Scanning Multivariate Conditional Densities with Probability Integral Transforms

This paper introduces new ways to construct probability integral transforms of random vectors that complement the approach of Diebold, Hahn, and Tay (1999) for evaluating multivariate conditional density forecasts. Our approach enables us to “scan” multivariate densities in various different ways. A simple bivariate normal example is given that illustrates how “scanning” a multivariate density from particular angles leads to tests with no power or high power. An empirical example is also given that applies several different probability integral transforms to specification testing of Engle’s (2002) dynamic conditional correlation model for multivariate financial returns time series with multivariate normal and t errors.