Misspecification Testing in a Class of Conditional Distributional Models

Abstract

We propose a specification test for a wide range of parametric models for the 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 a restricted estimate that imposes the structure implied by the model. The procedure is straightforward to implement, is consistent against fixed alternatives, has nontrivial power against local deviations of order n ^{− 1/2} from the null hypothesis, and does not require the choice of smoothing parameters. In an empirical application, we use our test to study the validity of various models for the conditional distribution of wages in the United States.

Keywords:

• Bootstrap
• Cramer–von Mises distance
• Distributional regression
• Location-scale model
• Quantile regression
• Wage distribution

Acknowledgments

We thank the associate editor, the referees, Roger Koenker, and Blaise Melly for their helpful comments. Financial support by Deutsche Forschungsgemeinschaft (SFB 823, project A1) is gratefully acknowledged.

Notes

Escanciano and Goh (2010) pointed out that for the case of iid data, which we consider in this article, the properties of their procedure are superior to those of tests based on subsampling, as, for example, in Escanciano and Velasco (2010), and hence we do not consider the latter for our simulations.

NOTE: KX denotes the Koenker and Xiao (2002) specification test based on Khmaladzation, and EG denotes the bootstrap specification test for quantile regression by Escanciano and Goh (2010). Suffixes denote the model being tested: location shift (LS), location-scale shift (LSS), quantile regression (QR), or distributional regression (DR). Left column specifies the true DGP, as described in the main text.

NOTE: Suffixes denote the specification being tested: location shift (LS), location-scale shift (LSS), quantile regression (QR), or distributional regression (DR). Left column specifies the set of explanatory variables used, as described in the main text.