Estimation and Inference in Univariate and Multivariate Log-GARCH-X Models When the Conditional Density is Unknown

Exponential models of Autoregressive Conditional Heteroscedasticity (ARCH) are of special interest, since they enable richer dynamics (e.g. contrarian or cyclical), provide greater robustness to jumps and outliers, and guarantee the positivity of volatility. The latter is not guaranteed in ordinary ARCH models, in particular when additional exogenous and/or predetermined variables (``X") are included in the volatility specification. We propose 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 not known. 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 are consistently estimated by (V)ARMA methods. We derive a simple formula for the bias-adjustment, and a closed-form expression for its asymptotic variance. Next, we show that adding exogenous or predetermined variables and/or increasing the dimension of the model does not change the structure of the problem. Accordingly, the univariate bias-adjustment is applicable not only in univariate log-GARCH-X models, but also in multivariate log-GARCH-X models. An empirical application illustrates the usefulness of the methods.