Testing Parametric versus Semiparametric Modeling in Generalized Linear Models

We consider a generalized partially linear model E(Y|X, T) = G{X T β + m(T)}, where G is a known function, β is an unknown parameter vector, and m is an unknown function. We introduce a test statistic that allows one to decide between a parametric and a semiparametric model: (a) m is linear (i.e., m(t) = t T γ for a parameter vector γ), and (b) m is a smooth (nonlinear) function. Under linearity (a), we show that the test statistic is asymptotically normal. Moreover, we prove that the bootstrap works asymptotically. Simulations suggest that (in small samples) the bootstrap outperforms the calculation of critical values from the normal approximation. The practical performance of the test is demonstrated in applications to data on East–West German migration and credit scoring.