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

Although QMLE is generally inconsistent, logistic regression relying on the binary choice model (BCM) with logistic errors is widely used, especially in machine learning contexts with many covariates and high-dimensional slope coefficients. This paper revisits the slope consistency of QMLE for BCMs. Ruud (1983) introduced a set of conditions under which QMLE may yield a constant multiple of the slope coefficient of BCMs 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 of slope consistency under the same set of conditions for any binary choice model identified as in Manski (1975, 1985). Our result implies that logistic regression yields a consistent estimate for the slope coefficient of BCMs under suitable conditions.