Backtesting portfolio value‐at‐risk with estimated portfolio weights

This article theoretically and empirically analyzes backtesting portfolio value‐at‐risk (VaR) with estimation risk in an intrinsically multi‐variate framework. It particularly takes into account the estimation of portfolio weights in forecasting portfolio VaR and its impact on backtesting. It shows that the estimation risk from estimating portfolio weights and that from estimating the multi‐variate dynamic model make the existing methods in a univariate framework inapplicable. It proposes a general theory to quantify estimation risk applicable to the present problem and suggests practitioners a simple but effective way to implement valid inference to overcome the effect of estimation risk in backtesting portfolio VaR. In particular, we apply our theory to the efficient mean‐variance‐skewness portfolio for a multi‐variate generalized autoregressive conditional heteroscedasticity model with multi‐variate general hyperbolic distributed innovations. Some Monte Carlo simulations and an empirical application demonstrate the merits of our method.