On the consistency of the logistic quasi-MLE under conditional symmetry

For estimating the parameters of a linear conditional mean, I show that the quasi-maximum likelihood estimator (QMLE) obtained under the nominal assumption that the error term is independent of the explanatory variables with a logistic distribution is consistent provided the conditional distribution of the error term is symmetric. No other restrictions are required for Fisher consistency; in particular, the error and covariates need not be independent, and so general heteroskedasticity of unknown form is allowed. Importantly, the influence function of the logistic quasi-log likelihood is bounded, making it more resilient to outliers than ordinary least squares. Inference using the logistic QMLE is straightforward using a robust asymptotic variance-covariance matrix estimator. (C) 2020 Elsevier B.V. All rights reserved.