ARCH Models and Financial Applications

Transcription

1 Christian Gourieroux ARCH Models and Financial Applications With 26 Figures Springer

2 Contents 1 Introduction The Development of ARCH Models Book Content 4 2 Linear and Nonlinear Processes Stochastic Processes Weak and Strict Stationarity A Few Examples Nonlinearities Portmanteau Statistic Some Implications of the White Noise Hypothesis Exercises 26 3 Univariate ARCH Models A Heteroscedastic Model of Order One Description of the Model Properties of the Innovation Process Properties of the Y Process Distribution of the Error Process General Properties of ARCH Processes Various Extensions Stationarity ofagarch(p, q) Process Kurtosis 38

3 vi Contents Yule-Walker Equations for the Square of a GARCH Process Exercises 39 i 4 Estimation and Tests Pseudo Maximum Likelihood Estimation Generalities The i.i.d. case Regression Model with Heteroscedastic Errors Regression Model with ARCH Errors Application to a GARCH Model Stochastic Variance Model Two Step Estimation Procedures Description of the Procedures Comparison of the Estimation Methods under Conditional Normality Efficiency Loss Analysis Forecast Intervals Homoscedasticity Test Regression Models with Heteroscedastic Errors The Test Statistic Interpretation 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 Exercises 65 5 Some Applications of Univariate ARCH Models Leptokurtic Aspects of Financial Series and Aggregation The Normality Assumption The Choice of a Time Unit ARCH Processes as an Approximation of Continuous Time Processes Stochastic Integrals Stochastic Differential Equations Some Equations and Their Solutions Continuous and Discrete Time Examples Simulated Estimation Methods The Random Walk Hypothesis Description of the Hypothesis The Classical Test Procedure of the Random Walk Hypothesis 85

4 Contents vii Limitations of the Portmanteau Tests Portmanteau Tests with Heteroscedasticity Threshold Models Definition and Stationarity Conditions Homoscedasticity Test Qualitative ARCH Models Nonparametric Approaches Integrated Models The IGARCH( 1,1) Model The Persistence Effect Weak and Strong Stationarity Example Exercises Multivariate ARCH Models Unconstrained Models Multivariate GARCH Models Positivity Constraints Stability Conditions An Example Spectral Decompositions Constrained Models Ill Diagonal Models Ill Models with Constant Conditional Correlations Models with Random Coefficients Model Based on a Spectral Decomposition Factor ARCH Models Estimation of Heteroscedastic Dynamic Models Pseudo Maximum Likelihood Estimators Asymptotic Properties of the Pseudo Maximum Likelihood Estimator Model with Constant Conditional Correlations Factor Models Efficient Portfolios and Hedging Portfolios Determination of an Efficient Portfolio Securities and Portfolios Mean Variance Criterion Mean Variance Efficient Portfolios Properties of the Set of Efficient Portfolios The Set of Efficient Portfolios Factors Asymmetric Information and Aggregation Incoherency of the Mean Variance Approach Study of the Basic Portfolios 138

5 viii Contents Aggregation Hedging Portfolios Determination of a Portfolio Mimicking a Series oflnterest A Model for the Call Seller Behavior The Firm Behavior Empirical Study of Performance Measures Performances of a Set of Assets Improving the Efficiency 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 Exercises Factor Models, Diversification and Efficiency Factor Models Linear Factor Representation Representation with Endogenous Factors Structure of the Conditional Moments Cofactors Characterization with the Matrix Defining the Endogenous Factors Arbitrage Theory Absence of Arbitrage Opportunities Diversification and Pricing Model Diversification and Risk Aversion Efficiency Tests and Diversification Ex-Ante Efficiency Ex-Post Efficiency Conditional and Historical Performance Measures The Dynamics of a Model with Endogenous Factors Tests for Ex-Ante Efficiency and Performances Exercises Equilibrium Models Capital Asset Pricing Model Description of the Model Market Portfolio.' The CAPM as a Factor Model Spectral Decomposition of the Moments Time Dependent Risk Aversion Test of the CAPM Some Difficulties 188

6 Contents ix Testing Procedures in a Static Framework Test for Efficiency of the Market Portfolio in a Dynamic Framework with Constant Betas Tests in the General Case Examples of Structural Models A Model with Speculative Bubbles The Consumption Based CAPM 202 References 207 Index 227