A sequential test for the specification of predictive densities

We develop a specification test of predictive densities, based on the fact that the generalized residuals of correctly specified predictive density models are independent and identically distributed uniform. The proposed sequential test examines the hypotheses of serial independence and uniformity in two stages, wherein the first-stage test of serial independence is robust to violation of uniformity. The approach of the data-driven smooth test is employed to construct the test statistics. The asymptotic independence between the two stages facilitates proper control of the overall type I error of the sequential test. We derive the asymptotic null distribution of the test, which is free of nuisance parameters, and we establish its consistency. Monte Carlo simulations demonstrate excellent finite sample performance of the test. We apply this test to evaluate some commonly used models of stock returns.