Maximum likelihood estimation of STAR and STAR‐GARCH models: theory and Monte Carlo evidence

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Abstract
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.