ML Estimators for SEM-GARCH Models: Relative Performance of Different Computational Algorithms

Though multivariate GARCH models are widely used in empirical research, their computational aspects still represent a major hurdle, especially when these specifications are introduced in structural models. One such extension namely the simultaneous equations model (SEM) with GARCH errors was considered by Engle and Kroner (1995). While there are many applications of the BEKK formulation proposed in the above article, the model as a whole has received little attention in the econometric literature. This paper uses analytical first and second order derivatives of the likelihood function of a SEM with GARCH errors to compute the corresponding ML estimators. We compare different gradient algorithms in a simulation framework and study the small sample behavior of alternative covariance matrix estimators. As it has been the case in several similar studies, it is found that using analytical results instead of numerical approximations yields better results.