This paper proposes a bootstrap test for the correct specification of
parametric conditional distributions. It extends Zheng's test (Zheng,
2000, Econometric Theory 16,
667–691) to allow for discrete dependent variables and for mixed
discrete and continuous conditional variables. We establish the asymptotic
null distribution of the test statistic with data-driven stochastic
smoothing parameters. By smoothing both the discrete and continuous
variables via the method of cross-validation, our test has the advantage
of automatically removing irrelevant variables from the estimate of the
conditional density function and, as a consequence, enjoys substantial
power gains in finite samples, as confirmed by our simulation results. The
simulation results also reveal that the bootstrap test successfully
overcomes the size distortion problem associated with Zheng's
test.We are grateful for the insightful
comments from three referees and a co-editor that greatly improved the
paper. Li's research is partially supported by the Private Enterprise
Research Center, Texas A&M University. Fan is grateful to the National
Science Foundation for research support.