An Oracle Inequality for Multivariate CAViaR Forecasting

This paper derives a finite-sample oracle inequality for a class of misspecified multivariate conditional autoregressive quantile forecasts. This inequality is used to establish that the M-estimator of a multivariate version of the conditional autoregressive Value-at-Risk model (CAViaR) achieves the best out-of-sample performance within its class in the check loss sense at a near optimal rate, even when the model is fully misspecified. An empirical application to backtesting global Growth-at-Risk shows that a combination of the AR-GARCH and CAViaR methodologies performs best out-of-sample in terms of the check loss