Hindawi Publishing Corporation Journal of Probability and Statistics Volume 2011, Article ID 718647, 39 pages doi:10.1155/2011/718647 Research Article Estimation and Properties of a Time-Varying GQARCH(1,1)-M Model Sofia Anyfantaki and Antonis Demos Athens University of Economics and Business, Department of International and European Economic Studies, 10434 Athens, Greece Correspondence should be addressed to Antonis Demos, demos@aueb.gr Received 16 May 2011; Accepted 14 July 2011 Academic Editor: Mike Tsionas Copyright q 2011 S. Anyfantaki and A. Demos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Time-varying GARCH-M models are commonly used in econometrics and financial economics. Yet the recursive nature of the conditional variance makes exact likelihood analysis of these models computationally infeasible. This paper outlines the issues and suggests to employ a Markov chain Monte Carlo algorithm which allows the calculation of a classical estimator via the simulated EM algorithm or a simulated Bayesian solution in only OT computational operations, where T is the sample size. Furthermore, the theoretical dynamic properties of a time-varying GQARCH1,1-M are derived. We discuss them and apply the suggested Bayesian estimation to three major stock markets. 1. Introduction Time series data, emerging from diverse fields appear to possess time-varying second con- ditional moments. Furthermore, theoretical results seem to postulate quite often, specific relationships between the second and the first conditional moment. For instance, in the stock market context, the first conditional moment of stock market’s excess returns, given some information set, is a possibly time-varying, linear function of volatility see, e.g., Merton 1, Glosten et al. 2. These have led to modifications and extensions of the initial ARCH model of Engle 3 and its generalization by Bollerslev 4, giving rise to a plethora of dynamic heteroscedasticity models. These models have been employed extensively to capture the time variation in the conditional variance of economic series, in general, and of financial time series, in particular see Bollerslev et al. 5 for a survey. Although the vast majority of the research in conditional heteroscedasticity is being processed aiming at the stylized facts of financial stock returns and of economic time series