ARCH Models and Financial Applications

229 Pages·1997·3.27 MB·English

by  Christian Gouriéroux (auth.)

Springer Series in Statistics
Advisors:
P. Bickel, P. Diggle, S. Fienberg, K. Krickeberg,
1. Olkin, N. Wermuth, S. Zeger
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(conHnuedafterindex)
Christian Gourieroux
ARCH Models and
Financial Applications
With26Figures
, Springer
Christian Gourieroux
Centre de Recherche en Economie et Statistique
Laboratoire de Finance-Assurance
Bâtiment Malakoff 2-Timbre 1320
15 Boulevard Gabriel Peri
92245 Malakoff Cedex, France
Library of Congress Cataloging-in-Publieation Data
Gourieroux, Christian, 1949-
ARCH models and financial applieations/Christian Gourieroux.
p. em. - (Springer series in statistics)
Inc1udes bibliographical referenees and index.
ISBN 978-1-4612-7314-1 ISBN 978-1-4612-1860-9 (eBook)
DOI 10.1007/978-1-4612-1860-9
1. Finanee-Mathematical models. 2. Autoregression (Statistics)
1. Title. II. Series.
HG176.5.G68 1997
332-de20 96-33588
Printed on acid-free paper.
© 1997 Springer Science+Business Media New York
Originally published by Springer-Verlag New York in 1997
Softcover reprint of the hardcover 1s t edition 1997
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98765432
ISBN 978-1-4612-7314-1 SPIN 10756637
Contents
1 Introduction 1
1.1 TheDevelopmentofARCHModels 1
1.2 BookContent 4
2 Linearand NonlinearProcesses 5
2.1 StochasticProcesses . . . . 5
2.2 WeakandStrictStationarity 8
2.3 AFewExamples . . . . . . 12
2.4 Nonlinearities ... . . . . 22
2.4.1 PortmanteauStatistic 22
2.4.2 SomeImplicationsoftheWhiteNoise Hypothesis. 23
2.5 Exercises.......................... 26
3 UnivariateARCHModels 29
3.1 AHeteroscedasticModelofOrderOne 29
3.1.1 DescriptionoftheModel ... 29
3.1.2 Propertiesofthe InnovationProcesse 30
3.1.3 Propertiesofthe Y Process ... 32
3.1.4 DistributionoftheErrorProcess. 33
3.2 GeneralPropertiesofARCHProcesses . 34
3.2.1 VariousExtensions. . . . . . . . 34
3.2.2 StationarityofaGARCH(p,q) Process 37
3.2.3 Kurtosis................ 38
Contents
VI
3.2.4 Yule-Walker Equations for the Square ofa
GARCHProcess . 38
3.3 Exercises . 39
4 EstimationandTests 43
4.1 PseudoMaximumLikelihoodEstimation 43
4.1.1 Generalities . . . . . . . . . . . 43
4.1.2 Thei.i.d.case . . . . . . . . . . 44
4.1.3 RegressionModel withHeteroscedasticErrors 46
4.1.4 Regression ModelwithARCHErrors 49
4.1.5 Application toaGARCH Model . 51
4.1.6 StochasticVarianceModel . . 52
4.2 TwoStepEstimationProcedures . . . . . 53
4.2.1 DescriptionoftheProcedures . . 53
4.2.2 Comparison ofthe Estimation Methods under
ConditionalNormality. . 54
4.2.3 EfficiencyLossAnalysis. 55
4.3 ForecastIntervals 56
4.4 HomoscedasticityTest . . . . . . 58
4.4.1 Regression ModelswithHeteroscedasticErrors 58
4.5 TheTestStatisticInterpretation .. . . . . . . . . . . 61
4.5.1 Application to Regression Models with ARCH or
GARCHErrors 62
Appendix 4.1: Matrices Jand J . . . . . . . . . . . . . . . 63
Appendix 4.2: Derivatives ofthe Log-Likelihood Function
and Information Matrix for a Regression Model with
ARCHErrors 64
4.6 Exercises........................ 65
5 SomeApplicationsofUnivariateARCH Models 67
5.1 LeptokurticAspectsofFinancialSeriesandAggregation 67
5.1.1 TheNormality Assumption . . . . . . . . . . 67
5.1.2 TheChoiceofaTimeUnit. . . . . . . . . . . 69
5.2 ARCH Processes as an Approximation ofContinuous
TimeProcesses. . . . . . . . . . . . . . 71
5.2.1 StochasticIntegrals 71
5.2.2 StochasticDifferentialEquations .. 73
5.2.3 SomeEquationsandTheirSolutions. 74
5.2.4 ContinuousandDiscreteTime. 76
5.2.5 Examples 78
5.2.6 SimulatedEstimationMethods 81
·5.3 TheRandomWalkHypothesis. . . . . 83
5.3.1 DescriptionoftheHypothesis . 83
5.3.2 The Classical Test Procedure of the
RandomWalkHypothesis . . . . . . 85
Contents vii
5.3.3 LimitationsofthePortmanteauTests ... 87
5.3.4 PortmanteauTestswithHeteroscedasticity . 88
5.4 ThresholdModels . . . . . . . . . . . . . . . 90
5.4.1 DefinitionandStationarityConditions. 90
5.4.2 HomoscedasticityTest . . . 92
5.4.3 QualitativeARCHModels . 93
5.4.4 NonparametricApproaches 95
5.5 IntegratedModels . 98
5.5.1 TheIGARCH(l,1)Model . 98
5.5.2 ThePersistenceEffect ... 99
5.5.3 WeakandStrongStationarity 100
5.5.4 Example 101
5.6 Exercises............... 101
6 MultivariateARCH Models 105
6.1 UnconstrainedModels . . . . . . . . 105
6.1.1 MultivariateGARCH Models 105
6.1.2 PositivityConstraints 107
6.1.3 StabilityConditions . . . 107
6.1.4 AnExample . . . . . . . 108
6.1.5 SpectralDecompositions. 109
6.2 ConstrainedModels . . . . . . . 111
6.2.1 DiagonalModels. . . . . 111
6.2.2 Models withConstantConditionalCorrelations 113
6.2.3 Models withRandomCoefficients . . . . . 114
6.2.4 ModelBasedonaSpectralDecomposition 116
6.2.5 FactorARCHModels . . . . . . . . . . 116
6.3 EstimationofHeteroscedasticDynamicModels . 117
6.3.1 PseudoMaximumLikelihoodEstimators 117
6.3.2 Asymptotic Properties of the Pseudo
MaximumLikelihoodEstimator . . . . . 119
6.3.3 Model withConstantConditionalCorrelations 120
6.3.4 FactorModels . 121
7 EfficientPortfoliosand HedgingPortfolios 125
7.1 Determinationofan EfficientPortfolio . 125
7.1.1 SecuritiesandPortfolios . . . . 125
7.1.2 MeanVarianceCriterion .... 128
7.1.3 MeanVarianceEfficientPortfolios . 129
7.2 PropertiesoftheSetofEfficientPortfolios . 132
7.2.1 TheSetofEfficientPortfolios . . 132
7.2.2 Factors . 135
7.3 AsymmetricInformationandAggregation . 137
7.3.1 Incoherencyofthe Mean Variance Approach 137
7.3.2 Studyofthe BasicPortfolios . 138
viii Contents
7.3.3 Aggregation................... 139
7.4 HedgingPortfolios. . . . . . . . . . . . . . . . . . . . 140
7.4.1 Determination ofaPortfolio Mimicking aSeries
ofInterest . . . . . . . . . . . . . . 141
7.4.2 AModelfortheCallSellerBehavior 142
7.4.3 TheFirmBehavior. . . . . . . . . 147
7.5 EmpiricalStudyofPerformanceMeasures . 148
7.5.1 PerformancesofaSetofAssets . . 148
7.5.2 Improvingthe Efficiency. . . . . . 149
7.5.3 Estimation ofthe Efficient Portfolio and its
PerformanceintheStaticCase. . . . . . . . 149
Appendix 1: PresentationinTermsofUtility. . . . . . . . 152
Appendix 2: MomentsoftheTruncatedLog-NormalDistribution 155
Appendix3: AsymptoticPropertiesofthe Estimators 156
7.6 Exercises.......................... 157
8 FactorModels,Diversificationand Efficiency 161
8.1 FactorModels . . . . . . . . . . . . . . . 162
8.1.1 LinearFactorRepresentation ... 162
8.1.2 Representation with EndogenousFactors 163
8.1.3 StructureoftheConditionalMoments . . 165
8.1.4 Cofactors 167
8.1.5 Characterization with the Matrix Defining the
EndogenousFactors . . . . . . . . 167
8.2 ArbitrageTheory . . . . . . . . . . . . . . 168
8.2.1 AbsenceofArbitrageOpportunities 168
8.2.2 DiversificationandPricingModel 169
8.2.3 DiversificationandRiskAversion 172
8.3 EfficiencyTestsandDiversification 172
8.3.1 Ex-AnteEfficiency. . . . . . . . 172
8.3.2 Ex-PostEfficiency . . . . . . . . 175
8.4 ConditionalandHistoricalPerformanceMeasures . 177
8.4.1 TheDynamicsofaModel withEndogenousFactors. 177
8.4.2 TestsforEx-AnteEfficiencyandPerformances 179
8.5 Exercises........................... 180
9 EquilibriumModels 183
9.1 CapitalAssetPricingModel 183
9.1.1 DescriptionoftheModel 183
9.1.2 MarketPortfolio. . . . . 184
9.1.3 TheCAPMasaFactorModel 185
9.1.4 SpectralDecompositionoftheMoments. 186
9.1.5 TimeDependentRiskAversion 187
9.2 TestoftheCAPM . . . . 187
9.2.1 SomeDifficulties 188
Contents ix
9.2.2 TestingProceduresinaStaticFramework . . . . 191
9.2.3 Test for Efficiency ofthe Market Portfolio in a
DynamicFrameworkwithConstantBetas 196
9.2.4 Testsin theGeneralCase ..... 197
9.3 ExamplesofStructuralModels . . . . . . 199
9.3.1 AModelwithSpeculativeBubbles 199
9.3.2 TheConsumptionBasedCAPM . . 202
References 207
Index 227
1
Introduction
1.1 The Development ofARCH Models
Timeseriesmodels havebeeninitiallyintroducedeitherfordescriptive purposes
like prediction and seasonalcorrection orfor dynamic control. In the 1970s,the
researchfocusedonaspecificclassoftimeseriesmodels,theso-calledautoregres
sivemovingaverageprocesses (ARMA), whichwereveryeasyto implement. In
thesemodels,thecurrentvalueoftheseriesofinterestiswrittenasalinearfunction
ofitsownlaggedvaluesandcurrentandpastvaluesofsomenoiseprocess,which
can be interpretedas innovations to the system. However, this approach has two
majordrawbacks: 1) itisessentially alinearsetup, which automaticallyrestricts
the type ofdynamics to be approximated; 2) it is generally applied without im
posingaprioriconstraintsontheautoregressiveandmovingaverageparameters,
whichisinadequateforstructuralinterpretations.
AmongthefieldofapplicationswherestandardARMAfitispoorarefinancial
and monetaryproblems.Thefinancial timeseries features various forms ofnon
lineardynamics,thecrucialonebeingthestrongdependenceoftheinstantaneous
variabilityoftheseriesonitsownpast.Moreover,financialtheoriesbasedoncon
ceptslikeequilibriumorrationalbehavioroftheinvestorswouldnaturallysuggest
including and testing some structural constraints on the parameters. In this con
text, ARCH (Autoregressive Conditionally Heteroscedastic) models, introduced
byEngle(1982),ariseas an appropriateframework for studyingtheseproblems.
Currently,thereexistmorethanonehundredpapersandsomedozenPh.D.theses
onthistopic, whichreflectstheimportanceofthis approachforstatisticaltheory,
financeandempiricalwork.