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

2025-05-05

AI Summary

This paper addresses the consistency of quasi-maximum likelihood estimation (QMLE) slope coefficients in binary choice models. Filling a theoretical gap left by Ruud (1983), it provides the first rigorous proof—under Horowitz’s (1992) identification condition—that the QMLE slope estimator converges in probability to the true slope direction up to a nonzero scale factor. This result holds irrespective of the specific link function and, in particular, establishes directional consistency of logistic regression even under high-dimensional covariates. The contributions are threefold: (1) it strengthens the theoretical foundation of QMLE for binary response models; (2) it delivers a crucial statistical guarantee for widely used classification methods in machine learning—especially logistic regression—regarding consistent estimation of the coefficient direction; and (3) it extends the applicability of pseudo-likelihood approaches to nonstandard settings where full model specification or distributional assumptions may be violated.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

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.

Problem

• Proving slope consistency of QMLE for binary choice models
• Filling the gap in Ruud's conditions for QMLE consistency
• Establishing logistic regression consistency for slope coefficients

Innovation

• Proves slope consistency for QMLE in binary models
• Uses Horowitz (1992) identification conditions
• Validates logistic regression for slope estimation