A consistent specification test for dynamic quantile models

Correct specification of a conditional quantile model implies that a particular conditional moment is equal to zero. We nonparametrically estimate the conditional moment function via series regression and test whether it is identically zero using uniform functional inference. Our approach is theoretically justified via a strong Gaussian approximation for statistics of growing dimensions in a general time series setting. We propose a novel bootstrap method in this nonstandard context and show that it significantly outperforms the benchmark asymptotic approximation in finite samples, especially for tail quantiles such as Value-at-Risk (VaR). We use the proposed new test to study the VaR and CoVaR (Adrian and Brunnermeier (2016)) of a collection of US financial institutions.