Semiparametric Single-index Poisson Regression Model with Unobserved Heterogeneity

We propose a two-step semiparametric pseudo-maximum likelihood procedure forsingle-index regression models where the conditional variance is a known function of theregression and an additional parameter. The Poisson single-index regression model withmultiplicative unobserved heterogeneity is an example of such a semiparametric model.Our procedure is based on linear exponential densities with nuisance parameter. Thenuisance parameter is estimated in a preliminary step and its estimate is used to buildthe pseudo-likelihood criterion for the second step. This pseudo-likelihood criterioncontains a nonparametric estimate of the index regression and therefore a rule forchoosing the smoothing parameter is needed. We propose an automatic and naturalrule based on the joint maximization of the pseudo-likelihood with respect to the indexparameter and the smoothing parameter. We derive the asymptotic properties of thesemiparametric estimator of the index parameter and the asymptotic behavior of our`optimal' smoothing parameter. The ¯nite sample performances of our methodologyare analyzed in a simulation experiment. An application to real data is also provided.