Comparing multivariate volatility forecasts by direct and indirect approaches

The evaluation of multivariate volatility models can be done through a direct or indirect approach. The former uses statistical loss func- tions (LFs) and a proxy providing consistent estimates of the unob- served volatility. The latter uses utility LFs or other instruments like the Value-at-Risk (VaR) and its backtesting procedures. Com- monly, existing studies employ these procedures separately, focusing mostly on the multivariate generalized autoregressive conditional het- eroskadisticity (MGARCH) models. This work aims to investigate and compare the two approaches in the 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 intradaily returns. With reference to the direct approach, we em- pirically rank the set of competing models by means of four consistent statistical LFs and by deteriorating the quality of the volatility proxy. As regards to the indirect approach, we use standard backtesting pro- cedures to evaluate if the number of VaR violations is acceptable and these violations are independently distributed over time.