  We propose a two-step pseudo-maximum likelihood procedure for semiparametric
single-index regression models where the conditional variance is a known
function of the regression and an additional parameter. The Poisson
single-index regression with multiplicative unobserved heterogeneity is an
example of such models. Our procedure is based on linear exponential densities
with nuisance parameter. The pseudo-likelihood criterion we use 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 sample
performances of our methodology are analyzed using simulated and real data.
