  This paper investigates the size performance of Wald tests for CAViaR models
(Engle and Manganelli, 2004). We find that the usual estimation strategy on
test statistics yields inaccuracies. Indeed, we show that existing density
estimation methods cannot adapt to the time-variation in the conditional
probability densities of CAViaR models. Consequently, we develop a method
called adaptive random bandwidth which can approximate time-varying conditional
probability densities robustly for inference testing on CAViaR models based on
the asymptotic normality of the model parameter estimator. This proposed method
also avoids the problem of choosing an optimal bandwidth in estimating
probability densities, and can be extended to multivariate quantile regressions
straightforward.
