Comparing multivariate volatility forecasts by direct and indirect approaches

Multivariate volatility models can be evaluated via direct and indirect approaches. The former uses statistical loss functions (LFs) and a proxy to provide consistent estimates of the unobserved volatility. The latter uses utility LFs or other instruments, such as value-at-risk and its backtesting procedures. Existing studies commonly employ these procedures separately, focusing mostly on the multivariate generalized autoregressive conditional heteroscedasticity (MGARCH) models. This work investigates and compares the two approaches in a model selection context. An extensive Monte Carlo simulation experiment is carried out, including MGARCH models based on daily returns and, extending the current literature, models that directly use the realized covariance, obtained from intraday returns. With reference to the direct approach, we rank the set of competing models empirically by means of four consistent statistical LFs and by reducing the quality of the volatility proxy. For the indirect approach, we use standard backtesting procedures to evaluate whether the number of value-at-risk violations is acceptable, and whether these violations are independently distributed over time.