Rates of convergence for estimating regression coefficients in heteroskedastic discrete response models

In this paper, we consider estimation of discrete response models exhibiting conditional heteroskedasticity of a multiplicative form, where the latent error term is assumed to be the product of an unknown scale function and a homoskedastic error term. It is first shown that for estimation of the slope coefficients in a binary choice model under this type of restriction, the semiparametric information bound is zero, even when the homoskedastic error term is parametrically specified. Hence, it is impossible to attain the parametric convergence rate for the parameters of interest. However, for ordered response models where the response variable can take at least three different values, the parameters of interest can be estimated at the parametric rate under the multiplicative heteroskedasticity assumption. Two estimation procedures are proposed. The first estimator, based on a parametric restriction on the homoskedastic component of the error term, is a two-step maximum likelihood estimators, where the unknown scale function is estimated nonparametrically in the first stage. The second procedure, which does not require the parametric restriction, estimates the parameters by a kernel weighted least-squares procedure. Under regularity conditions which are standard in the literature, both estimators are shown to be n-consistent and asymptotically normal.