Volatility Reprojection and Forecasting Performance -- An EMM Approach toward the Multivariate Stochastic Volatility Model

While the conditional volatility of time series is always dependent of the model specification, the {\\em ex post} or realized volatility series is often constructed on a model-free basis. The common proxies of daily volatility in the literature are the squared daily asset returns and the sum of squared intra-daily asset returns. In this paper, we propose to construct the underlying volatility series in a multivariate stochastic volatility (SV) model framework using the reprojection technique proposed by Gallant and Tauchen (1998). The reprojected underlying volatility series is obtained via a two-step procedure: in the first step the efficient method of moment (EMM) proposed by Gallant and Tauchen (1996) is employed to estimate the multivariate SV model of asset returns, and in the second step the underlying volatility reprojection technique is applied to the estimated multivariate SV model. The reprojected volatility series, a representation for unobservables in terms of observables, is consistent with the model specification and at the same time advantageous in volatility forecasting. While unavailability is among many issues realted to the use of intra-daily high-frequency asset returns, we show analytically that the squared daily asset return residuals as proxy of \\emph{ex-post} volatility directly leads to extremely low explanatory power in the common regression analysis of volatility forecasting. However, the performance of volatility forecasting based on reprojected volatility series is substantially improved. We also illustrate that the volatility forecasting performance based on multivariate SV model further improves over that based the univariate SV models due to the correlated movements of asset return volatility.