Finite-sample properties of the maximum likelihood estimator for the binary logit model with random covariates

We examine the finite sample properties of the maximum likelihood estimator for the binary logit model with random covariates. Previous studies have either relied on large-sample asymptotics or have assumed non-random covariates. Analytic expressions for the first-order bias and second-order mean squared error function for the maximum likelihood estimator in this model are derived, and we undertake numerical evaluations to illustrate these analytic results for the single covariate case. For various data distributions, the bias of the estimator is signed the same as the covariate's coefficient, and both the absolute bias and the mean squared errors increase symmetrically with the absolute value of that parameter. The behaviour of a bias-adjusted maximum likelihood estimator, constructed by subtracting the (maximum likelihood) estimator of the first-order bias from the original estimator, is examined in a Monte Carlo experiment. This bias-correction is effective in all of the cases considered, and is recommended for use when this logit model is estimated by maximum likelihood using small samples.