Testing Parametric Versus Semiparametric Modelling in Generalized Linear Models

We consider a generalized partially linear model E(Y|X,T) = G{X'b + m(T)} where G is a known function, b is an unknown parameter vector, and m is an unknown function. The paper introduces a test statistic which allows to decide between a parametric and a semiparametric model: (i) m is linear, i.e.m(t) = t'g for a parameter vector g, (ii) m is a smooth (nonlinear) function. Under linearity (i) it is shown that the test statistic is asymptotically normal. Moreover, for the case of binary responses, it is proved that the bootstrap works asymptotically. Simulations suggest that (in small samples) bootstrap outperforms the calculation of critical values from the normal approximation. The practical performance of the test is shown in applications to data on East--West German migration and credit scoring