On the Structure, Asymptotic Theory and Applications of STAR-GARCH Models

Non-linear time series models, especially regime-switching models, have become increasingly popular in the economics, finance and financial econometrics literature. However, much of the research has concentrated on the empirical applications of various models, with little theoretical or statistical analysis associated with the structure of the models or asymptotic theory. Some structural and statistical properties have recently been established for the Smooth Transition Autoregressive (STAR) - Generalised Autoregresssive Conditional Heteroscedasticity (GARCH), or STAR-GARCH, model, including the necessary and sufficient conditions for the existence of moments, and the sufficient condition for consistency and asymptotic normality of the (Quasi)-Maximum Likelihood Estimator ((Q)MLE). While these moment conditions are straightforward to verify in practice, they may not be satisfied for the GARCH model if the underlying long run persistence is close to unity. A less restrictive condition for consistency and asymptotic normality may alleviate this problem. The paper establishes a weak sufficient, or log-moment, condition for consistency and asymptotic normality of (Q)MLE for STAR-GARCH. This condition can easily be extended to any non-linear conditional mean model with GARCH errors, subject to reasonable regularity conditions. Although the log-moment condition cannot be verified as easily as the second and fourth moment conditions, it allows the long run persistence of the GARCH process to exceed one. Monte Carlo experiments show that the log-moment condition is more reliable in practice than the second and fourh moment conditions when the underlying long run persistence is close to unity. These experiments also show that the correct specification of the conditional mean is crucial in obtaining unbiased estimates for the GARCH component. The sufficient conditions for consistency and asymptotic normality are verified empirically using S&P 500 returns, 3-month US Treasury Bill returns, and exchange rates between Australia and the USA. The effects of outliers and extreme observations on the empirical moment conditions are also analysed in detail.