Semiparametric single-index Poisson regression model with unobserved heterogeneity

We propose a two-step semiparametric pseudo-maximum likelihood procedure for single-index regression models where the conditional variance depends on the regression function and an additional parameter. The Poisson single-index regres-sion model with multiplicative unobserved heterogeneity is an example of such a semiparametric model. Our procedure is based on linear exponential densities with nuisance parameter. The nuisance parameter is estimated in a preliminary step and its estimate is used to build the pseudo-likelihood criterion for the second step. This pseudo-likelihood criterion contains a nonparametric estimate of the index regression and therefore a rule for choosing the smoothing parameter is needed. We propose an automatic and natural rule based on the joint maximization of the pseudo-likelihood with respect to the index parameter and the smoothing parameter. We derive the asymptotic properties of the semiparametric estimator of the index parameter and the asymptotic behavior of our ‘optimal ’ smoothing parameter. The finite sam-ple performances of our methodology are analyzed in a simulation experiment. An application to real data is also provided