GARCH Models Structure, Statistical Inference and Financial Appl....pdf

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GARCH Models

Contents

Preface

Notation

1 Classical Time Series Models and Financial Series

1.1 Stationary Processes

1.2 ARMA and ARIMA Models

1.3 Financial Series

1.4 Random Variance Models

1.5 Bibliographical Notes

1.6 Exercises

Part I Univariate GARCH Models

2 GARCH(p, q) Processes

2.1 Definitions and Representations

2.2 Stationarity Study

2.2.1 The GARCH(1, 1) Case

2.2.2 The General Case

2.3 ARCH (∞) Representation

2.3.1 Existence Conditions

2.3.2 ARCH (∞) Representation of a GARCH

2.3.3 Long-Memory ARCH

2.4 Properties of the Marginal Distribution

2.4.1 Even-Order Moments

2.4.2 Kurtosis

2.5 Autocovariances of the Squares of a GARCH

2.5.1 Positivity of the Autocovariances

2.5.2 The Autocovariances Do Not Always Decrease

2.5.3 Explicit Computation of the Autocovariances of the Squares

2.6 Theoretical Predictions

2.7 Bibliographical Notes

2.8 Exercises

3 Mixing*

3.1 Markov Chains with Continuous State Space

3.2 Mixing Properties of GARCH Processes

3.3 Bibliographical Notes

3.4 Exercises

4 Temporal Aggregation and Weak GARCH Models

4.1 Temporal Aggregation of GARCH Processes

4.1.1 Nontemporal Aggregation of Strong Models

4.1.2 Nonaggregation in the Class of Semi-Strong GARCH Processes

4.2 Weak GARCH

4.3 Aggregation of Strong GARCH Processes in the Weak GARCH Class

4.4 Bibliographical Notes

4.5 Exercises

Part II Statistical Inference

5 Identification

5.1 Autocorrelation Check for White Noise

5.1.1 Behavior of the Sample Autocorrelations of a GARCH Process

5.1.2 Portmanteau Tests

5.1.3 Sample Partial Autocorrelations of a GARCH

5.1.4 Numerical Illustrations

5.2 Identifying the ARMA Orders of an ARMA-GARCH

5.2.1 Sample Autocorrelations of an ARMA-GARCH

5.2.2 Sample Autocorrelations of an ARMA-GARCH Process When the Noise is Not Symmetrically Distributed

5.2.3 Identifying the Orders (P,Q)

5.3 Identifying the GARCH Orders of an ARMA-GARCH Model

5.3.1 Corner Method in the GARCH Case

5.3.2 Applications

5.4 Lagrange Multiplier Test for Conditional Homoscedasticity

5.4.1 General Form of the LM Test

5.4.2 LM Test for Conditional Homoscedasticity

5.5 Application to Real Series

5.6 Bibliographical Notes

5.7 Exercises

6 Estimating ARCH Models by Least Squares

6.1 Estimation of ARCH(q) models by Ordinary Least Squares

6.2 Estimation of ARCH(q) Models by Feasible Generalized Least Squares

6.3 Estimation by Constrained Ordinary Least Squares

6.3.1 Properties of the Constrained OLS Estimator

6.3.2 Computation of the Constrained OLS Estimator

6.4 Bibliographical Notes

6.5 Exercises

7 Estimating GARCH Models by Quasi-Maximum Likelihood

7.1 Conditional Quasi-Likelihood

7.1.1 Asymptotic Properties of the QMLE

7.1.2 The ARCH(1) Case: Numerical Evaluation of the Asymptotic Variance

7.1.3 The Nonstationary ARCH(1)

7.2 Estimation of ARMA-GARCH Models by Quasi-Maximum Likelihood

7.3 Application to Real Data

7.4 Proofs of the Asymptotic Results*

7.5 Bibliographical Notes

7.6 Exercises

8 Tests Based on the Likelihood

8.1 Test of the Second-Order Stationarity Assumption

8.2 Asymptotic Distribution of the QML When ¥è0 is at the Boundary

8.2.1 Computation of the Asymptotic Distribution

8.3 Significance of the GARCH Coefficients

8.3.1 Tests and Rejection Regions

8.3.2 Modification of the Standard Tests

8.3.3 Test for the Nullity of One Coefficient

8.3.4 Conditional Homoscedasticity Tests with ARCH Models

8.3.5 Asymptotic Comparison of the Tests

8.4 Diagnostic Checking with Portmanteau Tests

8.5 Application: Is the GARCH(1,1) Model Overrepresented?

8.6 Proofs of the Main Results

8.7 Bibliographical Notes

8.8 Exercises

9 Optimal Inference and Alternatives to the QMLE*

9.1 Maximum Likelihood Estimator

9.1.1 Asymptotic Behavior

9.1.2 One-Step Efficient Estimator

9.1.3 Semiparametric Models and Adaptive Estimators

9.1.4 Local Asymptotic Normality

9.2 Maximum Likelihood Estimator with Misspecified Density

9.2.1 Condition for the Convergence of [omitted]

9.2.2 Reparameterization Implying the Convergence of [omitted]

9.2.3 Choice of Instrumental Density h

9.2.4 Asymptotic Distribution of [omitted]

9.3 Alternative Estimation Methods

9.3.1 Weighted LSE for the ARMA Parameters

9.3.2 Self-Weighted QMLE

9.3.3 Lp Estimators

9.3.4 Least Absolute Value Estimation

9.3.5 Whittle Estimator

9.4 Bibliographical Notes

9.5 Exercises

Part III Extensions and Applications

10 Asymmetries

10.1 Exponential GARCH Model

10.2 Threshold GARCH Model

10.3 Asymmetric Power GARCH Model

10.4 Other Asymmetric GARCH Models

10.5 A GARCH Model with Contemporaneous Conditional Asymmetry

10.6 Empirical Comparisons of Asymmetric GARCH Formulations

10.7 Bibliographical Notes

10.8 Exercises

11 Multivariate GARCH Processes

11.1 Multivariate Stationary Processes

11.2 Multivariate GARCH Models

11.2.1 Diagonal Model

11.2.2 Vector GARCH Model

11.2.3 Constant Conditional Correlations Models

11.2.4 Dynamic Conditional Correlations Models

11.2.5 BEKK-GARCH Model

11.2.6 Factor GARCH Models

11.3 Stationarity

11.3.1 Stationarity of VEC and BEKK Models

11.3.2 Stationarity of the CCC Model

11.4 Estimation of the CCC Model

11.4.1 Identifiability Conditions

11.4.2 Asymptotic Properties of the QMLE of the CCC-GARCH model

11.4.3 Proof of the Consistency and the Asymptotic Normality of the QML

11.5 Bibliographical Notes

11.6 Exercises

12 Financial Applications

12.1 Relation between GARCH and Continuous-Time Models

12.1.1 Some Properties of Stochastic Differential Equations

12.1.2 Convergence of Markov Chains to Diffusions

12.2 Option Pricing

12.2.1 Derivatives and Options

12.2.2 The Black–Scholes Approach

12.2.3 Historic Volatility and Implied Volatilities

12.2.4 Option Pricing when the Underlying Process is a GARCH

12.3 Value at Risk and Other Risk Measures

12.3.1 Value at Risk

12.3.2 Other Risk Measures

12.3.3 Estimation Methods

12.4 Bibliographical Notes

12.5 Exercises

Part IV Appendices

A Ergodicity, Martingales, Mixing

A.1 Ergodicity

A.2 Martingale Increments

A.3 Mixing

A.3.1 α-Mixing and β-Mixing Coefficients

A.3.2 Covariance Inequality

A.3.3 Central Limit Theorem

B Autocorrelation and Partial Autocorrelation

B.1 Partial Autocorrelation

B.2 Generalized Bartlett Formula for Nonlinear Processes

C Solutions to the Exercises

D Problems

References

Index

GARCH Models

GARCH Models Structure, Statistical Inference and Financial Applications Christian Francq University Lille 3, Lille, France Jean-Michel Zako¨ıan CREST, Paris, and University Lille 3, France A John Wiley and Sons, Ltd., Publication

This edition ﬁrst published 2010  2010 John Wiley & Sons Ltd Registered ofﬁce John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial ofﬁces, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. The right of the author to be identiﬁed as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloguing-in-Publication Data Francq, Christian. [Models GARCH. English] GARCH models : structure, statistical inference, and ﬁnancial applications / Christian Francq, Jean-Michel Zakoian. p. cm. Includes bibliographical references and index. ISBN 978-0-470-68391-0 (cloth) 1. Finance–Mathematical models. 2. Investments–Mathematical models. I. Zakoian, Jean-Michel. II. Title. HG106.F7213 2010 332.01’5195– dc22 2010013116 A catalogue record for this book is available from the British Library ISBN: 978-0-470-68391-0 Typeset in 9/11pt Times-Roman by Laserwords Private Limited, Chennai, India. Printed and bound in the United Kingdom by Antony Rowe Ltd, Chippenham, Wiltshire.

Contents Preface Notation 1 Classical Time Series Models and Financial Series 1.1 Stationary Processes 1.2 ARMA and ARIMA Models 1.3 Financial Series 1.4 Random Variance Models 1.5 Bibliographical Notes 1.6 Exercises Part I Univariate GARCH Models 2 GARCH(p, q) Processes 2.1 Deﬁnitions and Representations 2.2 Stationarity Study 2.2.1 The GARCH(1, 1) Case 2.2.2 The General Case 2.3 ARCH (∞) Representation ∗ 2.3.1 Existence Conditions 2.3.2 ARCH (∞) Representation of a GARCH 2.3.3 Long-Memory ARCH 2.4 Properties of the Marginal Distribution 2.4.1 Even-Order Moments 2.4.2 Kurtosis 2.5 Autocovariances of the Squares of a GARCH Positivity of the Autocovariances 2.5.1 2.5.2 The Autocovariances Do Not Always Decrease 2.5.3 Explicit Computation of the Autocovariances of the Squares 2.6 Theoretical Predictions 2.7 Bibliographical Notes 2.8 Exercises xi xiii 1 1 3 7 10 12 12 17 19 19 24 24 28 39 39 42 43 45 45 48 50 50 51 52 53 57 58

vi CONTENTS 3 Mixing* 3.1 Markov Chains with Continuous State Space 3.2 Mixing Properties of GARCH Processes 3.3 Bibliographical Notes 3.4 Exercises 4 Temporal Aggregation and Weak GARCH Models 4.1 Temporal Aggregation of GARCH Processes 4.1.1 Nontemporal Aggregation of Strong Models 4.1.2 Nonaggregation in the Class of Semi-Strong GARCH Processes 4.2 Weak GARCH 4.3 Aggregation of Strong GARCH Processes in the Weak GARCH Class 4.4 Bibliographical Notes 4.5 Exercises Part II Statistical Inference 5 Identiﬁcation 5.1 Autocorrelation Check for White Noise Portmanteau Tests Sample Partial Autocorrelations of a GARCH 5.1.1 Behavior of the Sample Autocorrelations of a GARCH Process 5.1.2 5.1.3 5.1.4 Numerical Illustrations Identifying the ARMA Orders of an ARMA-GARCH 5.2.1 5.2.2 Sample Autocorrelations of an ARMA-GARCH Sample Autocorrelations of an ARMA-GARCH Process When the Noise is Not Symmetrically Distributed Identifying the Orders (P , Q) 5.2.3 Identifying the GARCH Orders of an ARMA-GARCH Model 5.3.1 Corner Method in the GARCH Case 5.3.2 Applications 5.2 5.3 5.4 Lagrange Multiplier Test for Conditional Homoscedasticity 5.4.1 General Form of the LM Test 5.4.2 LM Test for Conditional Homoscedasticity 5.5 Application to Real Series 5.6 Bibliographical Notes 5.7 Exercises 6 Estimating ARCH Models by Least Squares 6.1 Estimation of ARCH(q) models by Ordinary Least Squares 6.2 Estimation of ARCH(q) Models by Feasible Generalized Least Squares 6.3 Estimation by Constrained Ordinary Least Squares Properties of the Constrained OLS Estimator 6.3.1 6.3.2 Computation of the Constrained OLS Estimator 6.4 Bibliographical Notes 6.5 Exercises 7 Estimating GARCH Models by Quasi-Maximum Likelihood 7.1 Conditional Quasi-Likelihood 7.1.1 Asymptotic Properties of the QMLE 7.1.2 The ARCH(1) Case: Numerical Evaluation of the Asymptotic Variance 63 63 68 76 76 79 79 80 81 82 85 88 89 91 93 93 94 97 97 98 100 101 104 106 108 109 109 111 111 115 117 120 122 127 127 132 135 135 137 138 138 141 141 143 147

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