A semiparametric single-index estimator for a class of estimating equation models

TL;DR

This paper introduces a two-step pseudo-maximum likelihood method for semiparametric single-index models, effectively estimating the index parameter with an automatic smoothing parameter selection, validated through simulations and real data.

Contribution

It develops a novel semiparametric estimator for single-index models with a new automatic smoothing parameter selection rule, extending existing methods.

Findings

Estimator has desirable asymptotic properties.

Method performs well in finite samples.

Applicable to models with known variance functions.

Abstract

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.