We develop a specification test of predictive densities based on that the generalized residuals of correctly specified predictive density models are i.i.d. 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 against violation to uniformity. The approach of 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 nuisance parameter free, and 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.
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