This paper develops a generalized autoregressive conditional correlation~GARCC! model when the standardized residuals follow a random coefficient vector autoregressive process+ As a multivariate generalization of the Tsay~1987, Journal of the American Statistical Association 82, 590-604! random coefficient autoregressive~RCA! model, the GARCC model provides a motivation for the conditional correlations to be time varying+ GARCC is also more general than the Engle~2002, Journal of Business & Economic Statistics 20, 339-350! dynamic conditional correlation~DCC! and the Tse and Tsui~2002, Journal of Business & Economic Statistics 20, 351-362! varying conditional correlation~VCC! models and does not impose unduly restrictive conditions on the parameters of the DCC model+ The structural properties of the GARCC model, specifically, the analytical forms of the regularity conditions, are derived, and the asymptotic theory is established+ The Baba, Engle, Kraft, and Kroner~BEKK! model of Engle and Kroner~1995, Econometric Theory 11, 122-150! is demonstrated to be a special case of a multivariate RCA process+ A likelihood ratio test is proposed for several special casesThe authors thank the co-editor, Bruce Hansen, and three referees for insightful suggestions and Manabu Asai,

Various univariate and multivariate models of volatility have been used to evaluate market risk, asymmetric shocks, thresholds, leverage effects, and Value-at-Risk in economics and finance. This article is concerned with market risk, and develops a constant conditional correlation vector ARMA-asymmetric GARCH (VARMA-AGARCH) model, as an extension of the widely used univariate asymmetric (or threshold) GJR model of Glosten et al. (1992), and establishes its underlying structure, including the unique, strictly stationary, and ergodic solution of the model, its causal expansion, and convenient sufficient conditions for the existence of moments. Alternative empirically verifiable sufficient conditions for the consistency and asymptotic normality of the quasi-maximum likelihood estimator are established under non-normality of the standardized shocks.Asymmetric effects, Asymptotic theory, Conditional volatility, Multivariate structure, Regularity conditions,

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Theoretical and practical interest in non-linear time series models, particularly regime switching models, have increased substantially in recent years. Given the abundant research activity in analysing time-varying volatility through Generalized Autoregressive Conditional Heteroscedasticity (GARCH) processes (see Engle, 1982; Bollerslev, 1986), it is important to analyse regime switching models with GARCH errors. A popular specification in this class is the (stationary) Smooth Transition Autoregressive-GARCH (STAR-GARCH) model. Little is presently known about the structure of the model, or the consistency, asymptotic normality and finite sample properties of the estimators. The paper develops the structural and statistical properties of the STAR-GARCH model, and investigates the finite sample properties of maximum likelihood estimation (MLE) of STAR and STAR-GARCH models through numerical simulation. The effects of fixing the threshold value and|or the transition rate for the STAR model, misspecification of the conditional mean and the transition function of the STAR-GARCH model, and the finite sample properties of the MLE for the STAR-GARCH model, are also examined. These numerical results are used as a guide in empirical research, with an application to Standard and Poor's Composite 500 Index returns for alternative STAR-GARCH models. Copyright © 2002 John Wiley & Sons, Ltd.