Estimation of SEM with GARCH errors

The interdependence of financial markets combined with their volatility make the multivariate GARCH model a suitable econometric framework for analysing their behaviour. However, the non-availability of analytical derivatives in a general context and the computational heaviness resulting from a numerical calculation still represent a major hurdle for the use of such models in practical applications. In a general simultaneous equation model with multivariate GARCH errors, analytical expressions of the score, the Hessian and the information matrices are derived and used for implementing QML and GMM estimation procedures. The asymptotic variances of these estimators are obtained using the same expressions and the asymptotic superiority of GMM over QML is shown in the non-normal case. A simulation study comparing different gradient algorithms for ML as well as the finite sample behaviour of ML and GMM shows that using analytical results instead of numerical approximations in the optimisation procedure yields better results and reiterates the superiority of GMM over QML in finite samples under non-normality.