Specification tests for non-Gaussian maximum likelihood estimators

We propose generalised DWH specification tests which simultaneously compare three or more likelihood-based estimators of conditional mean and variance parameters in multivariate conditionally heteroskedastic dynamic regression models. Our tests are useful for Garch models and in many empirically relevant macro and finance applications involving Vars and multivariate regressions. To design powerful and reliable tests, we determine the rank deficiencies of the differences between the estimators' asymptotic covariance matrices under the null of correct specification, and take into account that some parameters remain consistently estimated under the alternative of distributional misspecification. Finally, we provide finite sample results through Monte Carlo simulations.