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

TL;DR

This paper proves the slope consistency of the quasi-maximum likelihood estimator in binary choice models with logistic errors, clarifying when logistic regression provides reliable slope estimates.

Contribution

It provides a formal proof of slope consistency for QMLE in BCMs, filling a gap left by previous theoretical work.

Findings

QMLE yields consistent slope estimates under certain conditions

The proof confirms logistic regression's reliability for BCMs in specific settings

Conditions for slope consistency are explicitly characterized

Abstract

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. 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 the slope consistency of QMLE, which requires the existence of a positive multiple of the true slope that maximizes the population QMLE likelihood over an appropriately restricted parameter space. We close this gap by providing a formal proof of slope consistency under the same set of conditions for BCMs identified as in Manski (1975, 1985). Our result implies that, under suitable conditions, logistic regression yields a consistent estimate of the slope coefficient for BCMs.