast objectives/leads. The complete time series

is indexed Yt : t ∈N>0, t ≤ T where T is the sample length of the complete dataset

and Yt = y1t , ..., y N

t , xt (h), where y it , i = 1,2, ..., N . The estimation sample takes into

account observations indexed Yt : t ∈N>0,Ti −R +1 ≤ t ≤ Ti for i ∈N>0, i ≤ S with

T1 = R = T−S−hmax+1 the index of the last observation of the first estimation sample,

which coincides with the size of the rolling window, and Ti = T1 + i for i ∈N>0, i ≤ S


and hmax is the maximum forecast lead. In particular, we chose different values of S,

and R depending on the complete sample length of the yield curve data, T . We chose

hmax = 12 for all applications. The forecasting strategy for the indirect h-step ahead

forecasts for the supervised factor model, and indirect forecasts (equations (2.7) and

(2.8)) is the following:

(i) estimate the system parameters of the dynamic factor model using information

from time Ti −R +1 up to time Ti by maximizing the log-likelihood function

delivered by the Kalman filter;

(ii) indicating withαt the state vector containing the latent factors and the forecast

target, compute the smoothed estimate at time Ti of the state vector, i.e. αTi ;

(iii) iterate h times on the filtered state at time Ti , αTi , using the estimated parame-

ters to obtain the forecast:

xT+h|T = [01×k : 1]

ThαTi +

h−1∑j=0

Tjc

; (2.24)

for i = 1, . . . ,S.

Test of forecast performance

We use the conditional predictive ability (CPA) test proposed in Giacomini and White

(2006)10 using a quadratic loss function. This test also allows to compare nested

models, provided a rolling windows scheme for parameter estimation is used. The

autocorrelations of the loss differentials are taken into account by computing Newey

and West (1987) standard errors. We follow the “rule of thumb” in Clark and Mc-

Cracken (2011) and take a sample split ratio π= SR approximately equal to one.

Results

To assess the relative forecasting performance of the supervised factor models with

respect to the competing methods, we present mean squared prediction errors ratios

between forecasts from model (2.7) and the competing models. The results are sum-

marised in tables 2.1-2.9 in Appendix 2.8. In bold are the ratios lower than 1 which

indicate a better forecasting performance of the supervised factor model, eqn. (2.7),

compared to the competing method. We consider three applications:

(i) Gürkaynak et al. (2007) yield curve data, 7 maturities corresponding to 1, 2, 3,

4, 5, 6, and 7 years, sample size T = 617, rolling window of size R = T1 = 306,

10At http://www.runshare.org/CompanionSite/site.do?siteId=116 the authors provideMATLAB codes for the test.



number of forecasts S = 300. Observations range from August 1961 to Decem-

ber 2012. The 1-step ahead forecasts range from February 1987 to January 2012.

The 12-step ahead forecasts range from January 1988 to December 2012.

(ii) Gürkaynak et al. (2007) yield curve data, 30 maturities corresponding to 1,

2, ..., 29, and 30 years, sample size T = 325, rolling window of size R = T1 =189, number of forecasts S = 125. Observations range from December 1985

to December 2012. The 1-step ahead forecasts range from September 2001 to

January 2012. The 12-step ahead forecasts range from August 2002 to December

2012.

(iii) Diebold and Li (2006) yield curve data, 17 maturities corresponding to 3, 6,

9, 12, 15, 18, 21, 24, 30, 36, 48, 60, 72, 84, 96, 108, and 120 months, sample

size T = 346, rolling window of size R = T1 = 185, number of forecasts S = 150.

Observations range from February 1972 to November 2000. The 1-step ahead

forecasts range from July 1987 to December 1999. The 12-step ahead forecasts

range from June 1988 to November 2000.

In the tables we label the different forecasting models according to the following

convention.

• model 1. Principal component regression (PCR);

• model 2. Partial least squares regression (PLSR);

• model 3. AR(p) direct, equation (2.18);

• model 4. MA(q) direct, equation (2.19);

• model 5. AR(p) indirect, equation (2.21);

• model 6. MA(q) indirect, equation (2.22);

• model 7. Supervised dynamic factor model with Nelson-Siegel factor loadings

(2.8), and direct forecasts (2.2);

• model 8. Supervised dynamic factor model with Nelson-Siegel loadings (2.7),

and indirect forecasts (2.8);

• model 9. Supervised dynamic factor model with Nelson-Siegel loadings with

time varying λ=λ(t ), and indirect forecasts (2.10), (2.9);

• model 10. Unsupervised dynamic factor model with Nelson-Siegel loadings,

(2.7), (2.8), and forecast equation 2.20;

• model 11. Unsupervised dynamic factor model with Nelson-Siegel loadings

and time varying λ=λ(t ), (2.10), (2.9), and forecast equation 2.20;


• model 12. Factors extracted using the semiparametric factor model, Härdle

et al. (2012), and forecast equation 2.20;

• model 13. Forecast combination-Nelson Siegel (CF-NS), see Hillebrand et al.

(2012).

Looking at tables 2.1-2.9 in the Appendix, we can make the following remarks (divided

with respect to the three applications):

(i) Concerning the first application we note that the supervised factor model (2.7)

delivers better forecasts for output variables, namely, real disposable income

(RDI), industrial production (IP), and unemployment rate (UR), relative to

the different inflation variables, that is, consumer price index (CPI), personal

consumption expenditures (PCE), and producer price index (PPI). With regard

to the performance relative to other forecasting schemes the supervised factor

model generally performs similarly to or better than principal components

regression (model 1), partial least squares regression (model 2), a two-step fore-

casting procedure (model 12), the unsupervised scheme (model 10), and the

CF-NS procedure (model 13), as can be seen from tables 2.1-2.3. For inflation

related variables the direct and indirect MA(q) and AR(p) forecasts are hard to

beat, in particular for forecast lead h = 1. The supervised factor model performs

generally better than the unsupervised counterpart. Allowing for dynamics in

the λ improves the forecasts only for few forecast targets.

For real disposable income (RDI), the performance of model 8, relative to uni-

variate models, in the first half of the forecasting sample (corresponding to the

first 13 years of the Greenspan monetary regime, 1987-2000), is considerably

worse than that in the second half of the forecasting sample (corresponding to

the last 6 years of the Greenspan monetary regime and the first 6 years of the

Bernanke monetary regime, 2001-2012) as can be seen from tables 2.2 and 2.3.

(ii) The results for the second application are similar to the ones for the first appli-

cation, as can be seen from tables 2.4-2.6. For real disposable income (RDI), the

performance of model 8, relative to univariate models, in the first half of the

forecasting sample (corresponding to the last 6 years of the Greenspan mon-

etary regime, 2001-2006), is considerably worse than that in the second half

of the forecasting sample (corresponding to the first 6 years of the Bernanke

monetary regime, 2006-2012) as can be seen from tables 2.5 and 2.6.

(iii) Regarding the third application the differences between inflation and output

variables is less marked, as can be seen from tables 2.7-2.9. The supervised

framework delivers forecasts similar to or better than the unsupervised frame-

works. Unemployment rate (UR) and industrial production (IP) are the two

variables for which supervision is more beneficial. Allowing for dynamics in the

λ parameter does not improve the forecasts. For real disposable income (RDI),

2.5. CONCLUSIONS 59

the performance of model 8, relative to univariate models, in the first half of

the forecasting sample (corresponding to the first 6 years of the Greenspan

monetary regime, 1987-1993), is better than that in the second half of the fore-

casting sample (corresponding to the Greenspan monetary regime going from

1994-1999) as can be seen from tables 2.8 and 2.9.

Overall, from tables 2.1-2.9 we can see that the yield curve has more predictive power

for the periods 1987-1994 (the early Greenspan monetary regime) and 2006-2012

(the early Bernanke monetary regime) as compared to the period 1994-2006 (the late

Greenspan monetary regime). The good forecasting power of the yield curve in the

early Greenspan period is consistent with the findings in Giacomini and Rossi (2006).

Note that in this study we cannot make use of the supervision measure proposed in

Boldrini and Hillebrand (2015) as the observable variables are not guaranteed to be

stationary.

2.5 Conclusions

In this paper we study the forecasting power of the yield curve for some macroecomic

variables, for the US economy, in the framework of a supervised factor model. In this

model the factors are extracted conditionally on the forecast target. The model has a

linear state-space representation and standard Kalman filtering techniques apply.

We forecast macroeconomic variables using factors extracted from the yield curve.

We use the yield curve data from Gürkaynak et al. (2007) and Diebold and Li (2006)

and macroeconomic data from FRED. We use the Nelson and Siegel factor load-

ings and allow for dynamics in the factors. We forecast consumer price index (CPI),

personal consumer expenditures (PCE), producer price index (PPI), real disposable

income (RDI), unemployment rate (UR), and industrial production (IP).

We find that supervising the factor extraction can improve the forecasting perfor-

mance of the factor model. For this dataset and specification the supervised factor

model outperforms principal components regression, and partial least squares regres-

sion on most targets, in particular for UR, RDI, and IP. In forecasting inflation, both

measured by consumer price index, and producer price index, M A(q) and AR(p)

processes are difficult to beat, especially for one-step ahead forecasts. Allowing for

dynamics in the Nelson and Siegel factor loadings generally does not improve the

forecasts. The supervised factor model performs particularly well in forecasting un-

employment rate and real disposable income. Furthermore, supervising the factor

extraction leads in most cases to improved forecasts compared to unsupervised two-

step forecasting schemes.

We find that the yield curve has forecast power for unemployment rate, real

disposable income, and industrial production but less so for inflation measures. Sim-

ilarly to Giacomini and Rossi (2006), Rudebusch and Williams (2009), and Stock and

Watson (1999a) we find that the predictive ability of the yield curve is somewhat


unstable and has changed through the years. In particular, we find that the yield

curve has more predictive power for the periods 1987-1994 (the early Greenspan

monetary regime) and 2006-2012 (the early Bernanke monetary regime) as compared

to the period 1994-2006 (the late Greenspan monetary regime). The good forecasting

power of the yield curve in the early Greenspan period is consistent with the findings

in Giacomini and Rossi (2006).

2.6 Acknowledgements

This research was supported by the European Research Executive Agency in the

Marie-Sklodowska-Curie program under grant number 333701-SFM.

2.7. REFERENCES 61

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2.8. APPENDIX 65

2.8 Appendix

Tables

In this section we report mean square forecast errors (MSFE) ratios corresponding to

the empirical application (see Section 2.4). The results correspond to MSFE ratios

between model 8 and the competing models (see Section 2.4 for the description of

the different models involved). We consider different subsamples of the dataset. In

the tables below, three, two, and one stars refer to significance levels 0.01, 0.05, and

0.10 for the null hypothesis of equal conditional predictive ability for the Giacomini

and White (2006) test.

66 CHAPTER 2. THE FORECASTING POWER OF THE YIELD CURVETab

le2.1.M

SFE

ratios

for

wh

ole

forecastsam

ple

(Gü

rkaynak

etal.(2007)yield

data,7

matu

rities).

hm

od1

mod

2m

od3

mod

4m

od5

mod

6m

od7

mod

9m

od10

mod

11m

od12

mod

13

11,02

1,021,14

1,37*1,14

1,37*1

1,011

1,011,02

0,99*3

1,161,16

1,121,23

1,121,31**

1,141,04

1,151,16

1,161,04

CP

I6

1,031,03

1,051,07

1,021,03

11,01

1,031,03

1,030,99

90,98

0,981,01

1,021

10,98**

0,960,97**

0,98*0,98

0,96**12

10,99

11

1,01*1,01*

11,01

1,010,99

10,96**

11,01

11,33***

1,69***1,33***

1,69***1*

0,81*1*

0,991,01

0,6**3

11

10,99

0,951,01

10,88***

10,99

11

PC

E6

0,990,99

0,990,99

11

10,95*

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0,991

90,98

0,980,98

0,981

10,99

10,97**

0,98*0,98

0,9912

1,011,01

1,021,03

11

11

11

1,010,99

11,02

1,011,14

1,29*1,14

1,29*1

0,131

1,011,02

0,82***3

1,061,06

1,031,1

1,041,18

1,041,03

1,041,05

1,061,02

PP

I6

1,071,07

1,071,13*

1,021,02

1,031,01

1,051,07

1,070,99

91,01

11,02

1,011*

1*1

10,98

1,011,01

0,9612

1,051,05

1,071,08*

11

1,010,99*

1,021,04

1,050,96

11

11,01

1,021,01

1,021

0,931

11

0,973

1,081,08

1,14*1,18*

1,071,16*

1,11,01

1,111,11

1,081,06**

RD

I6

0,890,89

1,021,02

1,011,03

0,910,79**

0,950,92

0,891

90,83*

0,84*0,98

0,971

1,030,87**

0,76**0,94

0,890,84*

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0,73**0,75**

0,950,96

1,061,06

0,81*0,79*

0,860,82*

0,73**1

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0,971,12**

1,07**1,12**

1,07**1

0,79***1

0,990,98

1,07***3

0,950,95

1,1*0,96

10,76

0,94**0,85

0,95**0,96

0,950,82**

UR

60,9

0,90,98

0,960,99**

0,65**0,8**

0,920,84*

0,910,9

0,65***9

0,810,81

0,83**0,89

0,81**0,68***

0,75**1,08

0,75**0,86

0,820,65***

120,81**

0,81**0,82***

0,91**0,74**

0,77***0,73**

0,850,72**

0,87*0,81**

0,72**

10,95

0,941,07

1,041,07

1,040,83**

0,991

0,980,95

13

0,84**0,85**

0,990,96

0,910,68

0,6**0,51***

0,9**0,88*

0,85**0,66*

IP6

0,83**0,84**

0,991,11

0,950,86

0,7**0,99

0,86***0,88

0,84**0,8*

90,96**

0,97**1,13

1,31,06

1,210,88**

0,880,95***

1,030,97**

1,0812

1,091,1

1,331,28

1,081,52

1,02**1,22

1,061,18

1,11,36

Gü

rkaynak

etal.(2007)

yieldd

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aturities,fro

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2.8. APPENDIX 67Ta

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art

oft

he

Gre

ensp

an(1

987-

2006

)m

on

etar

yre

gim

e.


le2.3.M

SFE

ratios

for

secon

dh

alfoffo

recastsamp

le(G

ürkayn

aketal.(2007)

yieldd

ata,7m

aturities).

hm

od1

mod

2m

od3

mod

4m

od5

mod

6m

od7

mod

9m

od10

mod

11m

od12

mod

13

11,02

1,021,14

1,391,14

1,391

1,021

1,021,02

1,033

1,191,19

1,151,27

1,141,37**

1,171,05

1,181,19

1,191,06

CP

I6

1,041,04

1,051,07

1,021,03

1,021,01

1,041,04

1,041,01

91

0,991,01

1,021*

10,98**

0,95*0,98*

0,99*1

0,9812

1,031,02

0,990,99

1,011,01*

1,011,01

1,051,02*

1,030,99

11

11,46**

1,571,46**

1,571

0,761

11

0,61*3

1,021,01

1,011,01

0,991,01

1,010,77***

1,011,01

1,020,99

PC

E6

0,990,99

0,990,99

1,011,01

11,01

0,980,98

0,991

90,98

0,980,98*

0,981

10,99

0,990,97**

0,980,98

0,9912

0,990,99

11

0,990,99

0,991

0,981

0,990,98

11,02

1,021,16

1,311,16

1,311

0,111

1,021,02

0,82**3

1,071,06

1,031,11

1,051,2

1,041,03

1,051,06

1,071,03

PP

I6

1,091,08

1,081,15*

1,021,02

1,041,02

1,071,08

1,091

91,01

1,011,02

1,011**

1**1

10,99

1,011,01

0,9712

1,061,06

1,091,1**

11

1,011

1,041,06

1,060,97

11,01

1,011

1,031

1,031

0,911

1,011,01

0,993

1,191,19

1,141,18

1,081,17

1,161,01

1,191,19

1,191,05*

RD

I6

11

0,930,92

0,950,95

0,990,65***

1,020,98

11

90,94

0,950,83

0,830,89*

0,90,95

0,61***1,01

0,910,94

0,96*12

0,890,93

0,790,84

0,970,91

0,920,63***

1,030,86

0,891,01

10,94

0,931,2**

1,08**1,2**

1,08**1

0,72***1

0,950,94

1,09***3

0,930,93

1,070,91

10,68

0,95**0,77*

0,96**0,93

0,930,74**

UR

60,93

0,940,97

11

0,620,89

0,880,95**

0,950,94

0,6***9

0,890,89

0,870,93

0,81**0,71**

0,84*1,08

0,89*0,93

0,890,65**

120,91

0,920,87**

0,97**0,73*

0,85**0,81*

0,79**0,88

0,950,92

0,74**

10,94

0,941,12

1,031,12

1,030,77**

0,981

0,960,94

0,913

0,84*0,85*

0,95*0,89

0,870,6

0,5***0,41***

0,89**0,86

0,84*0,54**

IP6

0,870,87

0,971,14

0,930,84

0,65**0,98

0,9**0,89

0,870,72***

91,07*

1,071,17

1,381,07

1,310,88**

0,861,06**

1,11,07

1,04**12

1,251,25

1,441,52

1,071,7

1,02***1,19**

1,21,29

1,251,32

Gü

rkaynak

etal.(2007)

yieldd

ata.7m

aturities,fro

m1

to7

years.MSF

Eratio

sb

etween

mo

del8

and

mo

dels

1−13

for

CP

I,PC

E,P

PI,R

DI,U

R,an

dIP

for

allforecastin

glead

sh

.Avalu

elow

erth

anon

ein

dicates

alow

erM

SFEofm

odel8

w.r.t.m

odels

1−13.O

ne,tw

o,and

three

starsm

ean.10,.05,an

d.01

statisticalsignifi

cance,resp

ectively,fo

rth

eG

iacom

inian

dW

hite

(2006)testw

ithq

uad

raticlo

ssfu

nctio

n.T

he

nu

mb

ero

fforecasts

isS ′=

150(th

eseco

nd

halfo

fthe

S=300

com

plete

forecastsam

ple).T

he

1-stepah

eadfo

recastsran

gefro

mA

ugu

st1999to

Janu

ary2012.T

he

12-stepah

eadfo

recastsran

gefro

mJu

ly2000

toD

ecemb

er2012.T

he

forecastp

eriod

corresp

on

ds

top

artofth

eG

reensp

an(1987-2006)

and

Bern

anke

(2006-2012)m

on

etaryregim

es.

2.8. APPENDIX 69Ta

ble

2.4.

MSF

Era

tio

sfo

rw

ho

lefo

reca

stsa

mp

le(G

ürk

ayn

aket

al.(

2007

)yi

eld

dat

a,30

mat

uri

ties

).

hm

od1

mod

2m

od3

mod

4m

od5

mod

6m

od7

mod

9m

od10

mod

11m

od12

mod

13

11,

021

1,11

1,39

1,11

1,39

11

10,

991,

021,

093

1,2

1,18

1,17

1,28

1,15

1,44

**0,

711

1,21

11,

21,

1C

PI

61,

081,

061,

081,

091,

061,

060,

810,

991,

080,

991,

081,

079

1,03

1,02

0,95

0,92

1,01

*1,

010,

691,

011*

11,

031,

0212

1,04

**1,

03**

0,95

0,89

1,02

1,02

0,81

1,01

1,08

*0,

99*

1,05

**1,

02

11,

011,

011,

57*

1,86

1,57

*1,

861

0,98

1,01

0,98

1,01

0,61

31,

011,

011,

011,

010,

991

0,96

*0,

871,

010,

911,

011

PC

E6

0,97

0,97

0,97

0,96

1,01

1,01

0,98

0,98

0,95

10,

971,

019

0,99

11

11

10,

95*

0,95

0,96

0,97

0,99

112

0,97

0,97

0,88

0,96

11

0,92

1,04

0,95

0,99

0,97

1

11,

031,

021,

091,

251,

091,

251

11

0,99

1,02

0,83

*3

1,07

1,06

1,03

1,13

1,02

1,22

0,68

**0,

991,

060,

981,

071,

06P

PI

61,

091,

091,

071,

171,

051,

040,

870,

981,

050,

971,

091,

059

1,02

1,03

0,98

1,01

1,01

1,01

0,76

10,

961

1,02

1,01

121,

071,

081,

06*

1,14

**1,

021,

020,

840,

991,

031,

021,

071,

02

11,

021,

020,

991,

080,

991,

081

0,99

11,

061,

021,

013

1,22

*1,

24*

1,18

1,21

1,13

1,3

1,04

0,98

1,21

11,

22*

0,97

***

RD

I6

0,96

0,98

0,94

0,95

0,96

0,99

0,75

**1,

021,

011,

010,

970,

67**

*9

0,79

*0,

830,

750,

790,

860,

870,

29*

1,01

0,8*

*1

0,8*

0,54

**12

0,66

**0,

69**

0,78

0,83

0,88

0,89

0,99

***

0,99

0,68

***

0,99

0,66

**0,

48

10,

951,

061,

14*

0,98

1,14

*0,

981

10,

99**

0,93

0,95

0,99

*3

0,93

0,98

0,95

0,78

0,88

0,62

0,37

***

0,96

0,94

1,04

0,9

0,67

UR

60,

970,

970,

931,

220,

90,

640,

34**

0,92

0,94

0,94

**0,

960,

79

0,94

*0,

930,

890,

990,

810,

780,

24*

1,01

0,88

0,96

0,95

*0,

8712

1,06

*1,

05*

0,96

1,18

0,77

1,01

*0,

440,

930,

910,

931,

071,

1

10,

991,

061,

110,

961,

110,

961

1,04

11

0,98

0,98

30,

960,

950,

990,

920,

960,

650,

8*0,

980,

960,

980,

970,

66IP

60,

890,

870,

931,

310,

930,

930,

781,

020,

880,

970,

91,

029

1,14

1,1

1,15

1,56

1,06

1,46

0,79

1,02

0,98

0,97

1,15

*1,

6512

1,37

1,27

1,44

1,66

1,04

1,84

0,9

1,03

1,04

0,96

1,36

2,13

Gü

rkay

nak

etal

.(20

07)

yiel

dd

ata.

30m

atu

riti

es,f

rom

1to

30ye

ars.

MSF

Era

tio

sb

etw

een

mo

del

8an

dco

mp

etin

gm

od

els

for

CP

I,P

CE

,PP

I,R

DI,

UR

,an

dIP

for

all

fore

cast

ing

lead

sh

.Ava

lue

low

erth

ano

ne

ind

icat

esa

low

erM

SFE

ofm

od

el8

w.r

.t.m

od

els

1−1

3.O

ne,

two,

and

thre

est

ars

mea

n.1

0,.0

5,an

d.0

1st

atis

tica

lsig

nifi

can

ce,

resp

ecti

vely

,fo

rth

eG

iaco

min

ian

dW

hit

e(2

006)

test

wit

hq

uad

rati

clo

ssfu

nct

ion

.Th

en

um

ber

off

ore

cast

sis

S=

125.

Th

e1-

step

ahea

dfo

reca

sts

ran

gefr

om

Sep

tem

ber

2001

toJa

nu

ary

2012

.Th

e12

-ste

pah

ead

fore

cast

sra

nge

fro

mA

ugu

st20

02to

Dec

emb

er20

12.T

he

fore

cast

per

iod

corr

esp

on

ds

top

arto

fth

eG

reen

span

(198

7-20

06)a

nd

Ber

nan

ke(2

006-

2012

)m

on

etar

yre

gim

es.


le2.5.M

SFE

ratios

for

firsth

alfoffo

recastsamp

le(G

ürkayn

aketal.(2007)

yieldd

ata,30m

aturities).

hm

od1

mod

2m

od3

mod

4m

od5

mod

6m

od7

mod

9m

od10

mod

11m

od12

mod

13

11

11,26

1,561,26

1,561

1,011

0,991

1,13

1,151,15

1,151,19

1,11,25*

1,11

1,141

1,151,02

CP

I6

0,98*0,98*

0,97*0,96

11

11,01

0,971

0,98*1

91,01

1,011,09

1,08**1,01

1,010,97

1,010,99

1,011,01

1,0112

1,17***1,17***

1,3*1,16*

1,05**1,05**

1,061

1,15**0,99

1,17***1,06**

11,01

1,011,66

2,27*1,66

2,27*1

0,991,01

11,01

0,583

1,021,02

1,021,02

1,011

10,85

1,020,91

1,021

PC

E6

0,970,97

0,960,95

1,011

0,970,98

0,961

0,971,01

91

1,011,01

1,011

10,96

0,940,99

0,981

112

0,980,98

0,860,96

11*

0,9*1,06*

0,981,01

0,981

11

0,991,34**

1,49***1,34**

1,49***1

1,021

0,981

0,91**3

11

0,960,98

0,931,01

0,930,99

11

11,01

PP

I6

0,980,98

0,950,97

11

0,970,99

0,970,98

0,971

91,03

1,031,03

1,041,02

1,021,05

0,971,02

1,021,03

1,0212

1,071,07

1,141,08

1,051,05

1,021

1,051

1,071,05

11,01*

1,01**1,04

1,111,04

1,111

0,991

1,091,01**

0,963

1,27*1,28

1,341,31

1,36*1,39

1,380,93**

1,260,98

1,270,82***

RD

I6

1,121,06

1,28**1,31*

1,131,23*

1,131,05

1,131,04

1,120,45***

91,04

1,011,47***

1,46***1,15***

1,27**0,21

1,021,04

1,051,04

0,3***12

0,760,75

1,64***1,54***

1,13**1,36***

0,99***0,99

0,770,98***

0,760,23***

10,99

0,981,05

0,98**1,05

0,98**1

11

0,990,99

1,083

0,94**0,93***

0,960,95

1,02***0,83

0,63**0,98

0,97*1,09*

0,95**0,89

UR

60,82***

0,81***0,85

1,060,76

0,870,84**

0,95*0,89**

0,92*0,83***

0,949

0,69***0,69***

0,790,63***

0,51*0,75**

0,64*0,94

0,91**0,95

0,7***0,8***

120,58**

0,58**0,7

0,5*0,27

0,61**0,77**

0,98**0,88**

0,93*0,59***

0,62***

11,03

1,021,07

1,061,07

1,061

0,981,02*

1,02*1,03

1,073

1,041,04

1,21,21*

1,041,35*

1,110,96

1,081,02

1,041,12

IP6

0,930,93

1,36**1,6**

1,092,09*

1,32***1

1,060,94*

0,921,09

90,82

0,841,82

1,85***1,16

3,09**1,22**

11,03

1,020,82

1,0512

0,540,61

2,942,48**

0,723,53**

0,94***1

0,651

0,550,83

Gü

rkaynak

etal.(2007)

yieldd

ata.30m

aturities,fro

m1

to30

years.MSF

Eratio

sb

etween

mo

del8

and

com

petin

gm

od

elsfo

rC

PI,P

CE

,PP

I,RD

I,UR

,and

IPfo

rall

forecastin

glead

sh

.Avalu

elow

erth

ano

ne

ind

icatesa

lower

MSF

Eo

fmo

del8

w.r.t.m

od

els1−

13.On

e,two,an

dth

reestars

mean

.10,.05,and

.01statisticalsign

ifican

ce,resp

ectively,for

the

Giaco

min

iand

Wh

ite(2006)

testw

ithq

uad

raticlo

ssfu

nctio

n.T

he

nu

mb

ero

ffo

recastsis

S ′=62

(the

first

half

of

the

S=125

com

plete

forecast

samp

le).Th

e1-step

ahead

forecasts

range

from

Septem

ber

2001to

Octo

ber

2006.Th

e12-step

ahead

forecasts

range

from

Au

gust2002

toSep

temb

er2007.T

he

forecast

perio

dco

rrespo

nd

sto

parto

fthe

Green

span

(1987-2006)an

dB

ernan

ke(2006-2012)

mo

netary

regimes.

2.8. APPENDIX 71Ta

ble

2.6.

MSF

Era

tio

sfo

rse

con

dh

alfo

ffo

reca

stsa

mp

le(G

ürk

ayn

aket

al.(

2007

)yi

eld

dat

a,30

mat

uri

ties

).

hm

od1

mod

2m

od3

mod

4m

od5

mod

6m

od7

mod

9m

od10

mod

11m

od12

mod

13

11,

040,

991

1,27

11,

271

11

11,

031,

093

1,24

1,21

1,18

1,35

1,2

1,62

0,57

*1

1,26

0,99

1,23

1,15

CP

I6

1,12

1,1

1,14

1,16

1,09

1,1

0,75

0,99

1,13

0,99

1,12

1,1

91,

041,

020,

890,

851,

01**

1,01

0,6

11*

**1

1,04

1,02

120,

99*

0,98

**0,

840,

791

10,

731,

011,

050,

990,

99*

1

11

11,

31,

071,

31,

071

0,95

10,

921

0,74

30,

970,

970,

970,

980,

92**

0,99

0,81

**0,

960,

960,

940,

970,

99P

CE

60,

980,

971

0,99

1,02

*1,

021,

020,

970,

921,

040,

981,

03*

90,

950,

96**

0,97

0,97

11

0,9*

0,99

0,88

0,95

0,95

1,01

*12

0,94

0,95

0,95

0,96

0,99

0,99

10,

990,

85*

0,94

0,94

1***

11,

051,

040,

981,

130,

981,

131

0,99

10,

991,

040,

78*

31,

121,

11,

071,

221,

081,

370,

59**

*0,

991,

090,

971,

121,

09P

PI

61,

151,

141,

13**

1,28

*1,

071,

060,

830,

981,

090,

971,

151,

079

1,01

1,04

0,97

0,99

1,01

1,01

0,68

*1,

010,

931

1,01

1,01

121,

071,

081,

031,

16**

*1,

011,

010,

780,

981,

011,

031,

071,

01

11,

021,

020,

941,

050,

941,

051

0,99

11,

041,

031,

063

1,17

1,21

1,08

1,15

0,99

1,24

0,87

1,03

1,17

1,01

1,19

1,12

RD

I6

0,86

**0,

920,

780,

780,

870,

860,

6**

10,

930,

98**

0,88

**1,

089

0,69

**0,

750,

580,

62**

*0,

75**

*0,

74**

*0,

38**

10,

7***

0,97

0,7*

*1,

03*

120,

61**

0,67

0,62

**0,

67**

0,79

***

0,75

***

0,99

***

0,99

0,64

***

0,99

0,61

**1,

06

10,

941,

111,

20,

981,

20,

981

10,

98**

*0,

91*

0,92

0,94

*3

0,92

10,

950,

720,

830,

560,

32**

*0,

960,

92*

1,02

0,88

0,6

UR

61,

021,

020,

961,

260,

940,

60,

3***

0,91

0,96

0,94

10,

669

0,98

0,97

0,91

1,06

0,87

*0,

79*

0,23

**1,

02*

0,87

0,96

0,99

0,87

121,

11**

1,1*

*0,

981,

270,

86*

1,05

**0,

43*

0,93

0,91

***

0,93

*1,

121,

16*

10,

971,

071,

130,

921,

130,

921

1,06

0,99

***

10,

960,

953

0,93

0,92

**0,

930,

850,

940,

550,

73**

0,99

0,93

0,97

0,94

0,58

IP6

0,89

0,87

0,89

1,27

0,91

0,86

0,73

1,02

0,86

*0,

980,

91,

01**

91,

17*

1,13

*1,

12**

1,54

1,06

1,4

0,77

1,02

0,98

0,96

1,18

**1,

7212

1,47

1,34

1,41

1,64

1,06

1,81

0,9

1,04

1,07

0,95

1,46

2,29

Gü

rkay

nak

etal

.(20

07)

yiel

dd

ata.

30m

atu

riti

es,f

rom

1to

30ye

ars.

MSF

Era

tio

sb

etw

een

mo

del

8an

dco

mp

etin

gm

od

els

for

CP

I,P

CE

,PP

I,R

DI,

UR

,an

dIP

for

all

fore

cast

ing

lead

sh

.Ava

lue

low

erth

ano

ne

ind

icat

esa

low

erM

SFE

ofm

od

el8

w.r

.t.m

od

els

1−1

3.O

ne,

two,

and

thre

est

ars

mea

n.1

0,.0

5,an

d.0

1st

atis

tica

lsig

nifi

can

ce,

resp

ecti

vely

,fo

rth

eG

iaco

min

ian

dW

hit

e(2

006)

test

wit

hq

uad

rati

clo

ssfu

nct

ion

.Th

en

um

ber

off

ore

cast

sis

S′ =

63(t

he

seco

nd

hal

foft

he

S=

125

com

ple

tefo

reca

stsa

mp

le).

Th

e1-

step

ahea

dfo

reca

sts

ran

gefr

om

Nov

emb

er20

06to

Jan

uar

y20

12.T

he

12-s

tep

ahea

dfo

reca

sts

ran

gefr

om

Oct

ob

er20

07to

Dec

emb

er20

12.T

he

fore

cast

per

iod

corr

esp

on

ds

top

arto

fth

eB

ern

anke

(200

6-20

12)

mo

net

ary

regi

me.


le2.7.M

SFE

ratios

for

wh

ole

forecastsam

ple

(Dieb

old

and

Li(2006)yield

data,17

matu

rities).

hm

od1

mod

2m

od3

mod

4m

od5

mod

6m

od7

mod

9m

od10

mod

11m

od12

mod

13

11

0,981,12*

1,28***1,12*

1,28***1

11

11,01

0,84***3

10,99

1,01*1,01

1,021,02

0,980,99

0,990,99

1,01*1

CP

I6

0,960,94

1,061,1

1,021,02

0,98*1,02*

0,980,96

1,070,96

90,87

0,88**0,99

1,031,02

1,020,96***

1,01**0,91***

0,87*1,02

0,89***12

0,780,85**

1,041,06

1,01**1,01**

0,980,97

0,88**0,81**

1,020,85**

11,01

1,011,32*

1,69***1,32*

1,69***1

1,031

1,010,96

0,83

1,061,05

1,061,06

1,031,02

0,991,01*

1,041,06

1,011

PC

E6

0,960,96

0,970,98

11

1,011

0,940,97

0,921

90,95

0,94*0,95

0,941

10,96**

10,92**

0,950,9**

0,9912

10,97

1,03*1,02*

1,011,01

11

0,95*1

0,97***0,97*

11

0,99**1,08

1,27*1,08

1,27*1

1,131

1,011

0,873

1,031,03

1,031,03

1,011,1

10,91**

1,021,02

1,030,98

PP

I6

1,071,06

1,081,07

1,01**1,01**

11,01

1,071,07

1,081

90,97

0,960,96

11,01

1,010,99

1,010,97

0,980,98

0,9812

0,930,94

0,970,98*

0,990,99

0,991

0,94*0,94*

0,960,96

10,99

0,98*0,97

1,030,97

1,031

0,771

0,981,03

0,943

0,940,96

1,051,05

0,971,1

0,990,98

0,990,9

1,051,04

RD

I6

0,90,92

1,12**1,15*

1,11,14*

1,040,97

0,880,8

1,121

90,88**

0,91,19*

1,18*1,14

1,24*1,02

0,970,83

0,79***1,18*

1,0312

0,770,79

1,32**1,2**

1,221,41**

1,030,97

0,77*0,73**

1,29**1,02

11

11,05

1,051,05

1,051

0,99*1

0,98*0,99

1,04**3

0,930,91

1,111,05

1,021,03

0,990,84**

0,86*0,85*

0,98**0,85**

UR

60,88

0,861,1

1,041,03**

0,951

1,07*0,82

0,820,92

0,759

0,660,65

0,930,92

0,890,84

10,92

0,610,62

0,770,65

120,52

0,50,78

0,80,79

0,761,01

0,890,5

0,50,67**

0,63

11,01

1,021,08

1,11*1,08

1,11*1

0,991

11,01

1,07**3

0,980,96

1,09*1,12*

1,04*1,02

0,98*1,05

0,870,87

1,16**0,93

IP6

0,980,96

1,241,19

1,151,14

0,970,54**

0,90,88

1,231,06

90,84

0,821

1,131,06

1,020,95

1,070,79

0,791,08

1,0312

0,62**0,64***

0,670,65

0,990,87

0,95***1,02

0,63**0,63***

0,80,96

Dieb

oldan

dLi(2006)yield

data.17

matu

ritiescorresp

ond

ing

to3,6,9,12,15,18,21,24,30,36,48,60,72,84,96,108,an

d120

mon

ths.M

SFE

ratiosb

etween

mod

el8an

dm

od

els1−

13fo

rC

PI,P

CE

,PP

I,RD

I,UR

,and

IPfo

rallfo

recasting

leads

h.A

value

lower

than

on

ein

dicates

alow

erM

SFE

ofm

od

el8w

.r.t.the

com

petin

gm

od

els.On

e,tw

o,and

three

starsm

ean.10,.05,an

d.01

statisticalsignifi

cance,resp

ectively,for

the

Giaco

min

iand

Wh

ite(2006)

testw

ithq

uad

raticlo

ssfu

nctio

n.T

he

nu

mb

ero

fforecasts

isS=

150.Th

e1-step

ahead

forecastsran

gefrom

July

1987to

Decem

ber

1999.Th

e12-step

ahead

forecastsran

gefrom

Jun

e1988

toN

ovemb

er2000.T

he

forecastp

eriod

corresp

on

ds

top

artofth

eG

reensp

an(1987-2006)

mo

netary

regimes.

2.8. APPENDIX 73Ta

ble

2.8.

MSF

Era

tio

sfo

rfi

rsth

alfo

ffo

reca

stsa

mp

le(D

ieb

old

and

Li(2

006)

yiel

dd

ata,

17m

atu

riti

es).

hm

od1

mod

2m

od3

mod

4m

od5

mod

6m

od7

mod

9m

od10

mod

11m

od12

mod

13

11

1*1,

071,

21*

1,07

1,21

*1

11

11

0,84

**3

1,02

1,03

1,05

1,06

1,05

1,08

0,99

11,

011,

021,

031,

02C

PI

60,

950,

971,

091,

131,

021,

020,

981,

010,

990,

961,

080,

999

0,81

**0,

89**

0,95

1,01

1,02

1,02

0,94

**1

0,89

*0,

82**

10,

9512

0,71

**0,

87**

1,03

1,03

0,99

***

0,99

***

0,97

0,97

0,89

***

0,76

**0,

990,

93**

*

11,

011,

011,

311,

69**

1,31

1,69

**1

1,01

11,

010,

990,

843

1,07

1,06

1,07

1,08

1,05

1,02

0,99

1*1,

051,

071,

051*

PC

E6

0,96

0,95

0,97

0,98

11

0,99

10,

940,

960,

941

90,

94*

0,93

**0,

950,

941

10,

981

0,91

**0,

940,

930,

9912

0,99

*0,

96*

1,02

1,02

1,01

1,01

11,

01*

0,93

***

0,99

**0,

99*

0,96

**

11

11,

061,

261,

061,

261

1,14

11,

010,

990,

883

1,04

1,04

1,04

1,04

1,01

1,15

1*0,

93*

1,02

1,03

1,03

0,99

PP

I6

1,09

1,09

1,1

1,08

1,01

*1,

01*

1,01

1,01

1,09

1,1

1,08

1,02

***

90,

97*

0,98

0,94

0,99

11

0,99

*1,

010,

990,

980,

971,

0112

0,91

**0,

950,

950,

95**

0,98

0,98

11

0,95

0,94

**0,

931

10,

991

0,92

0,99

0,92

0,99

10,

631

0,97

10,

963

11,

020,

960,

970,

861,

030,

950,

991,

010,

890,

971,

11R

DI

60,

920,

910,

960,

990,

971,

010,

970,

990,

840,

730,

961,

059

0,98

0,95

1,02

1,01

0,98

1,08

0,91

***

1,01

0,8

0,76

0,99

1,12

120,

90,

881,

060,

910,

951,

15*

0,92

***

1,1

0,74

0,7

1,01

1,1

11,

03**

1,02

1,12

1,13

1,12

1,13

11,

011*

*0,

981,

071,

13**

*3

0,85

***

0,82

***

1,22

***

1,16

***

1,06

0,94

1,03

0,75

**0,

74**

0,75

**1,

040,

63**

UR

60,

780,

771,

2**

1,11

**1,

1***

1,01

1,04

1,17

**0,

710,

72*

1,05

**0,

69

0,55

*0,

54*

0,93

0,96

0,88

0,89

1,05

**0,

99**

0,51

*0,

52*

0,86

0,51

120,

410,

40,

730,

74**

0,73

0,76

1,07

0,8*

0,4

0,4*

0,69

**0,

49

11,

04*

1,04

*1,

091,

121,

091,

121

1,01

1**

11,

121,

053

0,91

0,89

*1,

2**

1,3*

**1,

25**

*1,

230,

961,

070,

77**

0,78

1,35

***

0,85

IP6

0,96

0,94

1,59

***

1,57

***

1,38

**1,

99**

0,95

**0,

53*

0,85

0,85

1,71

***

1,06

90,

84*

0,83

**1,

54**

1,6*

*1,

32**

2,09

**0,

92*

1,11

0,76

*0,

781,

82**

*0,

99**

120,

65**

0,71

**1,

261,

24*

1,43

1,89

**0,

91**

0,97

0,63

**0,

65**

1,52

0,92

**

Die

bo

ldan

dL

i(20

06)

yiel

dd

ata.

17m

atu

riti

esco

rres

po

nd

ing

to3,

6,9,

12,1

5,18

,21,

24,3

0,36

,48,

60,7

2,84

,96,

108,

and

120

mo

nth

s.M

SFE

rati

os

bet

wee

nm

od

el8

and

mo

del

s1−1

3fo

rC

PI,

PC

E,P

PI,

RD

I,U

R,a

nd

IPfo

ral

lfo

reca

stin

gle

ads

h.A

valu

elo

wer

than

on

ein

dic

ates

alo

wer

MSF

Eo

fmo

del

8w

.r.t

.th

eco

mp

etin

gm

od

els.

On

e,tw

o,an

dth

ree

star

sm

ean

.10,

.05,

and

.01

stat

isti

cals

ign

ifica

nce

,res

pec

tive

ly,f

or

the

Gia

com

inia

nd

Wh

ite

(200

6)te

stw

ith

qu

adra

tic

loss

fun

ctio

n.T

he

nu

mb

ero

ffo

reca

sts

isS′ =

75(t

he

firs

th

alfo

fth

eS=

150

com

ple

tefo

reca

stsa

mp

le).

Th

e1-

step

ahea

dfo

reca

sts

ran

gefr

om

July

1987

toSe

pte

mb

er19

93.T

he

12-s

tep

ahea

dfo

reca

sts

ran

gefr

om

Jun

e19

88to

Au

gust

1994

.Th

efo

reca

stp

erio

dco

rres

po

nd

sto

par

toft

he

Gre

ensp

an(1

987-

2006

)m

on

etar

yre

gim

e.


le2.9.M

SFE

ratios

for

secon

dh

alfoffo

recastsamp

le(D

iebo

ldan

dLi(2006)

yieldd

ata,17m

aturities).

hm

od1

mod

2m

od3

mod

4m

od5

mod

6m

od7

mod

9m

od10

mod

11m

od12

mod

13

11,01

0,961,2

1,37**1,2

1,37**1

11

11,02

0,85**3

0,960,95*

0,970,95

0,970,94

0,960,98*

0,97*0,96

0,980,96

CP

I6

10,89

1,011,03

1,021,02

1*1,03

0,970,98

1,05**0,9

91

0,871,06**

1,05**1,02

1,020,98*

1,02***0,93*

0,971,06***

0,8***12

0,960,82

1,081,13

1,061,06

0,990,98

0,850,9

1,060,75

11

11,37

1,68**1,37

1,68**1

1,061

1,010,88*

0,69*3

1,031,02

1,03*1,04*

0,971,01

11,01

1,011,03

0,91,01

PC

E6

0,970,97

0,960,98

1**1**

1,070,99

0,96**0,98

0,861

90,97

0,970,96

0,951

10,92**

0,990,95*

0,970,84**

1,0112

1,031,02

1,04*1,04*

1,011,01

1,010,98**

11,03

0,90,99

11,01

0,96**1,13

1,31,13

1,31

1,11

1,011,02

0,83**3

1,011

1,010,99

1,010,97

0,980,86*

1,011,01

1,020,96

PP

I6

1,020,99

1,041,05

1,021,02

0,971,02

0,991

1,050,93

90,99

0,921,01

1,021,03

1,031

10,93

0,951,01*

0,9112

0,990,91

1,041,05

1,031,03

0,97*1

0,90,93

1,030,86

10,98

0,96***1,04

1,081,04

1,081

1,011

0,991,07

0,933

0,88*0,88*

1,221,17

1,181,21

1,05**0,96

0,950,92*

1,180,96**

RD

I6

0,87*0,92

1,41,43

1,321,34

1,13**0,95

0,940,9*

1,380,94*

90,79*

0,841,46

1,421,37

1,491,15**

0,930,87

0,831,47

0,94*12

0,69**0,72

1,64*1,63*

1,571,72**

1,12***0,89*

0,790,75*

1,65*0,96

10,98

0,990,99

0,990,99

0,991

0,98*1*

0,98**0,92

0,963

1,041,04

1,010,95

0,97**1,15

0,94**0,97

1,040,99

0,92*1,4***

UR

61,16*

1,140,94

0,920,91

0,850,92***

0,91*1,15**

1,110,73

1,45*9

1,19**1,14**

0,94***0,84

0,910,76

0,91**0,79

1,17***1,11

0,631,54***

121,06

10,88***

0,97***0,98

0,760,91*

1,121,06

1,030,63

1,53**

10,98

1,011,07

1,091,07

1,091

0,981,01

0,990,91

1,093

1,151,14

0,920,88

0,78**0,77

1,011

1,161,14

0,91,14

IP6

1,071,01

0,70,65

0,740,46*

1,020,57

1,11,04

0,631,08

90,86

0,790,51*

0,63*0,69*

0,42**1,04

0,960,89

0,840,5*

1,1312

0,570,53**

0,34**0,33*

0,61*0,42*

1,06***1,15*

0,62**0,6*

0,4**1,08

Dieb

oldan

dLi(2006)yield

data.17

matu

ritiescorresp

ond

ing

to3,6,9,12,15,18,21,24,30,36,48,60,72,84,96,108,an

d120

mon

ths.M

SFE

ratiosb

etween

mod

el8an

dm

od

els1−

13fo

rC

PI,P

CE

,PP

I,RD

I,UR

,and

IPfo

rallfo

recasting

leads

h.A

value

lower

than

on

ein

dicates

alow

erM

SFE

ofm

od

el8w

.r.t.the

com

petin

gm

od

els.On

e,tw

o,and

three

starsm

ean.10,.05,an

d.01

statisticalsignifi

cance,resp

ectively,for

the

Giaco

min

iand

Wh

ite(2006)

testw

ithq

uad

raticlo

ssfu

nctio

n.T

he

nu

mb

ero

ffo

recastsis

S ′=75

(the

secon

dh

alfofth

eS=

150co

mp

letefo

recastsam

ple).T

he

1-stepah

eadfo

recastsran

gefro

mSep

temb

er1994

toD

ecemb

er1999.T

he

12-stepah

eadfo

recastsran

gefro

mSep

temb

er1994

toN

ovemb

er2000.T

he

forecastp

eriod

corresp

on

ds

top

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3FORECASTING THE GLOBAL MEAN SEA LEVEL

A CONTINUOUS-TIME STATE-SPACE APPROACH

Lorenzo Boldrini


75

76 CHAPTER 3. FORECASTING THE GLOBAL MEAN SEA LEVEL

Abstract

In this paper we propose a continuous-time, Gaussian, linear, state-space system

to model the relation between global mean sea level (GMSL) and the global mean

temperature (GMT), with the aim of making long-term projections for the GMSL. We

provide a justification for the model specification based on popular semi-empirical

methods present in the literature and on zero-dimensional energy balance models.

We show that some of the models developed in the literature on semi-empirical

models can be analysed within this framework. We use the sea-level data recon-

struction developed in Church and White (2011) and the temperature reconstruction

from Hansen et al. (2010). We compare the forecasting performance of the proposed

specification to the procedures developed in Rahmstorf (2007b) and Vermeer and

Rahmstorf (2009). Finally, we compute projections for the sea-level rise conditional

on the 21st century SRES temperature scenarios of the IPCC fourth assessment report.

Furthermore, we propose a bootstrap procedure to compute confidence intervals for

the projections, based on the method introduced in Rodriguez and Ruiz (2009).

3.1. INTRODUCTION 77

3.1 Introduction

Climate changes, the increase in the global temperature and sea level are long-

standing topics. Monitoring and predicting the rise in the sea level is of great impor-

tance due to its close relation with global climate changes and the socio-economic

effects it entails. In particular, the sea-level rise has direct consequences for popu-

lations living near the current mean sea level, Anthoff, Nicholls, Tol, and Vafeidis

(2006), Anthoff, Nicholls, and Tol (2010), Arnell, Tompkins, Adger, and Delaney (2005),

Sugiyama, Nicholls, and Vafeidis (2008). Physical and statistical models are needed

to measure the rate of change of the sea level and understand its relation to anthro-

pogenic and natural causes.

In this paper we propose a statistical framework to model the relation between

the global mean sea level (GMSL) and the global mean temperature (GMT), with

the aim of making long-term projections for the GMSL. The model belongs to the

class of semi-empirical models. We provide a justification for the model specifi-

cation based on popular semi-empirical methods present in the literature and on

zero-dimensional energy balance models. We show that some of the semi-empirical

models developed in the literature to study the relation between sea-level rise and

temperature can be analysed within this framework.

To date, there are two methods of estimating the sea-level rise as a function of

climate forcing. The conventional approach, used by the Intergovernmental Panel

on Climate Change (IPCC) climate assessments, is to use process-based models to

estimate contributions from the sea-level components and then sum them to obtain

an estimate of the sea-level increase, see for instance Meehl, Covey, Taylor, Delworth,

Stouffer, Latif, McAvaney, and Mitchell (2007a), Meehl, Stocker, Collins, Friedlingstein,

Gaye, Gregory, Kitoh, Knutti, Murphy, Noda, et al. (2007b), Pardaens, Lowe, Brown,

Nicholls, and De Gusmão (2011), Solomon, Plattner, Knutti, and Friedlingstein (2009).

Variations in the sea level originate from steric, eustatic, and non-climate changes.

By steric, we mean sea-level variations due to ocean volume changes, resulting from

temperature (thermosteric) and salinity (halosteric) variations. By eustatic, we mean

variations in the mass of the oceans as a result of water exchanges between the oceans

and other surface reservoirs (ice sheets, glaciers, land water reservoirs, and the at-

mosphere)1. By non-climate causes we mean variations in the quantity of water in

the oceans due to human impact, such as the building of dams and the extraction of

groundwater. However, the theoretical understanding of the different contributors is

incomplete, as IPCC models under-predict rates of sea-level increase.

The alternative way of making projections of the sea level is the class of semi-

empirical models. These models analyse statistical relationships using physically

plausible models of reduced complexity in which the sea-level rate of change de-

pends on the global temperature. The main idea behind semi-empirical models is

1In this paper we adopt the same definitions of steric and eustatic used in Cazenave and Nerem (2004).


that the steric and eustatic contributors to the sea level (the major ones) respond

to changes in the global temperature. The first semi-empirical model was proposed

by Gornitz, Lebedeff, and Hansen (1982) who specify a linear relation between sea

level and temperature. Some more recent models where developed by specifying

a differential equation, relating the sea level to temperature or other climate forc-

ing. Representative examples are Rahmstorf (2007b), Vermeer and Rahmstorf (2009),

Grinsted, Moore, and Jevrejeva (2010), Kemp, Horton, Donnelly, Mann, Vermeer, and

Rahmstorf (2011), Jevrejeva, Grinsted, and Moore (2009), Jevrejeva, Moore, and Grin-

sted (2010), Jevrejeva, Moore, and Grinsted (2012b), Jevrejeva, Moore, and Grinsted

(2012a).

All semi-empirical models project higher sea-level rise, for the 21st century, that

the last generation of process-based models, summarized in the IPCC Fourth As-

sessment Report, see for instance Moore, Grinsted, Zwinger, and Jevrejeva (2013),

Cazenave and Nerem (2004), Munk (2002), and Rahmstorf (2007b). A comprehensive

survey on the different process-based and semi-empirical models can be found in

Moore et al. (2013).

We propose a state-space approach to forecast the global mean sea level, condi-

tional on the global mean temperature. State-space systems allow to address the prob-

lems of smoothing, detrending, and parameter estimation in a unique framework.

We consider in particular continuous-time, linear, Gaussian state-space systems of

the type described in Bergstrom (1997). More specifically, the state vector follows

a multivariate, Gaussian, Ornstein–Uhlenbeck process. The discretised system pre-

serves its linearity and Kalman filtering techniques apply. In particular, the Kalman

filter is used for two tasks: the first one is to compute the likelihood function of the

state-space system, needed for parameter estimation, and the second one is to make

forecasts of the sea level, conditional on the temperature.

The statistical framework of state-space systems allows to distinguish between

measurement noise and model uncertainty, through the measurement and state

equations, respectively. Furthermore, in this setup it is possible to consider different

levels of measurement noise for different points in time. In fact, the reconstructions

of the sea level, in particular, are typically very noisy and the measurements uncer-

tainty reflects, for instance, the changes in the measurement instruments throughout

the decades, as well as the changes in the data sources. In this study we use the

sea-level reconstruction from Church and White (2011), who also provide an estimate

of the uncertainty of their sea-level estimates. In the state-space approach these

uncertainties (measured as standard deviations) are directly used to parameterise

the time-varying variances of the measurement errors of the sea-level time series.

Temperature data are taken from Hansen et al. (2010). Both sea level and temperature

data correspond to monthly reconstructions.

We provide a justification for the model specification based on popular semi-

empirical methods present in the literature and on zero-dimensional energy balance

3.2. MODEL SPECIFICATION 79

models. In more detail, we specified the system dynamics of sea level and tempera-

ture as well as the functional form linking these variables, consistently with the ones

suggested by the existing literature. We show that some of the semi-empirical models

developed in the literature to study the relation between sea level and temperature,

can be analysed within this framework.

Semi-empirical models are usually specified as differential equations, as in Rahm-

storf (2007b), Vermeer and Rahmstorf (2009), Grinsted et al. (2010), Kemp et al. (2011),

Jevrejeva et al. (2009), Jevrejeva et al. (2010), Jevrejeva et al. (2012b), and Jevrejeva et al.

(2012a). By specifying the state-space system in continuous time and then deriving

the exact discrete-time system, we can make inference on the structural parameters

driving the continuous-time process. A similar approach was used in Pretis (2015)

(forthcoming), in which the author shows the equivalence of a two-component en-

ergy balance model to a cointegrated system. He then shows the exact mapping

between the continuous-time system to the discrete-time one, that amounts to a

cointegrated vector autoregressive system.

In the literature on semi-empirical models, state-space system representations

and the Kalman filter are often used with the aim of assimilating noisy measure-

ments from different sources2. Such studies are for instance, Miller and Cane (1989)

and Chan, Kadane, Miller, and Palma (1996) who use the Kalman filter with an un-

derlying physical model to assimilate average sea-level anomalies from tide gauge

measurements and Cane, Kaplan, Miller, Tang, Hackert, and Busalacchi (1996) and

Hay, Morrow, Kopp, and Mitrovica (2013).

The paper is organised as follows: in Section 3.2, we explain the statistical frame-

work and introduce the model specification, showing how it relates to some impor-

tant models in the literature; in Section 3.3, we describe the dataset; in Section 3.4,

we illustrate the forecasting procedures; in Section 3.5, we provide some details on

the computational aspects of the analysis; in Section 3.6, we present results; finally,

Section 3.7 concludes.

3.2 Model specification

Energy balance models and temperature dynamics

In this section we present the foundations for the temperature process used in the

state-space model proposed in Section 3.2. This section draws heavily on North,

Cahalan, and Coakley (1981) and Imkeller (2001) and we refer the reader to these

papers for more details. The starting model, for the temperature, belongs to the class

of zero-dimensional energy balance models (EBMs), detailed in North et al. (1981),

2In this context, assimilation usually refers to the filtering of a variable, measured at a specific pointin time and geographic location, by taking into account information from measurements taken at neigh-bouring locations at the same point in time. If the filtered variables are output from the Kalman filter, thecontribution of variable j to the estimate of variable i is given by element (i , j ) of the Kalman gain matrix.


and whose review is based on the models introduced in Budyko (1968), Budyko

(1969), Budyko (1972), and Sellers (1969). These models are based on thermodynamic

concepts and global radiative heat balance, for the Earth system. This type of models

describe the global temperature process as (possibly stochastic) univariate, differen-

tial equations. In particular, the change in the Planet’s global temperature at time t is

seen as a function of the difference between the incoming and outgoing radiation.

The incoming (absorbed) radiation Ri n is caused by solar irradiance (i.e. the

sunlight reaching the Earth) and is affected by the reflectivity of the Planet. The in-

coming radiation is then a function of the solar constant3 σ0, the albedo coefficient4

α, and the radius R of the Earth, in particular: Ri n = σ0(1−α)πR2. The outgoing

radiation Rout is assumed, for simplicity, to be black-body5 radiation, obeying the

Stefan-Boltzmann law6. It is then a function of the absolute temperature T of the

Planet and its radius R, in particular: Rout = 4πR2kSB T 4.

The analysis of zero-dimensional EBMs begins with the concept of global ra-

diative heat balance. In radiative equilibrium the rate at which solar radiation is

absorbed by the Planet matches the rate at which infrared radiation is emitted by it.

The condition of radiative equilibrium is given by

Rout︷︸︸︷4πR2kSB T 4 =

Ri n︷︸︸︷σ0(1−α)πR2, (3.1)

where T is the effective radiating temperature of the Planet, and kSB = 0.56687 ·10−7[W m−2K 4], is the Stefan-Boltzmann constant. Note that both sides of equation

(3.1) are expressed in units of power, in particular in Watts [W ]. When the incoming ra-

diation does not match the outgoing radiation, the temperature of the Planet changes

in order to compensate the disequilibrium. The time-evolution of the temperature

can then be modelled with the following zero-dimensional EBM:

CdT (t )

d t= Ri n −Rout

= σ0(1−α)πR2 −4πR2kSB T 4(t ), (3.2)

where C , that has units of[

W ·sK

]=

[JK

], represents global thermal inertia and regulates

the speed of the temperature response. With T (t ) we make explicit the dependence

3The solar constant is a measure of the mean solar electromagnetic radiation per unit area that wouldbe incident on a plane perpendicular to the sun rays.

4The term albedo, Latin for white, describes the average reflection coefficient of an object. Thegreenhouse effect, for instance, can lower the albedo of the Earth, and cause global warming.

5A black body is an idealized physical object that absorbs all incident electromagnetic radiation. Thetotal energy per unit of time, per unit of surface area, radiated by a black body depends solely on itsabsolute temperature and obeys the Stefan-Boltzman law.

6The Stefan-Boltzmann law describes the power radiated from a black body in terms of its thermody-namic temperature. The thermodynamic temperature (absolute temperature) is commonly expressed inKelvin [K ], where 0[K ] =−273.15[°C] corresponds to the lowest achievable temperature, according to thethird principle of thermodynamics.


of temperature on time. Equation (3.2) can be written as

CdT (t )

d t= Qα−γT 4(t ),

(3.3)

where Q is a constant proportional to σ0, γ is a constant proportional to kSB , and

α= (1−α) is the co-albedo. Note that equation (3.3) allows to relax the black-body

assumption. In fact, for a so called grey body7 we have that the emissive power, per

unit surface area is I = εkSB T 4, with ε< 1.

Equation (3.3) is purely deterministic. To allow for random forcing, stochastic

EBM have been introduced, see for instance Fraedrich (1978) and Hasselmann (1976).

A stochastic EBM can be written in the following way:

CdT (t )

d t= Qα−γT 4(t )+W (t ), (3.4)

where W (t ) is a white noise random forcing8.

Depending on the time scale under examination, the solar constant funtion Q

can be allowed to be time-varying. For instance, the Milankovich cycle responsible for

the glaciations, i.e. a cyclical mutation in the eccentricity of the Earth orbit due to the

gravitational pull of other planets, has a period of approximately 105 years and can be

expressed as Q(t ) =Q0+si n(ωt ) withω= 10−5[1/year ], see Imkeller (2001). Similarly,

the co-albedo α can be assumed to be time-varying and, in particular, to depend on

the global temperature, i.e. α(T (t )). This is motivated, among other reasons, by the

ice-cap feedback. That is, the albedo of the Planet changes with the temperature as a

result of the shrinking or spreading of ice sheets on the Earth’s surface, that depends

on the global temperature.

Taking into consideration these arguments, equation (3.4) can be written in the

form

CdT (t )

d t= Ri n −Rout +W (t )

= Q(t )α(T (t ))−γT 4(t )+W (t ). (3.5)

Different specifications for Ri n and Rout have been suggested in the literature. Budyko

(1969), for instance, suggested that the infrared radiation to space, Rout , can be

represented as a linear function of the surface temperature T , that is:

4πR2σT 4(t ) ∼= 4πR2σ(δ1 +δ2T (t )), (3.6)

where δ1, and δ2 are constants, taking into account factors such as average cloudiness,

the effects of infrared absorbing gases, and the variability of water vapor. Sellers (1969)

7A body that does not absorb all incident radiation.8The white noise process is defined as E [W (t)] and E [W (t)W (t ′)] = q2δ(t − t ′) where δ(t − t ′) is a

Dirac delta, and q is a constant, see also Nicolis (1982).


suggested taking a linear approximation also of the albedo, or similarly, of the co-

albedo:

α(T (t )) =β1 +β2T (t ), (3.7)

where β1 and β2 are constants.

We now show that some important special cases of model (3.5) can be written in

the form

dT (t ) =

bT +κTµT (t )+aT S S(t )+aT T T (t )

d t +dηT (t ), (3.8)

where bT , aT S , aT T , and κT are constants, µT (t) is a time-varying process, S(t) is

the sea-level process, and ηT (t ) is a scaled Brownian motion with E [dηT (t )dηT (t )] =ΣT T d t . In particular, equation (3.8) is an exact representation of model (3.5) for these

specifications:

I. for a time-invariant solar constant Q, a constant co-albedo α, and taking a

linear approximation to Rout = γT 4(t) ∼= δ1 +δ2T (t), we obtain the following

relation between the components of equation (3.5) and the ones of equation

(3.8):

bT = 1

C(Qα−δ1), κTµT (t ) = 0, aT S = 0, aT T =− 1

Cδ2, dηT (t ) 6= 0;

(3.9)

II. for a time-varying solar constant Q(t), a constant co-albedo α, and taking a

linear approximation to Rout = γT 4(t ) ∼= δ1 +δ2T (t ), we obtain:

bT =− 1

Cδ1, κTµT (t ) = 1

CQ(t )α, aT S = 0, aT T =− 1

Cδ2, dηT (t ) 6= 0;

(3.10)

III. for a time-varying solar constant Q(t), a time varying co-albedo α(t), and a

linear approximation to Rout = γT 4(t ) ∼= δ1 +δ2T (t ), we have:

bT =− 1

Cδ1, κTµT (t ) = 1

CQ(t )α(t ), aT S = 0, aT T =− 1

Cδ2, dηT (t ) 6= 0;

(3.11)

IV. for a time-invariant solar constant Q, a time varying co-albedo, assuming

dependence on the temperature of the co-albedo α(T (t )) and linearity in the

temperature α(T (t )) =β1 +β2T (t ), and taking a linear approximation Rout =γT 4(t ) ∼= δ1 +δ2T (t ), we obtain:

bT = 1

C(Qβ1 −δ1), κTµT (t ) = 0, aT S = 0, aT T = 1

C(Qβ2 −δ2), dηT (t ) 6= 0.

(3.12)


The time-varying component κTµT (t) controls for changes in the albedo and/or

solar constant coefficients. That is, it accounts among other things, for the effects

of greenhouse gases and other phenomena that change the radiation-absorbing

properties of the Planet.

Semi-empirical models and sea-level dynamics

Some of the most representative semi-empirical models in the literature are here

briefly presented. Throughout this section we denote with t time, S(t) the global

mean sea level, and with T (t ) the global mean temperature. Parameters are indicated

with lower case letters. We consider the following five models:

I. Gornitz et al. (1982) suggest the following link between sea level and tempera-

ture:

S∗(t ) = aT ∗ (t − t0

)+b, (3.13)

where S∗ and T ∗ are the 5-year averages of the global sea level and temperature,

respectively. The parameters a and b are estimated by least-squares linear

regression and the time lag t0 is chosen to minimize the variance between

(3.13) and the sea-level curve.

II. Rahmstorf (2007b) suggests the following differential equation relating sea level

to temperature:

dS(t )

d t= r

(T (t )−T0

), (3.14)

where r is a parameter to be estimated. The sea-level rise above the previous

equilibrium state can be computed as

S(t ) = r∫ t

t0

(T (s)−T0

)d s. (3.15)

The statistical analysis in Rahmstorf (2007b) is comprised of several steps. First,

the GMSL and GMT series are processed to obtain annual means. Second, a

singular spectrum analysis filter, with a 15-year smoothing period, is applied to

the series of yearly averages. Third, data is divided into 5 years bins, in which

the average is taken. Lastly, the resulting sea-level series in first differences is

regressed on the resulting temperature in levels (with optional detrending of

both series before the regression). The data they use are the global mean sea

level from Church and White (2006) and the global temperature anomalies data

from GISTemp, Hansen, Ruedy, Sato, Imhoff, Lawrence, Easterling, Peterson,

and Karl (2001). See Holgate, Jevrejeva, Woodworth, and Brewer (2007) and

Rahmstorf (2007a) for comments on the statistical procedure.


III. Vermeer and Rahmstorf (2009) suggest the following extension of the previous

model:

dS(t )

d t= v1

(T (t )−T0

)+ v2dT (t )

d t. (3.16)

In this model the authors add the term v2dT (t )

d t to the Rahmstorf (2007b) model,

corresponding to an “instantaneous” sea-level response. The statistical method-

ology is similar to the one in Rahmstorf (2007b) and a thorough description of

it can be found in the online appendix to their paper.

IV. Kemp et al. (2011) propose the following model:

dS(t )

d t= k1

(T (t )−T0,0

)+k2(T (t )−T0(t )

)+k3dT (t )

d t,

dT0(t )

d t= T (t )−T0(t )

τ. (3.17)

The first term captures a slow response compared to the time scale of interest,

the second one captures intermediate time scales, where an initial linear rise

gradually saturates with time scale τ as the base temperature T0 catches up with

T (t). In Rahmstorf (2007b), equation (3.14), T0 was assumed to be constant.

The third term is the immediate response term introduced by Vermeer and

Rahmstorf (2009).

V. Grinsted et al. (2010) propose the following model:

Seq = g1T + g2,

dS(t )

d t= Seq −S(t )

τ, (3.18)

where Seq is the equilibrium sea level, for a given temperature. They assume a

linear approximation of the relation between sea level and temperature, due to

the closeness of the current sea level to the equilibrium in this climate period

(late Holocene-Anthropocene) and for small changes in the sea level. Equation

(3.18) can be integrated to give the sea level S over time, using the history of the

temperature T and knowledge of the initial sea level at the start of integration

S0. They impose constraints on the model, suggested by reasonable physical

assumptions.

We here show how some of the different model specifications for the relation between

sea level and temperature can be seen as special cases of the following stochastic

differential equation

dS(t ) =

bS +κSµS (t )+aSS S(t )+aST T (t )

d t +dηS (t ), (3.19)


where bS , κS , aSS , and aST are parameters, µS (t) is a time-dependent process, and

ηS (t ) is a scaled Brownian motion with E [dηS (t )dηS (t )] =ΣSS d t . In particular, mod-

els II and V can be written as particular cases of model (3.19) if some restrictions are

imposed on its components:

II. for the Rahmstorf (2007b) specification we have the following relation between

the components of equation (3.19) and equation (3.14):

bS =−r T0, κSµS (t ) = 0, aSS = 0, aST = r, dηS (t ) = 0; (3.20)

V. for the Grinsted et al. (2010) specification, the relation between the components

of equation (3.19) and (3.18) are:

bS = 1

τg2, κSµS (t ) = 0, aSS =−1

τ, aST = 1

τg1, dηS (t ) = 0. (3.21)

Note that in models II and V the component κSµS (t) is zero. This reflects the fact

that in both papers the series of temperature and sea level are detrended before the

parameter estimation. Instead, we prefer to model the trend component jointly with

the temperature and sea-level dynamics.

State-space system

Combining the temperature and sea-level dynamics, equations (3.8) and (3.19), we

obtain the following multivariate process

d

[S(t )

T (t )

]= bd t +

[aSS aST

aT S aT T

][S(t )

T (t )

]d t +

[κS 0

0 κT

][µS (t )

µT (t )

]d t +dη(t ),

(3.22)

where b = [bS : bT ]′, µ(t) = [µS (t) : µT (t)]′ is the trend component, and dη(t) =[dηS (t) : dηT (t)]′ where η(t) is a scaled, 2-dimensional, Brownian motion with

E [dη(t )dη(t )′] =Σd t and

Σ =[ΣSS ΣST

ΣT S ΣT T

], (3.23)

a positive semidefinite covariance matrix.

We consider two parametric forms for the trend componentµ(t ), namely a linear and

a quadratic trend:

(i) linear trend component,

dµ(t ) = λl d t , (3.24)

where λl = [λSl

:λTl ] are parameters to be estimated;


(ii) quadratic trend component,

dµ(t ) = λ(t )d t ,

dλ(t ) = νq d t , (3.25)

where λ(t ) = [λS (t ) :λT (t )]′ is a 2-dimensional process and νq = [νSq : νT

q ]′ are

parameters to be estimated.

The choice of the trend components was driven by the forecast performance of the

models, according to the forecasting exercise detailed in Section 3.4.

State equation

We now show how to obtain the exact discrete representation of the continuous-time

state-space system (3.22) with linear trend (3.24) (the derivation for the model with

the quadratic trend (3.25) is analogous). The system of equations (3.22)-(3.24) can

be written in compact form, delivering the following Gaussian, Ornstein-Uhlenbeck

(OU) process:

dα(t ) = cd t +Aα(t )d t +dξ(t ), (3.26)

where α(t ) = [S(t ),T (t ),µS (t ),µT (t )]′, µS (t ) and µT (t ) indicate the trends for the sea-

level and the temperature, respectively, dξ(t) = [dη(t)′ : 0′]′, c = [b′ :λ′l ]′ = [0′ :λ′

l ]′,and the autoregressive matrix has the form

A =

aSS aST 1 0

aT S aT T 0 1

0 0 0 0

0 0 0 0

, (3.27)

where we constrain the two parameters κS = κT = 1, as suggested in Bergstrom

(1997), in order to avoid identification issues. We set the intercept b = 0 because of

the presence of trend component. The exact discrete state-space representation can

be recovered from the continuous-time equations following, for instance, Bergstrom

(1997). The solution of the OU process (3.22) is

α(t ) = e tAα(0)+∫ t

0eA(t−s)cd s +

∫ t

0eA(t−s)dξ(s), (3.28)

whereα(0) is the initial value of the system. Note that the solution (3.28) always exists.

Denote with τ= 1, . . . ,n, n ∈N, the time instances at which the sea-level and the

temperature processes are sampled (measured), i.e. α(t = τ) =ατ. The relationship

between the state vector at time τ and time τ+1 derives from equation (3.28) and is

given by


ατ+1 = eAατ+∫ τ+1

τeA(τ+1−s)cd s +

∫ τ+1

τeA(τ+1−s)dξ(s). (3.29)

Equation (3.29) corresponds to a Gaussian, vector autoregressive process of this form

ατ+1 = c∗+A∗ατ+ξτ, (3.30)

with ξτ ∼ N (0,Σ∗), where

c∗ =∫ 1

0eA(1−s)cd s,

A∗ = eA,

Σ∗ =∫ 1

0eA(1−s)ΣeA′(1−s)d s. (3.31)

The constants aSS , aST , aT S , aT T , λ, and Σ are parameters to be estimated.

Measurement equation

The global mean sea level and the global mean temperature can be seen as stock vari-

ables, sampled at time instances τ and subject to measurement error, see for instance

Harvey and Stock (1993). Let Srτ and T r

τ be the reconstructed (or measured) sea-level

and temperature processes, respectively, and Sτ and Tτ the true (unobserved), latent

ones. The measurement equation for system (3.30) is thus:[Srτ

T rτ

]=

[SτTτ

]+

[εSτ

εTτ

], (3.32)

where ετ = [εSτ : εT

τ ]′ ∼ N(0,Hτ

)is a bivariate random vector of independent mea-

surement errors. Note that the variance-covariance matrix is allowed to vary through

time, in particular

Hτ =[σ2,Sτ 0

0 σ2,T

]. (3.33)

The variance of the measurement error for the sea level σ2,Sτ , is allowed to change

in time. In particular, in this work we use the sea-level reconstruction from Church

and White (2011). In their analysis, the authors provide uncertainty estimates of

the sea-level reconstruction at each point in time. The change in the uncertainty of

the reconstructed sea-level series reflects the change in time of the measurement

instruments, as well as the change in the data sources. Note that in this context, the

magnitude of the observation error variances controls the smoothness of the filtered

series of sea level and temperature. The parameterσ2,T is estimated together with the

other system parameters, whereas the sequence σ2,Sτ τ=1:n is fixed and corresponds


to the uncertainty values reported in Church and White (2011).

Combining equations (3.30) with (3.32) we obtain a linear, Gaussian, state-space

system (see for instance Brockwell and Davis (2009) and Durbin and Koopman

(2012)):

[Srτ

T rτ

]=

[1 0 0 0

0 1 0 0

]SτTτµSτ

µTτ

+ετ, ετ ∼ N(0,Hτ

),

Sτ+1

Tτ+1

µSτ+1

µTτ+1

= c∗+A∗

SτTτµSτ

µTτ

+ξτ, ξτ ∼ N(0,Σ∗)

. (3.34)

System (3.34) is linear in the state variables and Kalman filtering/smoothing tech-

niques apply, allowing to estimate the system parameters by maximum likelihood,

see for instance Durbin and Koopman (2012).

In the semi-empirical literature, dynamic models of sea level and temperature

are usually formulated in continuous time. A clear mapping between a multivari-

ate, Gaussian, Ornstein-Uhlenbeck process and its discrete-time analogue allows to

make inference on the parameters of the original process, introducing no bias due

to discretizations. A convenient aspect of specifying the model in state-space form

is that measurement noise and trends can be modelled in a joint framework. In this

way the problem of smoothing, detrending, and parameter inference can be handled

in a unified framework.

The dimensional analysis for the continuous-time and discrete-time systems is

provided in Appendix 3.9.

3.3 Data

• Temperatures. The temperature data are taken from the GISS dataset, Com-

bined Land-Surface Air and Sea-Surface Water Temperature Anomalies (Land-

Ocean Temperature Index, LOTI). The values are temperature anomalies, i.e.

deviations from the corresponding 1951-1980 means, Hansen et al. (2010)9.

The time series we use is composed of mean global monthly values. The values

of the original series are in centi-degrees Celsius ([c°C] = 10−2[°C]).

• Sea level. The sea-level data is from Church and White (2011)10. The authors

also provide uncertainty estimates for each measurement. They estimate the

rise in global average sea level from satellite altimeter data for 1993-2009 and

9Data can be downloaded from http://data.giss.nasa.gov/gistemp/.10Data can be downloaded from http://www.cmar.csiro.au/sealevel/sl_data_cmar.html.

http://data.giss.nasa.gov/gistemp/

http://www.cmar.csiro.au/sealevel/sl_data_cmar.html

3.3. DATA 89

from coastal and island sea-level measurements from 1880 to 2009. The mea-

surements of the original series are in millimetres [mm].

In our analysis we use monthly observations ranging from January 1880 to December

2009, range in which the two series overlap, for a sample size equal to 1560.

• IPCC temperature scenarios. These series correspond to reconstructed and

simulated annual temperatures, from 1900 to 2099 from the 2007 IPCC Fourth

Assessment Report, SRES scenarios. In particular, we use the A1b, A2, B1, and

commit groups of temperature scenarios11. The 4 groups correspond to dif-

ferent “storylines”12. The storylines “describe the relationships between the

forces driving greenhouse gas and aerosol emissions and their evolution during

the 21st century for large world regions and globally. Each storyline represents

different demographic, social, economic, technological, and environmental

developments that diverge in increasingly irreversible ways.” (Carter (2007,

page 9)). Each group has a different number of scenarios and in total there are

75 scenarios. The 4 scenarios groups can be described in the following way:

(i) A1b group. This group belongs to the A1 storyline and scenario family,

that is “a future World of very rapid economic growth, global population

that peaks in mid-century and declines thereafter, and rapid introduc-

tion of new and more efficient technologies.” (Carter (2007, page 9)) in

which an intermediate level of emissions has been assumed. There are 21

scenarios belonging to this group.

(ii) A2 group. “A very heterogeneous World with continuously increasing

global population and regionally oriented economic growth that is more

fragmented and slower than in other storylines.” (Carter (2007, page 9)).

There are 17 scenarios belonging to this group.

(iii) B1 group. “A convergent World with the same global population as in

the A1 storyline but with rapid changes in economic structures toward a

service and information economy, with reductions in materials intensity,

and the introduction of clean and resource-efficient technologies.” (Carter

(2007, page 9)). There are 21 scenarios belonging to this group.

(iv) commit group. In this group of scenarios, the World’s countries commit

to lower greenhouse gases emissions. There are 16 scenarios belonging to

this group.

The A1b and A2 groups reflect scenarios with an acceleration in temperature

growth (high temperature increase); the B1 group reflects scenarios of constant

11Data can be downloaded from http://www.ipcc-data.org/sim/gcm_global/index.html.12For a precise description of the storylines and scenarios see Carter (2007), http://www.ipcc-data.

org/guidelines/TGICA_guidance_sdciaa_v2_final.pdf.

http://www.ipcc-data.org/sim/gcm_global/index.html

http://www.ipcc-data.org/guidelines/TGICA_guidance_sdciaa_v2_final.pdf

http://www.ipcc-data.org/guidelines/TGICA_guidance_sdciaa_v2_final.pdf


temperature growth (medium temperature increase); the commit group reflects

scenarios of very low temperature growth (low temperature increase). The

measurements of the original series are in degrees Celsius [°C].

3.4 Forecasting

Model comparison

To assess the forecasting power of the state-space models proposed in Section 3.2,

we carry out the following forecasting exercise. Let n denote the complete sample

size (i.e. n = 1560, corresponding to monthly observations ranging from January

1880 to December 2009), n∗ < n the size of the estimation sample, h the forecast

horizon, f = n −n∗ the number of forecasts for a given estimation sample size. For

all forecasting methods, the setup of the exercise is the following:

1. estimate the system parameters using observations of sea level and tempera-

ture from time t = 1 up to t = n∗;

2. compute forecasts using temperature observations from time t = n∗ + 1 to

t = n.

In particular, for the state-space system (3.34) (and for the model with quadratic

trend) the steps to construct the forecasts are the following :

(i) estimate the system parameters by maximum likelihood, using observations of

sea level and temperature from time t = 1 up to t = n∗ (the likelihood function

is delivered by the Kalman filter, see for instance Durbin and Koopman (2012));

(ii) run the Kalman filter, using the estimated parameters, on the dataset composed

of observations of the sea level from time t = 1 to t = n∗ and observations of

the temperature from t = 1 to t = n.

(iii) the forecasts of the sea level are then the filtered values Sn∗+h with h = 1, . . . , f .

Observations of the sea level, from time t = n∗+1 to t = n, are treated as missing

values, see Durbin and Koopman (2012) for more details on how to modify the filter

in case of missing observations. Note that for a linear state-space system the Kalman

filter delivers the best linear predictions of the state vector, conditionally on the

observations. Moreover, if the innovations are Gaussian, the filtered states coincide

with conditional expectations, for more details on the optimality properties of the

Kalman filter see Brockwell and Davis (2009).

We select two benchmark forecasting methods to which we compare our specifica-

tions. In particular, we compare our model to the procedures developed in Rahmstorf

(2007b) and Vermeer and Rahmstorf (2009). The choice of these benchmarks reflects

their popularity in the literature and the replicability of the results in the papers due

to the availability of source codes.

3.4. FORECASTING 91

Rahmstorf (2007b) procedure

The first competing method is the one used in Rahmstorf (2007b). The model is based

on equations (3.14)-(3.15). The procedure can be summarized as follows:

(i) the sea-level and temperature series, from time t = 1 up to t = n∗, are smoothed

using singular spectrum analysis and embedding dimension equal to ned = 180

months, corresponding to 15 years (15×12 = 180), as used in their paper;

(ii) first differences of the smoothed sea-level series are then taken;

(iii) the smoothed series (temperature and first differences of the sea level) are then

divided into nbi n = 60 months bins, corresponding to 5 years (5×12 = 60), as

used in the paper, and in each bin the average is taken;

(iv) the time series of bin-averages are then detrended (fitting a linear trend);

(v) the bin-averages of sea level in first differences is then regressed onto the

bin-averages of the temperature in levels;

(vi) the estimated regression coefficients are then used to compute the values of the

sea level in first differences from the out-of-sample (smoothed) temperatures

(note that the information set used comprises the sea-level observations from

time t = 1 to t = n∗, and observations of the temperature from t = 1 to t = n);

(vii) the forecasts of the sea level are then obtained by summing the forecast sea

level in first differences.

We also compute forecasts with the combinations: ned = 60/nbi n = 60, ned = 60/nbi n =180, and ned = 180/nbi n = 180.

Vermeer and Rahmstorf (2009) procedure

The second competing method is the one used in Vermeer and Rahmstorf (2009).

Their model is based on equation (3.16). The procedure is similar to the previous one,

with the addition of an extra step, and it can be summarized as follows:

(i) the sea-level and temperature series, from time t = 1 up to t = n∗, are smoothed

using singular spectrum analysis and an embedding dimension equal to ed =180 months, corresponding to 15 years (15×12 = 180), as used in the paper;

(ii) first differences of the smoothed sea-level series are then taken;

(iii) the smoothed series (temperature and first differences of the sea level) are then

divided into 60 months bins, corresponding to 5 years (5×12 = 60), as used in

their paper, and in each bin the average is taken;

(iv) both time series of bin-averages are then detrended (by fitting a linear trend);


(v) the parameter v2, in equation (3.16) (λ in the notation of their paper), is then

selected as the value for which the correlation between the detrended bin-

averages of the smoothed temperature and the detrended bin-averages of the

first differences of the smoothed sea level, is maximized;

(vi) the bin-averages of the smoothed sea level in first differences are then regressed

on the bin-averages of the smoothed temperature in levels, corrected for the

rate of change of the temperature (that is the v2dT (t )

d t factor in equation (3.16));

(vii) the estimated regression coefficients are then used to compute the values of the

sea level in first differences from the out-of-sample (smoothed) temperatures,

corrected for the v2dT (t )

d t factor in equation (3.16), where the v2 used is the one

previously computed and the rate of change dT (t )d t is computed in the same way

as before but from the out-of-sample (smoothed) temperatures (note that the

information set used comprises the sea-level observations from time t = 1 to

t = n∗ and observations of the temperature from t = 1 to t = n);

(viii) the forecasts of the sea level are then obtained by summing the forecast sea

level in first differences.

We also compute forecasts with the combinations: ned = 60/nbi n = 60, ned = 60/nbi n =180, and ned = 180/nbi n = 180.

Performance measure

As a measure of the relative forecasting power between the models, we take the ratios

of the square roots of the mean squared forecast errors, from the different models.

For model j we have:

R jf =

√√√√ 1

f

f∑h=1

(S j

n∗+h −Srn∗+h

)2, (3.35)

where S jn∗+h denotes the sea-level forecast from the j -th model and Sr

n∗+h is the

observed sea level13. We select different values of n∗. Note that in computing R jf we

are giving equal weight to forecasts at different horizons. This strategy is motivated by

the final goal of the model, that is to make long-term projections of the sea level. To

compare the models, we take ratios between the R measures, equation (3.35), from

the different forecasting models/methods.

13The superscript r stands for “reconstructed”, as the observations are a reconstruction of the sea-levelseries, made from different measurements of the sea level.

3.4. FORECASTING 93

Forecasting conditional on AR4-IPCC temperature scenarios

In this subsection we explain the method used to make long-term projections for

the sea level, conditional on the IPCC temperature scenarios. First, note that the

measurements of sea level and temperature go from January 1880 to December

2009 and correspond to monthly averages, whereas the IPCC temperature scenarios,

ranging from 2010 to 2099, correspond to yearly values. In order to model the data

and the scenarios in the same framework, we transform the yearly values in monthly

ones. In particular, for each scenario we treat the temperature value corresponding

to a specific year, as an observation for the month of July for that year, treating the

values for the remaining months as missing values.

Denote with ntot the sample size of the assembled dataset made up of the monthly

observations of sea level and temperatures plus the IPCC temperature scenarios, in

particular we have ntot = 1560+1080 = 2640. To construct the sea-level forecasts,

conditional on the temperature scenarios, we follow these steps:

(i) estimate the system parameters by maximum likelihood using observations of

sea level and temperature from time t = 1 up to time t = n;

(ii) run the Kalman filter, using the estimated parameters, on the dataset composed

of observations of the sea level and temperature from time t = 1 to t = n, and

one temperature scenario from t = n +1 to t = ntot ;

(iii) the forecasts of the sea level are then the smoothed values Sn+h with h =1, . . . ,ntot −n.

The procedure is repeated for each of the 75 temperature scenarios. In order to

compute confidence intervals for the sea-level projections, we first use a bootstrap

procedure to obtain an empirical distribution function (EDF) for the forecasts, condi-

tioning on each scenario separately. We then aggregate these EDFs using the law of

total probabilities, assigning equal probability to the different scenarios. Denoting

with Bi the i -th IPCC temperature scenario, where i = 1, . . . , N (N = 75) and with h the

forecast horizon, the unconditional empirical distribution function for the sea-level

projections is

Pr(St+h ≤ s

)=

N∑i=1

Pr(St+h ≤ s|Bi

)Pr

(Bi

)= 1

N

N∑i=1

Pr(St+h ≤ s|Bi

), (3.36)

where Pr (St+h ≤ s|Bi ) is the conditional distribution function of the sea-level fore-

cast St+h , given a temperature scenario Bi , and Pr (Bi ) is the probability of the i -th

temperature scenario. The confidence intervals for the forecasts St+h are then ob-

tained by taking the 1st and 99th percentiles of the cumulative distribution function


Pr (St+h ≤ s). The EDFs Pr (St+h ≤ s|Bi ) are obtained using a bootstrap procedure,

and the probabilities Pr (Bi ) are set equal to 1/N . We thus assume equal probabilities

for the different temperature scenarios, in line with the literature.

The bootstrap procedure is detailed in Appendix 3.9 and it is a modification of the

method proposed in Rodriguez and Ruiz (2009). In Rodriguez and Ruiz (2009), they

consider a time invariant state-space system in which the system parameters do not

vary in time. In the present paper, however, we assume a time-varying measurement

noise variance for the sea level σ2,St (3.33), this introduces heteroskedasticity in the

innovations.


The parameters of the state-space system are estimated by maximum likelihood.

The likelihood function is delivered by the Kalman filter. We employ the univariate

Kalman filter derived in Koopman and Durbin (2000), as we assume a diagonal

covariance matrix for the innovations in the measurement equation. The maximum

of the likelihood function has no explicit form solution and numerical methods have

to be employed. We make use of two algorithms:



space according to a Gaussian search distribution, which changes according to


• BFGS. Broyden-Fletcher-Goldfarb-Shanno, see for instance Press et al. (2002).

This algorithm belongs to the class of quasi-Newton methods and requires the

computation of the gradient of the function to be minimized.

The CMA-ES algorithm performs very well when no good initial values are available

but it is slower to converge than the BFGS routine. The BFGS algorithm, on the

other hand, requires good initial values but converges considerably faster than the

CMA-ES algorithm (once good initial values have been obtained). Hence, we use the

CMA-ES algorithm to find good initial values and then the BFGS one to perform the

minimizations with the different sample sizes, needed in the forecasting exercise

detailed in Section 3.4.

To gain speed we choose C++ as the programming language, using routines from

the Numerical Recipes, Press et al. (2002). We compile and run the executables on a

Linux 64-bit operating system using the GCC compiler 15. The integrals appearing in

equations (3.31) can be computed analytically with the aid, for instance, of MATLABr


15See http://gcc.gnu.org/onlinedocs/ for more information on the Gnu Compiler Collection,GCC.



3.6. RESULTS AND DISCUSSION 95

symbolic toolbox. The generated code can then be directly converted into C++ code

with the command ccode.

3.6 Results and discussion

Model comparison results

To compute the out-of-sample forecasts for model (3.34) we use the Kalman filter,

treating the sea-level values as missing observations, for the time points at which we

want to forecast it. In the tables 3.1-3.4 in Appendix 3.9 are reported the ratios (3.35)

for different values of the estimation sample n∗ and forecast sample f . We label the

different forecasting models according to the following convention.

• model 1. State-space system (3.22) with linear trend (3.24), taking the filtered

values as forecasts;

• model 2. State-space system (3.22) with linear trend (3.24), taking the smoothed

values as forecasts;

• model 3. State-space system (3.22) with quadratic trend (3.25), taking the fil-

tered values as forecasts;

• model 4. State-space system (3.22) with quadratic trend (3.25), taking the

smoothed values as forecasts;

• model 5. Rahmstorf (2007b) procedure (see Section 3.4) with embedding di-

mension ned = 60 and number of bins nbi n = 60;







• model 9. Vermeer and Rahmstorf (2009) procedure (see Section 3.4) with em-

bedding dimension ned = 60 and number of bins nbi n = 60;

• model 10. Vermeer and Rahmstorf (2009) procedure (see Section 3.4) with

embedding dimension ned = 60 and number of bins nbi n = 180;


embedding dimension ned = 180 and number of bins nbi n = 60;



embedding dimension ned = 180 and number of bins nbi n = 180.

It can be seen from tables 3.1-3.4 that the state-space models 1-2 and 3-4 perform

quite well compared to models 5-12. In particular, the quadratic trend component (in

models 3-4) seems to help the forecasting performance of the state-space system. The

difference between the filtered and smoothed forecasts is negligible. These results

show that similar (or better) forecasts can be obtained using the state-space systems

presented in Section 3.2, compared to the two benchmark procedures outlined in

Section 3.4.

We also considered specifications without trend components, with stochastic

trends, and/or with various sets of coefficients restricted to zero. All these alternative

specifications were found to perform poorly compared to the ones presented in

this paper, in terms of forecasting performance. In particular, setting the coefficient

aSS = 0 or adding stochastic components in the trend process, considerably worsened

the forecast performance of the models.

Full sample estimation results

In this subsection we present the parameter estimates relative to models 1-2 and

models 3-4. The estimation results are contained in tables 3.5-3.6 (for models 1-

2) and in tables 3.7-3.9 (for models 3-4) and they are divided into estimates for

the continuous-time and discrete-time specifications. The tables can be found in

Appendix 3.9. We estimate the parameters using the complete dataset of sea-level and

temperature monthly observations, ranging from January 1880 to December 2009,

for a sample size equal to 1560. We group the comments according to the different

models (1-2 and 3-4):

• models 1-2. The estimated standard deviation of the temperature measure-

ment error is σT = 7.59[cK ] (0.0759[K ]), which is slightly lower than the aver-

age standard deviation of the sea-level measurement errors σS = 11.43[mm]

(0.01143[m]). The average σS is computed from the sequence of volatilities

σSττ=1:n , corresponding to the uncertainty estimates reported in Church and

White (2011). The autoregressive coefficients A∗,SS = 0.99 and A∗,T T = 0.92 are

both close to unity. The coefficient linking sea level to temperature is found

to be quite small, A∗,ST = 0.0054[mm/cK ] (0.00054[m/K ]), whereas the one

linking temperature to the sea level A∗,T S = 0.0489[cK /mm] (0.489[K /m]) is

quite large.

• models 3-4. The estimated standard deviation of the temperature measure-

ment error isσT = 7.41[cK ] (0.0741[K ]). The autoregressive coefficients A∗,SS =0.97 and A∗,T T = 0.89 are both close to unity. Interestingly, the coefficients

linking sea level to temperature, and vice versa, are found to have a negative


sign A∗,ST = −0.0035[mm/cK ] (−0.00035[m/K ]), A∗,T S = −0.0426[cK /mm]

(−0.426[K /m]).

The parameter aST , if left unrestricted, is estimated to be either positive or neg-

ative (depending on the model and the estimation sample) and of the order of

10−2[mm/cK ] (10−3[m/K ]) for a one month time-step. The low value of this pa-

rameter may be caused by long response times of the sea level to the temperature

and the fact that the time-step considered is quite small. Early studies indicate lags

in the order of 20 years, between temperature and sea-level rise, Gornitz et al. (1982).

One puzzling fact is the change in sign of aST and aT S between models 1-2 and 3-4.

We found that when the parameter aSS is left unconstrained, the coefficient link-

ing sea level and temperature is estimated to be very low. This may suggest long

response times of the sea level to the temperature changes, possible distortions in

the sea level and temperature reconstructions, and/or a misspecification of the func-

tional link between the two variables.

One interesting finding concerns the role of the sea-level measurement error vari-

anceσ2,Sτ in the state-space system. Namely, this parameter regulates the smoothness

of the filtered (and smoothed) sea-level series. Interestingly, if σ2,Sτ =σ2,S is left unre-

stricted and estimated together with the other parameters, the value obtained is very

close to zero. This causes the filtered (and smoothed) sea-level series to essentially

coincide with the observed ones. Intuitively, in this case the forecasts worsen.

Forecasting conditional on AR4-IPCC scenarios results

In this subsection we report the long-term sea-level projections, computed condition-

ally on the different temperature scenarios. See Section 3.4 for more details on the

forecasting procedure and Section 3.3 for a description of the temperature scenarios

used. The scenarios are depicted in figure 3.1. We make sea-level rise projections

using models 2 and 4, with respect to the (smoothed) sea-level value in 2009:

• model 2. The forecasts are quite sensitive to the temperature scenarios. De-

note with q0.01 and q0.99 the 1st and 99th percentiles of the distribution of

sea-level forecasts for the year 2099. If we condition on all of the 75 temper-

ature scenarios, taken with equal probability, the sea-level forecasts range

from q0.01 = 0.0948[m] to q0.99 = 0.3525[m] with a mean value of 0.2130[m],

see figure 3.2. This range changes if different scenario groups are considered

separately.

(i) A1b group: forecasts range from q0.01 = 0.1829[m] to q0.99 = 0.3537[m]

with a mean value of 0.2470[m];

(ii) A2 group: forecasts range from q0.01 = 0.2030[m] to q0.99 = 0.3654[m]



(iii) B1 group: forecasts range from q0.01 = 0.1485[m] to q0.99 = 0.2688[m]


(iv) commit group: forecasts range from q0.01 = 0.0868[m] to q0.99 = 0.1489[m]

with a mean value of 0.1174[m].

• model 4. The forecasts are not very sensitive to the temperature scenarios. In

particular, the forecasts relative to 2099 for the sea level range between q0.01 =0.1999[m] and q0.99 = 0.2817[m], with a mean value of 0.2410[m], conditioning

on all temperature scenarios, taken with equal probability, see figure 3.3. This

range does not change much if different groups are considered separately.

(i) A1b group: forecasts range from q0.01 = 0.1991[m] to q0.99 = 0.2696[m]


(ii) A2 group: from q0.01 = 0.1886[m] to q0.99 = 0.2704[m] with a mean value

of 0.2341[m];

(iii) B1 group: from q0.01 = 0.2121[m] to q0.99 = 0.2738[m] with a mean value

of 0.2428[m];

(iv) commit group: from q0.01 = 0.2072[m] to q0.99 = 0.2944[m] with a mean

value of 0.2507[m].

The difference between the average, smoothed sea level in 1990 and the smoothed

sea level in December 2009 is 0.0549[m]. Consequently, to compute the sea-level

changes with respect to the average 1990 level, 0.0549[m] has to be added to the

previous values. We make this remark because in several papers the sea-level rise

forecasts are reported with respect to the 1990 average, e.g. in Rahmstorf (2007b),

Vermeer and Rahmstorf (2009), and Grinsted et al. (2010).


Figure 3.1. AR4-IPCC tempterature scenarios.

1900 1950 2000 2050 2100

−1

0

1

2

3

4

years

temperature

anomalies(K

)

global mean temperature observationsAR4-IPCC-a1b scenariosAR4-IPCC-a2 scenariosAR4-IPCC-b1 scenariosAR4-IPCC-commit scenarios

Temperature observations and AR4-IPCC-SRES tempterature scenarios. The temperature observations areyearly averages, ranging from January 1880 to December 2009. The AR4-IPCC-SRES scenarios correspondto yearly simulated values, ranging from January 2010 to December 2099.

Figure 3.2. Forecasts based on A1b-A2-B1-commit-IPCC scenarios and model 2.

1900 1950 2000 2050 2100

−0.2

−0.1

0

0.1

0.2

0.3

years

sea-level

change(m

)

sea-level observationssea-level smoothed series and projections0.98 confidence interval

Forecasts based on IPCC-SRES (A1b, A2, B1, and commit groups) scenarios and model 2 (see Section3.4). The observations and the projections are monthly values. The sea-level observations range fromJanuary 1880 to December 2009. The projections correspond to smoothed monthly values and range fromJanuary 2010 to December 2099. The base sea-level value is the smoothed sea level in December 2009. Theconfidence bands correspond to a 98% confidence interval (see Section 3.4).


Figure 3.3. Forecasts based on A1b-A2-B1-commit-IPCC scenarios and model 4.

1900 1950 2000 2050 2100

−0.2

−0.1

0

0.1

0.2

0.3

years

sea-level

change(m

)

sea-level observationssea-level smoothed series and projections0.98 confidence interval

Forecasts based on IPCC-SRES (A1b, A2, B1, and commit groups) scenarios and model 4 (see Section3.4). The observations and the projections are monthly values. The sea-level observations range fromJanuary 1880 to December 2009. The projections correspond to smoothed monthly values and range fromJanuary 2010 to December 2099. The base sea-level value is the smoothed sea-level in December 2009. Theconfidence bands correspond to a 98% confidence interval (see Section 3.4).

3.7 Conclusions

In this paper we proposed a statistical framework to model and forecast the global

mean sea level, conditional on the global mean temperature. The specification is for-

mulated as a continuous-time state-space system. The state vector is composed of the

unobserved sea level and temperature processes, as well as trend components and,

jointly, follow an Ornstein-Uhlenbeck process. This process can be exactly discretised.

The measurement equation adds independent noise to the discretely sampled states.

The resulting system is linear and Kalman filtering techniques apply. In particular,

the Kalman filter is used to compute the likelihood function. Furthermore, we exploit

the ability of the Kalman filter to deal with missing observations, to make projections

for the sea level. The state-space specification also allows to model changes in the

accuracy of the reconstructed sea-level series. Specifically, this is achieved by allowing

the volatility parameter of the sea-level measurement error to be time-varying, and

matching it to the sea-level uncertainty values reported in Church and White (2011).

We find that this modelling scheme performs better, in forecasting, compared to one

in which the volatility of the measurement error of the sea level is estimated together

with the other parameters. If σ2,Sτ =σ2,S is left unrestricted and estimated together

with the other parameters, the value obtained is very close to zero. This causes the

filtered (and smoothed) sea-level series to essentially coincide with the observed one.

In this case the predictive ability of the model deteriorates. The advantage of using

the proposed state-space model is that there is no difference between the system

dynamics assumed for the variables of interest and the statistical model estimated

3.7. CONCLUSIONS 101

using real data.

The choice of the models was made according to their forecasting performance,

relative to selected benchmarks, namely the Rahmstorf (2007b) and Vermeer and

Rahmstorf (2009) methods. This model selection criterion was chosen because of the

final objective of semi-empirical models, that is making long-term projections for

the sea level.

We find that the magnitude of the parameter A∗,ST , linking the sea level to the

temperature, is estimated to be of the order of 10−2[mm/cK ] (10−3[m/K ]). Note

that A∗,ST relates the value of the unobserved temperature process at time τ to the

value of the sea level at time τ+1, where the time step corresponds to one month.

The low value of A∗,ST may be caused by long response times of the sea level to the

temperature and the fact that the time step is quite short. Early studies indicate lags

of the order of 20 years (240 months), between temperature and sea-level rise, see for

instance Gornitz et al. (1982).

When the parameter aSS is left unconstrained, the coefficient linking sea level and

temperature is estimated to be very low. However, if the parameter aSS is restricted to

be zero, the forecast performance of the model deteriorates considerably. One puz-

zling fact is the change in sign of A∗,ST and A∗,T S , between the linear and quadratic

trend specifications. In particular, both parameters are positive in the linear trend

specification and negative in the quadratic trend one. This finding is quite surprising,

considering that the quadratic trend model seems to forecast better than the linear

one. Concerning the model comparison exercise, the state-space specifications be-

have well compared to the Rahmstorf (2007b) and Vermeer and Rahmstorf (2009)

methods. The choice of the trend component influences somewhat the forecasting

performance of the model.

We make projections for the sea level from 2010 up to 2099. Under the linear trend

specification the forecasts are quite sensitive to the temperature scenarios, whereas

under the quadratic one the projections are quite similar across scenarios. Denote

with q0.01 and q0.99 the 1st and 99th percentiles, respectively, of the distribution of

sea-level rise forecasts for the year 2099 with respect to the (smoothed) sea-level value

in 2009. Conditionally on all the 75 temperature scenarios, the sea-level rise forecasts

range from q0.01 = 0.0948[m] to q0.99 = 0.3525[m], with a mean value of 0.2130[m]

under the linear trend specification, and from q0.01 = 0.1999[m] to q0.99 = 0.2817[m],

with a mean value of 0.2410[m], under the quadratic trend model. With respect to

the mean, smoothed 1990 sea-level value, the above results translate into sea-level

rise forecasts ranging from q0.01 = 0.1497[m] to q0.99 = 0.4074[m], with a mean value

of 0.2679[m] under the linear trend specification, and from q0.01 = 0.2548[m] to

q0.99 = 0.3366[m], with a mean value of 0.2959[m], under the quadratic trend model.

The projections obtained with the models proposed in this study are lower than

the ones obtained in Rahmstorf (2007b), Vermeer and Rahmstorf (2009), and Grin-

sted et al. (2010). In particular, their projections for the year 2100, with respect to the


mean sea level in 1990, range from 0.5[m] to 1.38[m] for Rahmstorf (2007b), from

0.72[m] to 1.81[m] for Vermeer and Rahmstorf (2009), and from 1.30[m] to 1.80[m]

(or between 0.95[m] to 1.48[m], depending on the temperature reconstruction) for

Grinsted et al. (2010). Note, however, that in these three papers the interpretation of

the range spanned by the forecasts is different from ours. Most likely, we obtain lower

estimates for the sea level because of the coefficient relating temperature to sea-level

A∗,ST , which in the state-space specifications is estimated to be quite small. This im-

plies longer estimated response times of the sea level to changes in the temperature,

with respect to the aforementioned studies.

Possible continuations of this work could be represented by considering alter-

native continuous-time stochastic processes to model sea level and temperature,

for instance geometric Brownian motions or more general Itô processes. Another

important aspect in this analysis was the specification of the trend components,

which played a key role in the forecasting of the sea level. It would be interesting

to study alternative specifications for the trend components and their relation to

different climate forcings, such as human induced changes in greenhouse gases and

aerosols concentrations.

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3.9. APPENDIX 107

3.9 Appendix

Details of univariate Kalman filter

In this section we give details on the Kalman filter and its univariate version. The

univariate Kalman filter was used for the computation of the likelihood function and

for the bootstrap procedure, needed to compute prediction intervals for the sea-level

projections, see Section 3.6. This subsection draws heavily on Durbin and Koopman

(2012).

Consider the following state-space system

yt = Zαt +εt ,

αt+1 = c+Tαt +ηt , (3.37)

where εt ∼ N (0,Ht ) takes values in Rp , with Ht a covariance matrix, ηt ∼ N (0,Q)

takes values in Rk , with Q a covariance matrix, yt ∈ Rp , αt ∈ Rk , and Z, T, and c are

parameter matrices and vectors of appropriate dimensions. Notice that the state-

space system (3.34) is of the same type as system (3.37). The standard Kalman filter

recursions for system (3.37) are

vt = yt −Zat ,

Ft = ZPt Z′+Ht ,

Kt = Pt Z′, (3.38)

at |t = at + Kt F−1t vt , Pt |t = Pt − Kt F−1

t K′t ,

at+1 = Tt at |t +c, Pt+1 = TPt |t T′+Q, (3.39)

for t = 1, . . . ,n, where Pt = E[[αt −at ][αt −at ]′

], Pt |t = E

[[αt −at |t ][αt −at |t ]′

], and

at = E [α|y0, . . . ,yt−1] and at |t = E [α|y0, . . . ,yt ] are the one-step-ahead prediction and

the filtered states, respectively. Koopman and Durbin (2000) derived a univariate

version of this algorithm in the case of diagonal variance-covariance matrices Ht . In

this case the system (3.37) can be represented as

yt ,i = ziαt ,i +εt ,i , t = 1, . . . ,n i = 1, . . . , p,

αt ,i+1 =αt ,i , t = 1, . . . ,n i = 1, . . . , p −1,

αt ,i+1 = c+Tαt ,i +ηt , t = 1, . . . ,n i = p. (3.40)

Where zi is the i − th row of matrix Z, yt ,i and εt ,i ∼ N (0,σ2t ,i ) are the i − th compo-

nents of yt and εt , respectively. The Kalman filter recursions for specification (3.40)


can be written as

vt ,i = yt ,i −zi at ,i , (3.41)

Ft ,i = zi Pt ,i z′i +σ2t ,i , (3.42)

kt ,i = Pt ,i z′i , (3.43)

at ,i+1 = at ,i + kt ,i F−1t ,i vt ,i

Pt ,i+1 = Pt ,i − kt ,i F−1t ,i k

′t ,i

for i = 1, . . . , p −1 t = 1, . . . ,n, (3.44)

at+1,1 = T(at ,i + kt ,i F−1t ,i vt ,i )+c

Pt+1,1 = T(Pt ,i − kt ,i F−1t ,i k

′t ,i )T′+Q

for i = p t = 1, . . . ,n. (3.45)

Notice that Ft ,i is a scalar. As a consequence the univariate recursions do not require

the inversion of p ×p matrices, as in the standard Kalman filter recursions (3.38)-

(3.39) and can lead to computational savings.

The state-space system (3.40) has two types of disturbances, namely εt ,i , and ηt .

The so called “innovation form” has a unique source of disturbance, that is vt ,i . The

innovation form is made up of the following equations:

yt ,i = zi at ,i + vt ,i ,

kt ,i = Pt ,i z′i ,

at ,i+1 = at ,i +kt ,i F−1t ,i vt ,i , for i = 1,2, . . . , p −1 t = 1,2, . . . ,n,

at+1,1 = T(at ,i +kt ,i F−1

t ,i vt ,i

)+c, for i = p t = 1,2, . . . ,n. (3.46)

The innovation form (3.46) of the state-space system (3.40) constitutes the basis for

the bootstrap procedure outlined in the following subsection.

3.9. APPENDIX 109

Bootstrap procedure

In this section we outline the bootstrap procedure used to compute the prediction

intervals for the sea-level projections, conditional on the IPCC scenarios, as described

in Section 3.6. We detail the algorithm with respect to the state-space system (3.37)

and the Kalman filter recursions (3.41)-(3.45). The algorithm is a modification of the

one proposed in Rodriguez and Ruiz (2009) that allows for time-varying measurement

error variances. Denote withθ the vector containing the parameters of the state-space

system (3.37). The algorithm we propose is made up of the following steps:

1. estimate the parameters of model (3.30) by maximum likelihood and obtain θ

and the sequence of innovations vt ,i i=1,...,pt=1,...,n ;

2. compute the centred innovations vct ,i i=1,...,p

t=1,...,n , obtained as vct ,i = vt ,i − vn,i , with

vn,i = (1/n)∑n

t=1 vt ,i ;

3. obtain the standardized innovations v st ,i i=1,...,p

t=1,...,n , computed as v st ,i =

vct ,ipFt ,i

;

4. obtain a sequence of bootstrap standardized innovations v∗t ,i i=1,...,p

t=1,...,n via ran-

dom draws with replacement from the randomly scaled standardized innova-

tions v st ,i ·εt ,i i=1,...,p

t=1,...,n , where εt ,i ∼ N (0,1);

5. compute a bootstrap replicate of the observations y∗t ,i i=1,...,p

t=1,...,n by means of the

innovation form (3.46) using v∗t ,i i=1,...,p

t=1,...,n and the estimated parameters θ;

6. estimate the corresponding bootstrap parameters θ∗ from the bootstrap repli-

cates;

7. run the Kalman filter with θ∗ using the original observations and one tempera-

ture scenario as described in Section 3.4.

Steps 1-7 are repeated N = 500 for each temperature scenario. As made clear in step

4 we make use of a wild bootstrap procedure as opposed to the simple re-sampling

method used in Rodriguez and Ruiz (2009). The wild bootstrap was originally pro-

posed by Wu (1986) and it is well known in the literature to perform better than a

simple resampling scheme in the presence of heteroskedasticity, see for instance

Liu et al. (1988) and Mammen (1993). In this paper the heteroskedasticity comes

from the time varying matrix Ht . Note that the variance of the innovations vt ,i is

given by Ft ,i = zi Pt ,i z′i +σ2t ,i where σ2

t ,i is time-varying. In the notation of equation

(3.33), it’s the parameter σ2,St that causes the innovations vt ,i , in equation (3.41), to

be heteroskedastic.


State-space system and dimensional analysis

In this section we rewrite the state-space system (3.22) with linear trend (3.24), making

clear the fundamental dimensions and the units of measurement of the quantities

involved. We make use of SI units apart from the time dimension, for which we use

months (or years). The time series of the sea level is in millimetres [mm] and the

temperature one is in centikelvin [cK ].

Continuous-time state equation

The continuous-time process driving the state equation has the following dimensions:

d

[mm]S(t )[cK ]T (t )[mm]

µS (t )[cK ]

µT (t )

=

0

0[mm

month

]λS[mm

month

]λT

[month]

d t

+

[1

month

]aSS

[mm

cK ·month

]aST

[1

month

]κS 0[

cKmm·month

]aT S

[1

month

]aT T 0

[1

month

]κT

0 0 0 0

0 0 0 0

[mm]S(t )[cK ]T (t )[mm]

µS (t )[cK ]

µT (t )

[month]

d t

+

1 0

0 1

0 0

0 0

[mm]

dηS (t )[cK ]

dηT (t )

, (3.47)

denoting with dη(t ) = [dηS (t ) : dηT (t )], we have E [dη(t )dη(t )′] =Σd t , withΣ a sym-

metric positive semidefinite matrix. To understand the units of measurement of the

components of Σ we can reason in the following way: first, write E [dη(t)dη(t)′] =E [

pΣdW(t)dW(t)′

pΣ′] where

pΣ represents a square root of the matrix Σ and

dW(t ) = [dW S (t ) : dW T (t )]′ is a two-dimensional Brownian motion such that

E [dW(t )dW(t )′] =[

1 ρ

ρ 1

]d t , (3.48)

where |ρ| < 1; second, denote

pΣ =

[ωSS ωST

ωT S ωT T

], (3.49)

3.9. APPENDIX 111

we can write

pΣdW(t ) =

[ωSS dW S (t )+ωST dW T (t )

ωT S dW S (t )+ωT T dW T (t )

], (3.50)

and

E[pΣdW(t )dW(t )′

pΣ′] =

[ΣSS ΣST

ΣT S ΣT T

]d t , (3.51)

where

ΣSS d t = E[

(ωSS dW S )2 + (ωST dW T )2 +2ωSSωST dW S dW T]

= (ωSS )2d t + (ωST )2d t +2ωSSωSTρd t

=[

(ωSS )2 + (ωST )2 +2ωSSωSTρ]

d t , (3.52)

and

ΣST d t = E[ωSSωT S (dW S (t ))2 +ωSTωT S dW S (t )dW T (t )

]+ E

[ωSSωT T dW S (t )dW T (t )+ωSTωT T dW T (t )dW T (t )]

]= ωSSωT S d t +ωSTωT Sρd t

+ ωSSωT Tρd t +ωSTωT T d t

=[ωSSωT S +ωSTωT Sρ+ωSSωT Tρ+ωSTωT T

]d t , (3.53)

from equation (3.52) we can deduce that ωSS and ωST have units of measurement

corresponding to [mm/p

month]; to see this set ωST = 0 in equation 3.52, we have

then ΣSS d t = (ωSS )2d t which has units of measurement of [mm2], as it is the expec-

tation of the square of a quantity with units [mm], this implies that (ωSS )2 and ΣSS

have units of [mm2/month], as time is measured in months [month]; from equation

(3.52) it is also clear that ωST has the same units as ωSS . Using the same line of rea-

soning we deduce that (ωT T )2, (ωT S )2, and ΣT T have units of [cK 2/month]. Having

recovered the units of measurement of ωSS , ωST , ωT S , and ωT T we can deduce from

equation (3.53) that the units of measurement ofΣST andΣT S are [(mm ·cK )/month].

In summary we obtain:

Σ =

[mm2

month

]ΣSS

[mm·cKmonth

]ΣST[

cK ·mmmonth

]ΣT S

[cK 2

month

]ΣT T

. (3.54)


Discrete-time state vector and measurement equation

The discretized state vector has the following dimensions:

[mm]Sτ+1[cK ]

Tτ+1[mm]

µSτ+1

[cK ]

µTτ+1

=

[mm]

c∗,S

[cK ]

c∗,T

[mm]

c∗,µS

[cK ]

c∗,µT

+

A∗,SS

[mmcK

]A∗,ST A∗,SµS

[mmcK

]A∗,SµT

[cK

mm

]A∗,T S A∗,T T

[cK

mm

]A∗,TµS

A∗,TµT

0 0 1 0

0 0 0 1

[mm]Sτ

[cK ]Tτ

[mm]

µSτ

[cK ]

µTτ

+

[mm]

ξSτ

[cK ]

ξTτ

0

0

, (3.55)

denoting ξτ = [ξSτ : ξT

τ ]′, we have

E[ξτξ

′τ

]=

[

mm2]

Σ∗,SS[mm·cK ]Σ∗,ST

[cK ·mm]Σ∗,T S

[cK 2

]Σ∗,T T

, (3.56)

where the units of measurement were obtained by following the same logic as in the

previous section.

Finally, the measurement equation has the following dimensions:

[mm]Srτ

[cK ]T rτ

=[

1 0 0 0

0 1 0 0

]

[mm]Sτ

[cK ]Tτ

[mm]

µSτ

[cK ]

µTτ

+

[mm]

εSτ

[cK ]

εTτ

. (3.57)

3.9. APPENDIX 113

Tables

In this section we report the tables with the results for the model comparison fore-

casting exercise and the parameter estimates for models 1-2 and models 3-4. See

Sections 3.4 and 3.6 for more details.


Table

3.1.Ratio

sb

etween

R1f

and

Rjf ,

j=1,...,12.

n ∗1535

15101485

14601435

14101385

13601335

1310f

2550

75100

125150

175200

225250

mo

del1

1,001,00

1,001,00

1,001,00

1,001,00

1,001,00

mo

del2

1,001,02

1,021,00

1,011,04

0,891,06

1,031,02

mo

del3

0,991,10

1,081,03

1,011,03

1,021,00

1,021,03

mo

del4

0,991,09

1,071,02

0,990,99

1,020,9

0,971,03

mo

del5

1,071,00

1,311,21

1,191,24

1,060,75

1,091,04

mo

del6

1,060,98

1,261,17

1,070,82

0,621,01

0,560,5

mo

del7

0,71,26

1,241,09

1,171,14

0,760,55

1,010,97

mo

del8

0,71,25

1,221,08

1,151,22

1,191,15

1,030,9

mo

del9

1,071,00

1,311,21

1,191,24

1,060,75

1,091,04

mo

del10

1,060,99

1,261,17

0,980,99

0,771,16

0,750,66

mo

del11

0,71,24

1,211,01

1,020,89

0,520,38

0,560,54

mo

del12

0,71,22

1,170,99

1,001,18

0,870,73

0,971,02

n ∗1535

15101485

14601435

14101385

13601335

1310f

2550

75100

125150

175200

225250

mo

del1

1,001,00

1,001,00

1,001,00

1,001,00

1,001,00

mo

del2

1,041,05

0,911,00

0,980,91

0,90,99

0,940,97

mo

del3

1,141,09

0,70,76

1,281,10

1,231,04

1,421,85

mo

del4

1,121,17

0,780,85

1,361,21

1,371,12

1,592,03

mo

del5

1,121,05

0,310,31

0,670,5

0,480,46

0,670,96

mo

del6

0,870,68

0,310,31

0,590,52

0,490,46

0,671,02

mo

del7

1,190,6

0,380,36

0,570,51

0,680,87

1,031,29

mo

del8

0,910,51

0,380,35

0,540,48

0,620,78

0,971,17

mo

del9

1,121,05

0,310,31

0,670,5

0,480,46

0,670,96

mo

del10

1,111,09

0,40,41

0,90,8

0,710,66

0,971,89

mo

del11

0,851,14

0,590,66

1,231,21

1,631,43

1,421,80

mo

del12

1,101,08

0,710,75

1,241,09

1,481,53

1,361,91

Ratio

sb

etween

R1f

(perfo

rman

cem

easure

(3.35)fo

rm

od

el1)an

dR

jf(p

erform

ance

measu

re(3.35)

for

mo

del

j),j=

1,...,12fo

rd

ifferentvalu

eso

fn ∗(th

elen

gtho

fthe

estimatio

nsam

ple)

and

f(th

en

um

ber

ofo

ut-o

f-samp

lefo

recasts).SeeSectio

n3.4

for

mo

red

etailso

nth

efo

recasting

pro

cedu

re.Th

eco

mp

leted

atasetis

mad

eu

po

fn=

n ∗+f=

1560o

bservatio

ns,ran

ging

from

Janu

ary1880

toD

ecemb

er2009.

3.9. APPENDIX 115

Tab

le3.

2.R

atio

sb

etw

een

R2 f

and

Rj f

,j=

1,..

.,12

.

n∗

1535

1510

1485

1460

1435

1410

1385

1360

1335

1310

f25

5075

100

125

150

175

200

225

250

mo

del

11,

000,

980,

981,

000,

990,

961,

120,

940,

980,

98m

od

el2

1,00

1,00

1,00

1,00

1,00

1,00

1,00

1,00

1,00

1,00

mo

del

30,

991,

081,

051,

031,

000,

981,

140,

941,

001,

01m

od

el4

0,99

1,07

1,04

1,02

0,98

0,95

1,14

0,84

0,95

1,01

mo

del

51,

070,

981,

281,

201,

171,

191,

190,

711,

071,

02m

od

el6

1,06

0,96

1,23

1,17

1,06

0,78

0,7

0,95

0,54

0,49

mo

del

70,

71,

231,

211,

091,

151,

090,

850,

520,

990,

95m

od

el8

0,7

1,23

1,20

1,08

1,14

1,17

1,34

1,08

1,00

0,89

mo

del

91,

070,

981,

281,

201,

171,

191,

190,

711,

071,

02m

od

el10

1,06

0,97

1,24

1,16

0,97

0,95

0,87

1,09

0,73

0,65

mo

del

110,

71,

221,

181,

011,

000,

850,

580,

360,

550,

53m

od

el12

0,7

1,19

1,14

0,99

0,99

1,13

0,98

0,69

0,94

1,00

n∗

1535

1510

1485

1460

1435

1410

1385

1360

1335

1310

f25

5075

100

125

150

175

200

225

250

mo

del

10,

960,

951,

101,

001,

021,

101,

111,

011,

061,

03m

od

el2

1,00

1,00

1,00

1,00

1,00

1,00

1,00

1,00

1,00

1,00

mo

del

31,

101,

040,

770,

761,

301,

211,

371,

051,

511,

91m

od

el4

1,07

1,12

0,86

0,85

1,39

1,33

1,52

1,13

1,69

2,09

mo

del

51,

071,

000,

340,

310,

680,

550,

530,

460,

710,

98m

od

el6

0,83

0,65

0,34

0,31

0,6

0,58

0,55

0,46

0,71

1,05

mo

del

71,

140,

570,

420,

360,

580,

570,

760,

871,

101,

33m

od

el8

0,87

0,49

0,42

0,35

0,55

0,53

0,69

0,79

1,03

1,20

mo

del

91,

071,

000,

340,

310,

680,

550,

540,

460,

710,

99m

od

el10

1,07

1,03

0,44

0,41

0,92

0,88

0,79

0,67

1,03

1,95

mo

del

110,

821,

080,

640,

661,

251,

331,

811,

441,

511,

86m

od

el12

1,06

1,03

0,78

0,75

1,26

1,20

1,65

1,54

1,44

1,97

Rat

ios

bet

wee

nR

2 f(p

erfo

rman

cem

easu

re(3

.35)

for

mo

del

2)an

dR

j f(p

erfo

rman

cem

easu

re(3

.35)

for

mo

del

j),

j=

1,..

.,12

for

dif

fere

ntv

alu

eso

fn∗

(th

ele

ngt

ho

fth

e

esti

mat

ion

sam

ple

)an

df

(th

en

um

ber

ofo

ut-

of-

sam

ple

fore

cast

s).S

eeSe

ctio

n3.

4fo

rm

ore

det

ails

on

the

fore

cast

ing

pro

ced

ure

.Th

eco

mp

lete

dat

aset

ism

ade

up

of

n=

n∗ +

f=

1560

ob

serv

atio

ns,

ran

gin

gfr

om

Jan

uar

y18

80to

Dec

emb

er20

09.


Table

3.3.Ratio

sb

etween

R3f

and

Rjf ,

j=1,...,12.

n ∗1535

15101485

14601435

14101385

13601335

1310f

2550

75100

125150

175200

225250

mo

del1

1,010,91

0,930,97

0,990,97

0,981,00

0,980,97

mo

del2

1,010,93

0,950,97

1,001,02

0,881,06

1,000,99

mo

del3

1,001,00

1,001,00

1,001,00

1,001,00

1,001,00

mo

del4

1,001,00

0,990,99

0,980,96

1,000,9

0,951,00

mo

del5

1,070,91

1,221,17

1,171,21

1,040,75

1,071,01

mo

del6

1,070,9

1,171,13

1,060,8

0,611,01

0,540,48

mo

del7

0,711,15

1,151,06

1,151,11

0,750,55

0,990,95

mo

del8

0,711,14

1,141,04

1,141,19

1,171,15

1,010,88

mo

del9

1,070,91

1,221,17

1,171,21

1,040,75

1,071,01

mo

del10

1,070,9

1,181,13

0,970,96

0,761,16

0,730,65

mo

del11

0,711,13

1,130,98

1,000,86

0,510,38

0,550,52

mo

del12

0,711,11

1,090,96

0,981,15

0,860,73

0,950,99

n ∗1285

12601235

12101185

11601135

11101085

1060f

275300

325350

375400

425450

475500

mo

del1

0,870,91

1,421,31

0,780,91

0,810,96

0,70,54

mo

del2

0,910,96

1,301,31

0,770,83

0,730,95

0,660,52

mo

del3

1,001,00

1,001,00

1,001,00

1,001,00

1,001,00

mo

del4

0,981,07

1,111,12

1,071,10

1,111,07

1,121,10

mo

del5

0,980,96

0,440,4

0,520,45

0,390,44

0,470,52

mo

del6

0,760,62

0,450,4

0,460,47

0,40,44

0,470,55

mo

del7

1,040,54

0,540,48

0,450,47

0,560,83

0,730,7

mo

del8

0,80,47

0,540,46

0,420,44

0,50,75

0,680,63

mo

del9

0,980,96

0,440,4

0,520,45

0,390,44

0,470,52

mo

del10

0,970,99

0,570,54

0,710,72

0,580,63

0,681,02

mo

del11

0,751,04

0,840,86

0,961,10

1,321,37

1,000,97

mo

del12

0,970,99

1,010,98

0,970,99

1,201,46

0,961,03

Ratio

sb

etween

R3f

(perfo

rman

cem

easure

(3.35)fo

rm

od

el3)an

dR

jf(p

erform

ance

measu

re(3.35)

for

mo

del

j),j=

1,...,12fo

rd

ifferentvalu

eso

fn ∗(th

elen

gtho

fthe

estimatio

nsam

ple)

and

f(th

en

um

ber

ofo

ut-o

f-samp

lefo

recasts).SeeSectio

n3.4

for

mo

red

etailso

nth

efo

recasting

pro

cedu

re.Th

eco

mp

leted

atasetis

mad

eu

po

fn=

n ∗+f=

1560o

bservatio

ns,ran

ging

from

Janu

ary1880

toD

ecemb

er2009.

3.9. APPENDIX 117

Tab

le3.

4.R

atio

sb

etw

een

R4 f

and

Rj f

,j=

1,..

.,12

.

n∗

1535

1510

1485

1460

1435

1410

1385

1360

1335

1310

f25

5075

100

125

150

175

200

225

250

mo

del

11,

010,

920,

940,

981,

011,

010,

981,

121,

030,

97m

od

el2

1,01

0,93

0,96

0,98

1,02

1,06

0,88

1,18

1,06

0,99

mo

del

31,

001,

001,

011,

011,

031,

041,

001,

121,

061,

00m

od

el4

1,00

1,00

1,00

1,00

1,00

1,00

1,00

1,00

1,00

1,00

mo

del

51,

070,

911,

231,

181,

201,

261,

040,

841,

131,

01m

od

el6

1,07

0,9

1,18

1,14

1,08

0,83

0,61

1,12

0,57

0,48

mo

del

70,

711,

151,

161,

071,

181,

150,

750,

611,

050,

95m

od

el8

0,71

1,15

1,15

1,05

1,16

1,24

1,17

1,28

1,06

0,88

mo

del

91,

070,

911,

231,

181,

201,

261,

040,

841,

131,

01m

od

el10

1,07

0,9

1,19

1,14

0,99

1,00

0,76

1,30

0,77

0,64

mo

del

110,

711,

141,

140,

991,

030,

90,

510,

420,

580,

52m

od

el12

0,71

1,11

1,09

0,97

1,01

1,19

0,86

0,82

1,00

0,99

n∗

1535

1510

1485

1460

1435

1410

1385

1360

1335

1310

f25

5075

100

125

150

175

200

225

250

mo

del

10,

890,

851,

281,

170,

730,

830,

730,

890,

630,

49m

od

el2

0,93

0,9

1,17

1,17

0,72

0,75

0,66

0,88

0,59

0,48

mo

del

31,

020,

930,

90,

90,

940,

910,

90,

930,

90,

91m

od

el4

1,00

1,00

1,00

1,00

1,00

1,00

1,00

1,00

1,00

1,00

mo

del

51,

000,

890,

40,

360,

490,

410,

350,

410,

420,

47m

od

el6

0,78

0,58

0,4

0,36

0,43

0,43

0,36

0,41

0,42

0,5

mo

del

71,

060,

510,

490,

430,

420,

430,

50,

770,

650,

64m

od

el8

0,81

0,44

0,49

0,41

0,4

0,4

0,45

0,7

0,61

0,58

mo

del

91,

000,

90,

40,

360,

490,

410,

350,

410,

420,

47m

od

el10

0,99

0,92

0,51

0,49

0,66

0,66

0,52

0,59

0,61

0,93

mo

del

110,

760,

970,

750,

770,

91,

001,

191,

280,

90,

89m

od

el12

0,99

0,92

0,91

0,87

0,91

0,9

1,08

1,36

0,86

0,94

Rat

ios

bet

wee

nR

4 f(p

erfo

rman

cem

easu

re(3

.35)

for

mo

del

4)an

dR

j f(p

erfo

rman

cem

easu

re(3

.35)

for

mo

del

j),

j=

1,..

.,12

for

dif

fere

ntv

alu

eso

fn∗

(th

ele

ngt

ho

fth

e

esti

mat

ion

sam

ple

)an

df

(th

en

um

ber

ofo

ut-

of-

sam

ple

fore

cast

s).S

eeSe

ctio

n3.

4fo

rm

ore

det

ails

on

the

fore

cast

ing

pro

ced

ure

.Th

eco

mp

lete

dat

aset

ism

ade

up

of

n=

n∗ +

f=

1560

ob

serv

atio

ns,

ran

gin

gfr

om

Jan

uar

y18

80to

Dec

emb

er20

09.


Table 3.5. Parameter estimates for model 1-2.

(a) Continuous-time, model 1-2, parameter estimates.

σT aSS aST aT S aT T[cK

] [1

month

] [mm

cK ·month

] [cK

mm·month

] [1

month

]value 7,59 -0,0112 0,0056 0,0512 -0,0816

std. (0,2311) (0,0016) (0,002) (0,0075) (0,0075)t-ratio [32,84] [-7] [2,8] [6,83] [-10,88]

λSl

λTl

√ΣSS ΣST

√ΣT T[

mmmonth

] [cK

month

] [mmp

month

] [mm·cKmonth

] [cKp

month

]value 0,0012 -0,0022 1,11 0,34 5,87

std. (0,0001) (0,0008) (0,1151) (0,3733) (0,2703)t-ratio [12] [-2,75] [9,64] [0,91] [21,72]

Parameter estimates for model 1-2 in the continuous-time representation (equations (3.22), (3.24), and(3.32)). Standard deviations are reported in parentheses, and the t-ratios in square brackets. The parameterswere estimated using information from time t = 1 (January 1880) to time t = n (December 2009). Thestandard deviations were obtained from the bootstrap procedure described in Appendix 3.9.

(b) Continuous-time, model 1-2, parameter estimates (alternative measurement units).

σT aSS aST aT S aT T[K

] [1

year

] [mm

cK ·month

] [cK

mm·month

] [1

year

]value 0,0759 -0,1344 0,00672 6,144 -0,9792

λSl

λTl

√ΣSS ΣST

√ΣT T[

myear

] [K

year

] [mpyear

] [m·Kyear

] [Kp

year

]value 0,000014 -0,00026 0,0038 0,000041 0,20

Parameter estimates for model 1-2 in the continuous-time representation (equations (3.22), (3.24), and(3.32)). The parameters were estimated using information from time t = 1 (January 1880) to time t = n(December 2009).

3.9. APPENDIX 119


(a) Discrete-time, model 1-2, parameter estimates.

σT A∗,SS A∗,ST A∗,SµSA∗,SµT

A∗,T S A∗,T T A∗,TµS[cK

] [mmcK

] [mmcK

] [cK

mm

] [cK

mm

]value 7,59 0,99 0,0054 0,99 0,0027 0,0489 0,92 0,0248

std. (0,2311) (0,0015) (0,0019) (0,0008) (0,001) (0,007) (0,0068) (0,0036)t-ratio [32,82] [641,96] [2,86] [1275,33] [2,86] [6,98] [135,24] [6,92]

A∗,TµTcS cT cµ

Scµ

T √Σ∗,SS Σ∗,ST

√Σ∗,T T

[mm][cK

][mm]

[cK

][mm]

[mm · cK

] [cK

]value 0,96 0,0006 -0,001 0,0012 -0,0022 1,11 0,44 5,64

std. (0,0035) (0,0001) (0,0004) (0,0001) (0,0008) (0,1108) (0,3661) (0,2535)t-ratio [273,14] [8,28] [-2,56] [8,25] [-2,57] [9,98] [1,21] [22,24]

Parameter estimates for model 1-2 in the dicrete-time representation (equations (3.34)). Standard devi-ations are reported in parentheses, and the t-ratios in square brackets. The parameters were estimatedusing information from time t = 1 (January 1880) to time t = n (December 2009). The standard deviationswere obtained from the bootstrap procedure described in Appendix 3.9.

(b) Discrete-time, model 1-2, parameter estimates (alternative measurement units).


A∗,T S A∗,T T A∗,TµS[K

] [mK

] [mK

] [Km

] [Km

]value 0,0759 0,99 0,00054 0,99 0,00027 0,489 0,92 0,248

A∗,TµTcS cT cµ

Scµ

T √Σ∗,SS Σ∗,ST

√Σ∗,T T

[m][K

][m]

[K

][m]

[m ·K

] [K

]value 0,96 0,0000006 -0,00001 0,0000012 -0,000022 0,00111 0,0000044 0,0564

Parameter estimates for model 1-2 in the discrete-time representation (equations (3.34)). The parameterswere estimated using information from time t = 1 (January 1880) to time t = n (December 2009).



(a) Continuous-time, model 3-4, parameter estimates.

σT aSS aST aT S aT T[cK

] [1

month

] [mm

cK ·month

] [cK

mm·month

] [1

month

]value 7,41 -0,0288 -0,0037 -0,0458 -0,1169

std. (0,27) (0,0014) (0,0026) (0,006) (0,0106)t-ratio [27,44] [-20,57] [-1,42] [-7,63] [-11,03]

λSq λT

q

√ΣSS ΣST

√ΣT T[

mmmonth3

] [cK

month3

] [mmp

month

] [mm·cKmonth

] [cKp

month

]value 0,000003 0,000015 1,25 1,46 5,91

std. (0,0000004) (0,000002) (0,19) (0,47) (0,30)t-ratio [7,5] [7,5] [6,43] [3,12] [19,97]

Parameter estimates for model 3-4 in the continuous-time representation (equations (3.22), (3.25), and(3.32)). Standard deviations are reported in parentheses, and the t-ratios in square brackets. The parameterswere estimated using information from time t = 1 (January 1880) to time t = n (December 2009). Thestandard deviations were obtained from the bootstrap procedure described in Appendix 3.9.

(b) Continuous-time, model 3-4, parameter estimates (alternative measurement units).

σT aSS aST aT S aT T[K

] [1

year

] [m

K ·year

] [K

m·year

] [1

year

]value 0,0741 -0,3456 -0,00444 -5,496 -1,4028

λSq λT

q

√ΣSS ΣST

√ΣT T[

myear 3

] [K

year 3

] [mpyear

] [m·Kyear

] [Kp

year

]value 0,00000043 0,000021 0,0043 0,00018 0,20

Parameter estimates for model 3-4 in the continuous-time representation (equations (3.22), (3.25), and(3.32)). The parameters were estimated using information from time t = 1 (January 1880) to time t = n(December 2009).

3.9. APPENDIX 121

Table 3.8. Discrete-time, model 3-4, parameter estimates.


A∗,SλS[cK

] [mmcK

] [mmcK

] [month

]value 7,41 0,97 -0,0035 0,99 -0,0018 0,4952

std. (0,27) (0,0013) (0,0024) (0,0007) (0,0012) (0,0002)t-ratio [27,44] [746,15] [-1,46] [1414,29] [-1,5] [2476]

A∗,SλTA∗,T S A∗,T T A∗,TµS

A∗,TµTA∗,TλS[

month·mmcK

] [cK

mm

] [cK

mm

] [month·cK

mm

]value -0,0006 -0,0426 0,89 -0,0218 0,94 -0,0074

std. (0,0004) (0,0053) (0,0092) (0,0028) (0,0048) (0,0009)t-ratio [-1,5] [-8,04] [96,74] [-7,79] [196,63] [-8,22]

A∗,TλTcS cT cµ

Scµ

T[month

][mm]

[cK

][mm]

[cK

]value 0,4811 0,00000044 0,000002 0,000001 0,000007

std. (0,0016) (0,00000007) (0,00000039) (0,0000002) (0,000001)t-ratio [300,69] [6,29] [6,05] [6,65] [7,32]

cλS

cλT √

Σ∗,SS Σ∗,ST√Σ∗,T T[

mmmonth

] [cK

month

][mm]

[mm · cK

] [cK

]value 0,000003 0,000015 1,25 1,46 5,91

std. (0,0000004) (0,000002) (0,19) (0,47) (0,3)t-ratio [6,63] [7,32] [6,58] [3,11] [19,7]

Parameter estimates for model 3-4 in the discrete-time representation (equations (3.22), (3.25), and (3.32)).Standard deviations are reported in parentheses, and the t-ratios in square brackets. The parameters wereestimated using information from time t = 1 (January 1880) to time t = n (December 2009). The standarddeviations were obtained from the bootstrap procedure described in Appendix 3.9.


Table 3.9. Discrete-time, model 3-4, parameter estimates (alternative measurement units).


A∗,SλS[K

] [mK

] [mK

] [year

]value 0,074 0,97 -0,00035 0,99 -0,00018 0,041

A∗,SλTA∗,T S A∗,T T A∗,TµS

A∗,TµTA∗,TλS[

year ·mK

] [Km

] [Km

] [year ·K

m

]value -0,000005 -0,426 0,89 -0,218 0,94 -0,0062

A∗,TλTcS cT cµ

Scµ

T[year

][m]

[K

][m]

[K

]value 0,040 0,00000000044 0,00000002 0,000000001 0,00000007

cλS

cλT √

Σ∗,SS Σ∗,ST√Σ∗,T T[

myear

] [K

year

][m]

[m ·K

] [K

]value 0,000000036 0,0000018 0,0013 0,000014 0,0591

Parameter estimates for model 3-4 in the discrete-time representation (equations (3.22), (3.25), and (3.32)).The parameters were estimated using information from time t = 1 (January 1880) to time t = n (December2009).

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