An Adaptive Specification Test For Semiparametric Models

This paper introduces a new test of a semiparametric model of a conditional density function against a fully nonparametric alternative. This test is motivated by the fact that many important econometric models need to be estimated through maximum likelihood type procedures, e.g. semiparametric limited dependent variable models. This specification is also important for prediction purposes. Our test statistic combines the methodology of goodness of fit tests and nonparametric methods and the specific difficulty we focus here comes from the fact that we consider a semiparametric null hypothesis and the test statistic depends on a bandwidth parameter that needs to be estimated from the data. In order to handle the previous difficulties we introduce a data adaptive testing procedure that enables us to select the bandwidth parameter in such a way that it maximizes the power of the test. It is also shown that this procedure handles properly the bias problem generated by the introduction of a semiparametric model under the null. The distribution of the standarized test statistic is approximated under the null by both bootstrap and subsampling methods and its power is studied against local alternatives to the null hypothesis. We discuss practical issues for the application statistics and illustrate in an intensive monte carlo study both the feasibility and the performance of the procedure in finite samples of moderate size