Testing for additivity in partially linear regression with possibly missing responses

We consider a partially linear regression model with multivariate covariates and with responses that are allowed to be missing at random. This covers the usual settings with fully observed data and the nonparametric regression model as special cases. We first develop a test for additivity of the nonparametric part in the complete data model. The test statistic is based on the difference between two empirical estimators that estimate the errors in two ways: the first uses a local polynomial smoother for the nonparametric part; the second estimates the additive components by a marginal integration estimator derived from the local polynomial smoother. We present a uniform stochastic expansion of the empirical estimator based on the marginal integration estimator, and we derive the asymptotic distribution of the test statistic. The transfer principle of Koul et al. (2012) then allows a direct adaptation of the results to the case when responses are missing at random. We examine the performance of the tests in a small simulation study.