Predictive density and conditional confidence interval accuracy tests

This paper outlines testing procedures for assessing the relative out-of-sample predictive accuracy of multiple conditional distribution models. The tests that are discussed are based on either the comparison of entire conditional distributions or the comparison of predictive confidence intervals. We also briefly survey existing related methods in the area of predictive density evaluation, including methods based on the probability integral transform and the Kullback-Leibler Information Criterion. The procedures proposed in this paper are similar in many ways to [Andrews', 1997. A conditional Kolmogorov test. Econometrica 65, 1097-1128.] conditional Kolmogorov test and to [White's, 2000. A reality check for data snooping. Econometrica 68, 1097-1126.] reality check. In particular, a predictive density test is outlined that involves comparing square (approximation) errors associated with models i,i=1,…,n, by constructing weighted averages over U of E((Fi(u|Zt,θi†)-F0(u|Zt,θ0))2), where F0(·|·) and Fi(·|·) are true and model-i distributions, u∈U, and U is a possibly unbounded set on the real line. A conditional confidence interval version of this test is also outlined, and appropriate bootstrap procedures for obtaining critical values when predictions used in the formation of the test statistics are obtained via rolling and recursive estimation schemes are developed. An empirical illustration comparing alternative predictive models for U.S. inflation is given for the predictive confidence interval test.