This paper presents a parameter covariance matrix estimator which is consistent even when the disturbances of a linear regression model are heteroskedastic. This estimator does not depend on a formal model of the structure of the heteroskedasticity. By comparing the elements of the new estimator to those of the usual covariance estimator, one obtains a direct test for heteroskedasticity, since in the absence of heteroskedasticity, the two estimators will be approximately equal, but will generally diverge otherwise. The test has an appealing least squares interpretation.

A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity

http://public.econ.duke.edu/~boller/Published_Papers/joe_86.pdf

Generalized autoregressive conditional heteroskedasticity

Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root?

This paper introduces an ARCH model (exponential ARCH) that (1) allows correlation between returns and volatility innovations (an important feature of stock market volatility changes), (2) eliminates the need for inequality constraints on parameters, and (3) allows for a straightforward interpretation of the "persistence" of shocks to volatility. In the above respects, it is an improvement over the widely-used GARCH model. The model is applied to study volatility changes and the risk premium on the CRSP Value-Weighted Market Index from 1962 to 1987. Copyright 1991 by The Econometric Society.

Conditional heteroskedasticity in asset returns: a new approach

Any series of observations ordered along a single dimension, such as time, may be thought of as a time series. The emphasis in time series analysis is on studying the dependence among observations at different points in time. What distinguishes time series analysis from general multivariate analysis is precisely the temporal order imposed on the observations. Many economic variables, such as GNP and its components, price indices, sales, and stock returns are observed over time. In addition to being interested in the contemporaneous relationships among such variables, we are often concerned with relationships between their current and past values, that is, relationships over time.

Time Series Analysis

List of figures Acknowledgement Preface Notation and conventions List of abbreviations 1. Introduction 2. Univariate time series models 3. State space models and the Kalman filter 4. Estimation, prediction and smoothing for univariate structural time series models 5. Testing and model selection 6. Extensions of the univariate model 7. Explanatory variables 8. Multivariate models 9. Continuous time Appendices Selected answers to exercises References Author index Subject index.

Forecasting, Structural Time Series Models and the Kalman Filter

In this book, Andrew Harvey sets out to provide a unified and comprehensive theory of structural time series models. Unlike the traditional ARIMA models, structural time series models consist explicitly of unobserved components, such as trends and seasonals, which have a direct interpretation. As a result the model selection methodology associated with structural models is much closer to econometric methodology. The link with econometrics is made even closer by the natural way in which the models can be extended to include explanatory variables and to cope with multivariate time series. From the technical point of view, state space models and the Kalman filter play a key role in the statistical treatment of structural time series models. The book includes a detailed treatment of the Kalman filter. This technique was originally developed in control engineering, but is becoming increasingly important in fields such as economics and operations research. This book is concerned primarily with modelling economic and social time series, and with addressing the special problems which the treatment of such series poses. The properties of the models and the methodological techniques used to select them are illustrated with various applications. These range from the modellling of trends and cycles in US macroeconomic time series to to an evaluation of the effects of seat belt legislation in the UK.

Publisher Summary The class of stochastic volatility (SV) models has its roots in both, mathematical finance and financial econometrics. In fact, several variations of SV models originated from research looking at very different issues. Volatility plays a central role in the pricing of derivative securities. The Black-Scholes model for the pricing of an European option is by far the most widely used formula even when the underlying assumptions are known to be violated. The Black-Scholes model predicts a flat term structure of volatilities. In reality, the term structure of at-the-money implied volatilities is typically upward sloping when short term volatilities are low and the reverse when they are high. The Black-Scholes model is taken as a reference point from which several notions of volatility are presented. Several stylized facts regarding volatility and option prices are also presented. Both sections set the scene for a formal framework defining stochastic volatility. The chapter introduces the statistical models of stochastic volatility.

5 Stochastic volatility

Changes in variance, or volatility, over time can be modelled using the approach based on autoregressive conditional heteroscedasticity (ARCH). However, the generalizations to multivariate series can be difficult to estimate and interpret. Another approach is to model variance as an unobserved stochastic process. Although it is not easy to obtain the exact likelihood function for such stochastic variance models, they tie in closely with developments in finance theory and have certain statistical attractions. This article sets up a multivariate model, discusses its statistical treatment and shows how it can be modified to capture common movements in volatility in a very natural way. The model is then fitted to daily observations on exchange rates.

https://faculty.washington.edu/ezivot/econ589/2297980.pdf

Multivariate stochastic variance models

This new edition of A.C. Harvey's clearly written, upper-level text has been revised and several sections have been completely rewritten. There is new material on a number of topics, including unit roots, ARCH, and cointegration."The Econometric Analysis of Time Series "focuses on the statistical aspects of model building, with an emphasis on providing an understanding of the main ideas and concepts in econometrics rather than presenting a series of rigorous proofs. It explores the way in which recent advances in time series analysis have affected the development of a theory of dynamic econometrics, sets out an integrated approach to the problems of estimation and testing based on the method of maximum likelihood, and presents a coherent strategy for model selection.A.C. Harvey is Professor of Econometrics at the London School of Economics.

Andrew Harvey

Papers

Forecasting, Structural Time Series Models and the Kalman Filter

Forecasting, Structural Time Series Models and the Kalman Filter

5 Stochastic volatility

Multivariate stochastic variance models

The econometric analysis of time series