Optimal Multi-Step VAR Forecasting Averaging

This paper proposes frequentist multiple-equation least squares averaging approaches for multi-step forecasting with vector autoregressive (VAR) models. The proposed VAR forecasting averaging methods are based on the multivariate Mallows model averaging (MMMA) and multivariate leave-h-out cross-validation averaging (MCVAh) criteria (with h denoting the forecast horizon), which are valid for iterative and direct multi-step forecasting averaging, respectively. Under the framework of stationary VAR processes of infinite order, we provide theoretical justifications by establishing asymptotic unbiasedness and asymptotic optimality of the proposed forecasting averaging approaches. Specifically, MMMA exhibits asymptotic optimality for one-step ahead forecast averaging, whereas for direct multi-step forecasting averaging the asymptotically optimal combination weights are determined separately for each forecast horizon based on the MCVAh procedure. The finite-sample behaviour of the proposed averaging procedures under misspecification is investigated via simulation experiments. An empirical application to a three-variable monetary VAR, based on the U.S. data, is also provided to present our methodology