ARCH Models and Financial Applications

1 Introduction. - 1.1 The Development of ARCH Models. - 1.2 Book Content. - 2 Linear and Nonlinear Processes. - 2.1 Stochastic Processes. - 2.2 Weak and Strict Stationarity. - 2.3 A Few Examples. - 2.4 Nonlinearities. - 2.4.1 Portmanteau Statistic. - 2.4.2 Some Implications of the White Noise Hypothesis. - 2.5 Exercises. - 3 Univariate ARCH Models. - 3.1 A Heteroscedastic Model of Order One. - 3.1.1 Description of the Model. - 3.1.2 Properties of the Innovation Process ? - 3.1. 3 Properties of the Y Process. - 3.1. 4 Distribution of the Error Process. - 3.2 General Properties of ARCH Processes. - 3.2.1 Various Extensions. - 3.2.2 Stationarity of a GARCH(p, q) Process. - 3.2.3 Kurtosis. - 3.2.4 Yule-Walker Equations for the Square of a GARCH Process. - 3.3 Exercises. - 4 Estimation and Tests. - 4.1 Pseudo Maximum Likelihood Estimation. - 4.1.1 Generalities. - 4.1.2 The i.i. d. case. - 4.1. 3 Regression Model with Heteroscedastic Errors. - 4.1. 4 Regression Model with ARCH Errors. - 4.1. 5 Application to a GARCH Model. - 4.1. 6 Stochastic Variance Model. - 4.2 Two Step Estimation Procedures. - 4.2.1 Description of the Procedures. - 4.2.2 Comparison of the Estimation Methods under Conditional Normality. - 4.2.3 Efficiency Loss Analysis. - 4.3 Forecast Intervals. - 4.4 Homoscedasticity Test. - 4.4.1 Regression Models with Heteroscedastic Errors. - 4.5 The Test Statistic Interpretation. - 4.5.1 Application to Regression Models with ARCH or GARCH Errors. - Appendix 4.1: Matrices I and J.- Appendix 4.2: Derivatives of the Log-Likelihood Function and Information Matrix for a Regression Model with ARCH Errors. - 4.6 Exercises. - 5 Some Applications of Univariate ARCH Models. - 5.1 Leptokurtic Aspects of Financial Series and Aggregation. - 5.1.1 The Normality Assumption. - 5.1.2 The Choice of a Time Unit. - 5.2 ARCH Processes as an Approximation of Continuous Time Processes. - 5.2.1 Stochastic Integrals. - 5.2.2 Stochastic Differential Equations. - 5.2.3 Some Equations and Their Solutions. - 5.2.4 Continuous and Discrete Time. - 5.2.5 Examples. - 5.2. 6 Simulated Estimation Methods. - 5.3 The Random Walk Hypothesis. - 5.3.1 Description of the Hypothesis. - 5.3.2 The Classical Test Procedure of the Random Walk Hypothesis. - 5.3.3 Limitations of the Portmanteau Tests. - 5.3. 4 Portmanteau Tests with Heteroscedasticity. - 5.4 Threshold Models. - 5.4.1 Definition and Stationarity Conditions. - 5.4.2 Homoscedasticity Test. - 5.4.3 Qualitative ARCH Models. - 5.4.4 Nonparametric Approaches. - 5.5 Integrated Models. - 5.5.1 The IGARCH(1,1) Model. - 5.5.2 The Persistence Effect. - 5.5.3 Weak and Strong Stationarity. - 5.5.4 Example. - 5.6 Exercises. - 6 Multivariate ARCH Models. - 6.1 Unconstrained Models. - 6.1.1 Multivariate GARCH Models. - 6.1.2 Positivity Constraints. - 6.1. 3 Stability Conditions. - 6.1. 4 An Example. - 6.1. 5 Spectral Decompositions. - 6.2 Constrained Models. - 6.2.1 Diagonal Models. - 6.2.2 Models with Constant Conditional Correlations. - 6.2.3 Models with Random Coefficients. - 6.2.4 Model Based on a Spectral Decomposition. - 6.2.5 Factor ARCH Models. - 6.3 Estimation of Heteroscedastic Dynamic Models. - 6.3.1 Pseudo Maximum Likelihood Estimators. - 6.3.2 Asymptotic Properties of the Pseudo Maximum Likelihood Estimator. - 6.3.3 Model with Constant Conditional Correlations. - 6.3. 4 Factor Models. - 7 Efficient Portfolios and Hedging Portfolios. - 7.1 Determination of an Efficient Portfolio. - 7.1.1 Securities and Portfolios. - 7.1.2 Mean Variance Criterion. - 7.1. 3 Mean Variance Efficient Portfolios. - 7.2 Properties of the Set of Efficient Portfolios. - 7.2.1 The Set of Efficient Portfolios. - 7.2.2 Factors. - 7.3 Asymmetric Information and Aggregation. - 7.3.1 Incoherency of the Mean Variance Approach. - 7.3.2 Study of the Basic Portfolios. - 7.3.3 Aggregation. - 7.4 Hedging Portfolios. - 7.4.1 Determination of a Portfolio Mimicking a Series of Interest. - 7.4.2 A Model for the Call Seller Behavior. - 7.4.3 The Firm Behavior. - 7.5 Empirical Study of Performance Measures. - 7.5.1 Performances of a Set of Assets. - 7.5.2 Improving the Efficiency. - 7.5.3 Estimation of the Efficient Portfolio and its Performance in the Static Case. - Appendix 1: Presentation in Terms of Utility. - Appendix 2: Moments of the Truncated Log-Normal Distribution. - Appendix 3: Asymptotic Properties of the Estimators. - 7.6 Exercises. - 8 Factor Models, Diversification and Efficiency. - 8.1 Factor Models. - 8.1.1 Linear Factor Representation. - 8.1.2 Representation with Endogenous Factors. - 8.1. 3 Structure of the Conditional Moments. - 8.1. 4 Cofactors. - 8.1. 5 Characterization with the Matrix Defining the Endogenous Factors. - 8.2 Arbitrage Theory. - 8.2.1 Absence of Arbitrage Opportunities. - 8.2.2 Diversification and Pricing Model. - 8.2.3 Diversification and Risk Aversion. - 8.3 Efficiency Tests and Diversification. - 8.3.1 Ex-Ante Efficiency. - 8.3.2 Ex-Post Efficiency. - 8.4 Conditional and Historical Performance Measures. - 8.4.1 The Dynamics of a Model with Endogenous Factors. - 8.4.2 Tests for Ex-Ante Efficiency and Performances. - 8.5 Exercises. - 9 Equilibrium Models. - 9.1 Capital Asset Pricing Model. - 9.1.1 Description of the Model. - 9.1.2 Market Portfolio. - 9.1. 3 The CAPM as a Factor Model. - 9.1. 4 Spectral Decomposition of the Moments. - 9.1. 5 Time Dependent Risk Aversion. - 9.2 Test of the CAPM. - 9.2.1 Some Difficulties. - 9.2.2 Testing Procedures in a Static Framework. - 9.2.3 Test for Efficiency of the Market Portfolio in a Dynamic Framework with Constant Betas. - 9.2.4 Tests in the General Case. - 9.3 Examples of Structural Models. - 9.3.1 A Model with Speculative Bubbles. - 9.3.2 The Consumption Based CAPM.