Slope Consistency of Quasi-Maximum Likelihood Estimator for Binary Choice Models

Abstract: This paper revisits the slope consistency of QMLE for binary choice models. Ruud (1983, \emph{Econometrica}) introduced a set of conditions under which QMLE may yield a constant multiple of the slope coefficient of binary choice models asymptotically. However, he did not fully establish slope consistency of QMLE, which requires the existence of a positive multiple of slope coefficient identified as an interior maximizer of the population QMLE likelihood function over an appropriately restricted parameter space. We fill this gap by providing a formal proof for slope consistency under the same set of conditions for any binary choice model identified as in Horowitz (1992, \emph{Econometrica}). Our result implies that the logistic regression, which is used extensively in machine learning to analyze binary outcomes associated with a large number of covariates, yields a consistent estimate for the slope coefficient of binary choice models under suitable conditions.