Moment tests for density forecast evaluation in the presence of parameter estimation uncertainty

Density forecast (DF) possesses appealing properties when it is correctly specified for the true conditional distribution. Although a number of parametric specification tests have been introduced for the DF evaluation (DFE) in the parameter-free context, econometric DF models are typically parameter-dependent. In this paper, we first use a generalized probability integral transformation-based moment test to unify these existing tests, and then apply the Newey–Tauchen method (the West–McCracken method) to correct this unified test as a generalized full-sample (out-of-sample) test in the parameter-dependent context. Unlike the corrected tests, the uncorrected tests could be substantially undersized (oversized) when they are directly applied to the full-sample (out-of-sample) DFE in the presence of parameter estimation uncertainty. We also use a simulation to show the usefulness of the corrected tests in rectifying the size distortion problem, and apply the corrected tests to an empirical study of stock index returns.