The main problem in the combination of volatility forecasts is that the volatility cannot be directly observed and hence loss functions such as the MSFE cannot be directly used unless a suitable proxy of the conditional variance is defined. A common approach is to use the squared returns but these offer a noisy measure of the volatility and, in many settings, their use can give rise to a non-consistent ranking of the candidate models. A more accurate approximation can be obtained by referring to the concept of realized volatility even if, at very high frequencies, micro-structure market frictions can distort such a measure of the unobserved volatility. Also, in many applications, high frequency data on the phenomenon of interest are not available and so realized volatility measures cannot be computed. For example, this usually happens when working with macroeconomic data for which the maximum sampling frequency is monthly. The aim of this paper is twofold. First, we present a novel approach to the combination of volatility forecasts in which the optimal combination weights are estimated by the Generalized Method of Moments (GMM) imposing appropriate conditions on the standardized residuals implied by a given set of combination weights. The aim is to constrain the standardized residuals to be as close as possible to a sequence of i.i.d. zero mean, unit variance random variables. Second, we suggest a Wald type test of forecast encompassing for volatility models and derive its asymptotic distribution. The finite sample properties of the proposed test statistics are investigated by means of a simulation study.
