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PROVIDES A COMPREHENSIVE AND UPDATED STUDY OF GARCH MODELS AND THEIR APPLICATIONS IN FINANCE, COVERING NEW DEVELOPMENTS IN THE DISCIPLINE This book provides a comprehensive and systematic approach to understanding GARCH time series models and their applications whilst presenting the most advanced results concerning the theory and practical aspects of GARCH. The probability structure of standard GARCH models is studied in detail as well as statistical inference such as identification, estimation, and tests. The book also provides new coverage of several extensions such as multivariate models, examines financial applications, and explores the very validation of the models used. GARCH Models: Structure, Statistical Inference and Financial Applications, Second Edition features a new chapter on Parameter-Driven Volatility Models, which covers Stochastic Volatility Models and Markov Switching Volatility Models. A second new chapter entitled Alternative Models for the Conditional Variance contains a section on Stochastic Recurrence Equations and additional material on EGARCH, Log-GARCH, GAS, MIDAS, and intraday volatility models, amongst others. The book is also updated with a more complete discussion of multivariate GARCH; a new section on Cholesky GARCH; a larger emphasis on the inference of multivariate GARCH models; a new set of corrected problems available online; and an up-to-date list of references. This book also: Features up-to-date coverage of the current research in the probability, statistics, and econometric theory of GARCH models Covers significant developments in the field, especially in multivariate models Contains completely renewed chapters with new topics and results Handles both theoretical and applied aspects Applies to researchers in different fields (time series, econometrics, finance) Includes numerous illustrations and applications to real financial series Presents a large collection of exercises with corrections Supplemented by a supporting website featuring R codes, Fortran programs, data sets and problems with corrections GARCH Models: Structure, Statistical Inference and Financial Applications, Second Edition is an authoritative, state-of-the-art reference that is ideal for graduate students, researchers, and practitioners in business and finance seeking to broaden their skills of understanding of econometric time series models.

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PROVIDES A COMPREHENSIVE AND UPDATED STUDY OF GARCH MODELS AND THEIR APPLICATIONS IN FINANCE, COVERING NEW DEVELOPMENTS IN THE DISCIPLINE This book provides a comprehensive and systematic approach to understanding GARCH time series models and their applications whilst presenting the most advanced results concerning the theory and practical aspects of GARCH. The probability structure of standard GARCH models is studied in detail as well as statistical inference such as identification, estimation, and tests. The book also provides new coverage of several extensions such as multivariate models, examines financial applications, and explores the very validation of the models used. GARCH Models: Structure, Statistical Inference and Financial Applications, Second Edition features a new chapter on Parameter-Driven Volatility Models, which covers Stochastic Volatility Models and Markov Switching Volatility Models. A second new chapter entitled Alternative Models for the Conditional Variance contains a section on Stochastic Recurrence Equations and additional material on EGARCH, Log-GARCH, GAS, MIDAS, and intraday volatility models, amongst others. The book is also updated with a more complete discussion of multivariate GARCH; a new section on Cholesky GARCH; a larger emphasis on the inference of multivariate GARCH models; a new set of corrected problems available online; and an up-to-date list of references. This book also: Features up-to-date coverage of the current research in the probability, statistics, and econometric theory of GARCH models Covers significant developments in the field, especially in multivariate models Contains completely renewed chapters with new topics and results Handles both theoretical and applied aspects Applies to researchers in different fields (time series, econometrics, finance) Includes numerous illustrations and applications to real financial series Presents a large collection of exercises with corrections Supplemented by a supporting website featuring R codes, Fortran programs, data sets and problems with corrections GARCH Models: Structure, Statistical Inference and Financial Applications, Second Edition is an authoritative, state-of-the-art reference that is ideal for graduate students, researchers, and practitioners in business and finance seeking to broaden their skills of understanding of econometric time series models.

Table of Contents

Preface to the Second Edition xi Preface to the First Edition xiii Notation xv 1 Classical Time Series Models and Financial Series 1 1.1 Stationary Processes 1 1.2 ARMA and ARIMA Models 3 1.3 Financial Series 6 1.4 Random Variance Models 10 1.5 Bibliographical Notes 11 1.6 Exercises 12 Part I Univariate GARCH Models 2 GARCH(p, q) Processes 17 2.1 Definitions and Representations 17 2.2 Stationarity Study 22 2.2.1 The GARCH(1,1) Case 22 2.2.2 The General Case 26 2.3 ARCH(∞) Representation* 36 2.3.1 Existence Conditions 36 2.3.2 ARCH(∞) Representation of a GARCH 39 2.3.3 Long-Memory ARCH 40 2.4 Properties of the Marginal Distribution 41 2.4.1 Even-Order Moments 42 2.4.2 Kurtosis 45 2.5 Autocovariances of the Squares of a GARCH 46 2.5.1 Positivity of the Autocovariances 47 2.5.2 The Autocovariances Do Not Always Decrease 48 2.5.3 Explicit Computation of the Autocovariances of the Squares 48 2.6 Theoretical Predictions 50 2.7 Bibliographical Notes 54 2.8 Exercises 55 3 Mixing* 59 3.1 Markov Chains with Continuous State Space 59 3.2 Mixing Properties of GARCH Processes 64 3.3 Bibliographical Notes 71 3.4 Exercises 71 4 Alternative Models for the Conditional Variance 73 4.1 Stochastic Recurrence Equation (SRE) 74 4.2 Exponential GARCH Model 77 4.3 Log-GARCH Model 82 4.3.1 Stationarity of the Extended Log-GARCH Model 83 4.3.2 Existence of Moments and Log-Moments 86 4.3.3 Relations with the EGARCH Model 88 4.4 Threshold GARCH Model 90 4.5 Asymmetric Power GARCH Model 96 4.6 Other Asymmetric GARCH Models 98 4.7 A GARCH Model with Contemporaneous Conditional Asymmetry 99 4.8 Empirical Comparisons of Asymmetric GARCH Formulations 101 4.9 Models Incorporating External Information 109 4.10 Models Based on the Score: GAS and Beta-t-(E)GARCH 113 4.11 GARCH-type Models for Observations Other Than Returns 115 4.12 Complementary Bibliographical Notes 119 4.13 Exercises 120 Part II Statistical Inference 5 Identification 125 5.1 Autocorrelation Check for White Noise 125 5.1.1 Behaviour of the Sample Autocorrelations of a GARCH Process 126 5.1.2 Portmanteau Tests 128 5.1.3 Sample Partial Autocorrelations of a GARCH 129 5.1.4 Numerical Illustrations 129 5.2 Identifying the ARMA Orders of an ARMA-GARCH 132 5.2.1 Sample Autocorrelations of an ARMA-GARCH 132 5.2.2 Sample Autocorrelations of an ARMA-GARCH Process When the Noise is Not Symmetrically Distributed 136 5.2.3 Identifying the Orders (P, Q) 138 5.3 Identifying the GARCH Orders of an ARMA-GARCH Model 140 5.3.1 Corner Method in the GARCH Case 141 5.3.2 Applications 141 5.4 Lagrange Multiplier Test for Conditional Homoscedasticity 143 5.4.1 General Form of the LM Test 143 5.4.2 LM Test for Conditional Homoscedasticity 147 5.5 Application to Real Series 149 5.6 Bibliographical Notes 151 5.7 Exercises 158 6 Estimating ARCH Models by Least Squares 161 6.1 Estimation of ARCH(q) models by Ordinary Least Squares 161 6.2 Estimation of ARCH(q) Models by Feasible Generalised Least Squares 165 6.3 Estimation by Constrained Ordinary Least Squares 168 6.3.1 Properties of the Constrained OLS Estimator 169 6.3.2 Computation of the Constrained OLS Estimator 170 6.4 Bibliographical Notes 171 6.5 Exercises 171 7 Estimating GARCH Models by Quasi-Maximum Likelihood 175 7.1 Conditional Quasi-Likelihood 175 7.1.1 Asymptotic Properties of the QMLE 177 7.1.2 The ARCH(1) Case: Numerical Evaluation of the Asymptotic Variance 180 7.1.3 The Non-stationary ARCH(1) 181 7.2 Estimation of ARMA–GARCH Models by Quasi-Maximum Likelihood 183 7.3 Application to Real Data 187 7.4 Proofs of the Asymptotic Results* 188 7.5 Bibliographical Notes 211 7.6 Exercises 212 8 Tests Based on the Likelihood 217 8.1 Test of the Second-Order Stationarity Assumption 217 8.2 Asymptotic Distribution of the QML When

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