ARCH Models and Financial Applications

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.

Klappentext

The classical ARMA models have limitations when applied to the field of financial and monetary economics. Financial time series present nonlinear dynamic characteristics and the ARCH models offer a more adaptive framework for this type of problem. This book surveys the recent work in this area from the perspective of statistical theory, financial models, and applications and will be of interest to theorists and practitioners. From the view point of statistical theory, ARCH models may be considered as specific nonlinear time series models which allow for an exhaustive study of the underlying dynamics. It is possible to reexamine a number of classical questions such as the random walk hypothesis, prediction interval building, presence of latent variables etc., and to test the validity of the previously studied results. There are two main categories of potential applications. One is testing several economic or financial theories concerning the stocks, bonds, and currencies markets, or studying the links between the short and long run. The second is related to the interventions of the banks on the markets, such as choice of optimal portfolios, hedging portfolios, values at risk, and the size and times of block trading.

Inhalt
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.- 2.5 Exercises.- 3 Univariate ARCH Models.- 3.1 A Heteroscedastic Model of Order One.- 3.2 General Properties of ARCH Processes.- 3.3 Exercises.- 4 Estimation and Tests.- 4.1 Pseudo Maximum Likelihood Estimation.- 4.2 Two Step Estimation Procedures.- 4.3 Forecast Intervals.- 4.4 Homoscedasticity Test.- 4.5 The Test Statistic Interpretation.- 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.- 5.2 ARCH Processes as an Approximation of Continuous Time Processes.- 5.3 The Random Walk Hypothesis.- 5.4 Threshold Models.- 5.5 Integrated Models.- 5.6 Exercises.- 6 Multivariate ARCH Models.- 6.1 Unconstrained Models.- 6.2 Constrained Models.- 6.3 Estimation of Heteroscedastic Dynamic Models.- 7 Efficient Portfolios and Hedging Portfolios.- 7.1 Determination of an Efficient Portfolio.- 7.2 Properties of the Set of Efficient Portfolios.- 7.3 Asymmetric Information and Aggregation.- 7.4 Hedging Portfolios.- 7.5 Empirical Study of Performance Measures.- 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.- 8.2 Arbitrage Theory.- 8.3 Efficiency Tests and Diversification.- 8.5 Exercises.- 9 Equilibrium Models.- 9.1 Capital Asset Pricing Model.- 9.2 Test of theCAPM.- 9.3 Examples of Structural Models.