Misspecification testing in a class of conditional distributional models

We propose a specification test for a wide range of parametric models for conditional distribution function of an outcome variable given a vector of covariates. The test is based on the Cramer-von Mises distance between an unrestricted estimate of the joint distribution function of the data, and an restricted estimate that imposes the structure implied by the model. The procedure is straightforward to implement, is consistent against fixed alternatives, has non-trivial power against local deviations from the null hypothesis of order n^(-1/2), and does not require the choice of smoothing parameters. We also provide an empirical application using data on wages in the US