Estimation and inference in univariate and multivariate log-GARCH-X models when the conditional density is unknown

A general framework for the estimation and inference in univariate and multivariate Generalised log-ARCH-X (i.e. log-GARCH-X) models when the conditional density is unknown is proposed. The framework employs (V)ARMA-X representations and relies on a bias-adjustment in the log-volatility intercept. The bias is induced by (V)ARMA estimators, but the remaining parameters can be estimated in a consistent and asymptotically normal manner by usual (V)ARMA methods. An estimator of the bias and a closed-form expression for the asymptotic variance is derived. Adding covariates and/or increasing the dimension of the model does not change the structure of the problem, so the univariate bias-adjustment procedure is applicable not only in univariate log-GARCH-X models estimated by the ARMA-X representation, but also in multivariate log-GARCH-X models estimated by VARMA-X representations. Extensive simulations verify the properties of the log-moment estimator, and an empirical application illustrates the usefulness of the methods.