Christian Gourieroux ARCH Models and Financial Applications With 26 Figures Springer
Contents 1 Introduction 1 1.1 The Development of ARCH Models 1 1.2 Book Content 4 2 Linear and Nonlinear Processes 5 2.1 Stochastic Processes 5 2.2 Weak and Strict Stationarity 8 2.3 A Few Examples 12 2.4 Nonlinearities 22 2.4.1 Portmanteau Statistic 22 2.4.2 Some Implications of the White Noise Hypothesis.. 23 2.5 Exercises 26 3 Univariate ARCH Models 29 3.1 A Heteroscedastic Model of Order One 29 3.1.1 Description of the Model 29 3.1.2 Properties of the Innovation Process 30 3.1.3 Properties of the Y Process 32 3.1.4 Distribution of the Error Process 33 3.2 General Properties of ARCH Processes 34 3.2.1 Various Extensions 34 3.2.2 Stationarity ofagarch(p, q) Process 37 3.2.3 Kurtosis 38
vi Contents 3.2.4 Yule-Walker Equations for the Square of a GARCH Process 38 3.3 Exercises 39 i 4 Estimation and Tests 43 4.1 Pseudo Maximum Likelihood Estimation 43 4.1.1 Generalities 43 4.1.2 The i.i.d. case 44 4.1.3 Regression Model with Heteroscedastic Errors... 46 4.1.4 Regression Model with ARCH Errors 49 4.1.5 Application to a GARCH Model 51 4.1.6 Stochastic Variance Model 52 4.2 Two Step Estimation Procedures 53 4.2.1 Description of the Procedures 53 4.2.2 Comparison of the Estimation Methods under Conditional Normality 54 4.2.3 Efficiency Loss Analysis 55 4.3 Forecast Intervals 56 4.4 Homoscedasticity Test 58 4.4.1 Regression Models with Heteroscedastic Errors... 58 4.5 The Test Statistic Interpretation 61 4.5.1 Application to Regression Models with ARCH or GARCH Errors 62 Appendix 4.1: Matrices / and / 63 Appendix 4.2: Derivatives of the Log-Likelihood Function and Information Matrix for a Regression Model with ARCH Errors 64 4.6 Exercises 65 5 Some Applications of Univariate ARCH Models 67 5.1 Leptokurtic Aspects of Financial Series and Aggregation.. 67 5.1.1 The Normality Assumption 67 5.1.2 The Choice of a Time Unit 69 5.2 ARCH Processes as an Approximation of Continuous Time Processes 71 5.2.1 Stochastic Integrals 71 5.2.2 Stochastic Differential Equations 73 5.2.3 Some Equations and Their Solutions 74 5.2.4 Continuous and Discrete Time 76 5.2.5 Examples 78 5.2.6 Simulated Estimation Methods 81 5.3 The Random Walk Hypothesis 83 5.3.1 Description of the Hypothesis 83 5.3.2 The Classical Test Procedure of the Random Walk Hypothesis 85
Contents vii 5.3.3 Limitations of the Portmanteau Tests 87 5.3.4 Portmanteau Tests with Heteroscedasticity 88 5.4 Threshold Models 90 5.4.1 Definition and Stationarity Conditions 90 5.4.2 Homoscedasticity Test 92 5.4.3 Qualitative ARCH Models 93 5.4.4 Nonparametric Approaches 95 5.5 Integrated Models 98 5.5.1 The IGARCH( 1,1) Model 98 5.5.2 The Persistence Effect 99 5.5.3 Weak and Strong Stationarity 100 5.5.4 Example 101 5.6 Exercises 101 6 Multivariate ARCH Models 105 6.1 Unconstrained Models 105 6.1.1 Multivariate GARCH Models 105 6.1.2 Positivity Constraints 107 6.1.3 Stability Conditions 107 6.1.4 An Example 108 6.1.5 Spectral Decompositions 109 6.2 Constrained Models Ill 6.2.1 Diagonal Models Ill 6.2.2 Models with Constant Conditional Correlations... 113 6.2.3 Models with Random Coefficients 114 6.2.4 Model Based on a Spectral Decomposition 116 6.2.5 Factor ARCH Models 116 6.3 Estimation of Heteroscedastic Dynamic Models 117 6.3.1 Pseudo Maximum Likelihood Estimators 117 6.3.2 Asymptotic Properties of the Pseudo Maximum Likelihood Estimator 119 6.3.3 Model with Constant Conditional Correlations... 120 6.3.4 Factor Models 121 7 Efficient Portfolios and Hedging Portfolios 125 7.1 Determination of an Efficient Portfolio 125 7.1.1 Securities and Portfolios 125 7.1.2 Mean Variance Criterion 128 7.1.3 Mean Variance Efficient Portfolios 129 7.2 Properties of the Set of Efficient Portfolios 132 7.2.1 The Set of Efficient Portfolios 132 7.2.2 Factors 135 7.3 Asymmetric Information and Aggregation 137 7.3.1 Incoherency of the Mean Variance Approach... 137 7.3.2 Study of the Basic Portfolios 138
viii Contents 7.3.3 Aggregation 139 7.4 Hedging Portfolios 140 7.4.1 Determination of a Portfolio Mimicking a Series oflnterest 141 7.4.2 A Model for the Call Seller Behavior 142 7.4.3 The Firm Behavior 147 7.5 Empirical Study of Performance Measures 148 7.5.1 Performances of a Set of Assets 148 7.5.2 Improving the Efficiency 149 7.5.3 Estimation of the Efficient Portfolio and its Performance in the Static Case 149 Appendix 1: Presentation in Terms of Utility 152 Appendix 2: Moments of the Truncated Log-Normal Distribution. 155 Appendix 3: Asymptotic Properties of the Estimators 156 7.6 Exercises 157 8 Factor Models, Diversification and Efficiency 161 8.1 Factor Models 162 8.1.1 Linear Factor Representation 162 8.1.2 Representation with Endogenous Factors 163 8.1.3 Structure of the Conditional Moments 165 8.1.4 Cofactors 167 8.1.5 Characterization with the Matrix Defining the Endogenous Factors 167 8.2 Arbitrage Theory 168 8.2.1 Absence of Arbitrage Opportunities 168 8.2.2 Diversification and Pricing Model 169 8.2.3 Diversification and Risk Aversion 172 8.3 Efficiency Tests and Diversification 172 8.3.1 Ex-Ante Efficiency 172 8.3.2 Ex-Post Efficiency 175 8.4 Conditional and Historical Performance Measures 177 8.4.1 The Dynamics of a Model with Endogenous Factors. 177 8.4.2 Tests for Ex-Ante Efficiency and Performances... 179 8.5 Exercises 180 9 Equilibrium Models 183 9.1 Capital Asset Pricing Model 183 9.1.1 Description of the Model 183 9.1.2 Market Portfolio.' 184 9.1.3 The CAPM as a Factor Model 185 9.1.4 Spectral Decomposition of the Moments 186 9.1.5 Time Dependent Risk Aversion 187 9.2 Test of the CAPM 187 9.2.1 Some Difficulties 188
Contents ix 9.2.2 Testing Procedures in a Static Framework 191 9.2.3 Test for Efficiency of the Market Portfolio in a Dynamic Framework with Constant Betas 196 9.2.4 Tests in the General Case 197 9.3 Examples of Structural Models 199 9.3.1 A Model with Speculative Bubbles 199 9.3.2 The Consumption Based CAPM 202 References 207 Index 227