Arch Models and Financial Applications

ISBN-13: 9780387948768 / Angielski / Twarda / 1997 / 229 str.

Christian Gourieroux; C. Gourieroux

Opis

1.1 The DevelopmentofARCH Models Time series models have been initially introduced either for descriptive purposes like prediction and seasonal correction or for dynamic control. In the 1970s, the researchfocusedonaspecificclassoftimeseriesmodels, theso-calledautoregres sive moving average processes (ARMA), which were very easy to implement. In thesemodels, thecurrentvalueoftheseriesofinterestiswrittenasalinearfunction ofits own laggedvalues andcurrentandpastvaluesofsomenoiseprocess, which can be interpreted as innovations to the system. However, this approach has two major drawbacks: 1) it is essentially a linear setup, which automatically restricts the type of dynamics to be approximated; 2) it is generally applied without im posing a priori constraintson the autoregressive and moving average parameters, which is inadequatefor structural interpretations. Among the field ofapplications where standard ARMA fit is poorare financial and monetary problems. The financial time series features various forms ofnon lineardynamics, the crucialone being the strongdependenceofthe instantaneous variabilityoftheseriesonitsownpast. Moreover, financial theoriesbasedoncon ceptslikeequilibriumorrationalbehavioroftheinvestorswouldnaturallysuggest including and testing some structural constraints on the parameters. In this con text, ARCH (Autoregressive Conditionally Heteroscedastic) models, introduced by Engle (1982), arise as an appropriate framework for studying these problems. Currently, there existmorethan onehundredpapers and some dozenPh.D. theses on this topic, which reflects the importance ofthis approach for statistical theory, finance and empirical work. 2 1. Introduction From the viewpoint ofstatistical theory, the ARCH models may be considered as some specific nonlinear time series models, which allow for aquite exhaustive studyoftheunderlyingdynamics.Itisthereforepossibletoreexamineanumberof classicalquestions like the random walkhypothesis, prediction intervals building, presenceoflatentvariables factors] etc., and to test the validity ofthe previously established results.

Recenzje

From the reviews:

RISKBOOK.COM

"Gourieroux offers a nice balance of theory and application in this book on ARCH modeling in finance...The book is well written and has extensive references. Its focus on finance will appeal to financial engineers and financial risk managers."

Spis treści

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.