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

Yoosoon Chang, Joon Y. Park, Guo Yan

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.

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

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.
Paper Structure (9 sections, 4 theorems, 34 equations)

This paper contains 9 sections, 4 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Slope Consistency
  4. Concluding Remarks
  5. Proof of Lemma \ref{['cast-positive']}
  6. Step 1:
  7. Step 2:
  8. Step 3:
  9. Proof of Theorem \ref{['main']}

Key Result

Lemma 2.1

Let Assumption regularity hold. Then $\hat{\theta} \to_p \theta_\ast$.

Theorems & Definitions (4)

  • Lemma 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3