# (Why) is a logistic regression maximum likelihood estimator consistent?

A nice property of maximum likelihood estimators is that, while they can be biased, they are consistent for $$iid$$ observations.

In a logistic regression, unless the conditional distributions all have the same probability parameter, we lose the "identically" distributed of $$iid$$. Nonetheless, logistic regressions tend to have their parameters estimated by maximizing the likelihood.

(Why) does this not lead to an inconsistent maximum likelihood estimator?

Related post: Logistic regression panel data fixed effects.

• That's what I thought, too, but evidently not. @JohnMadden
– Dave
Commented Jan 30, 2023 at 23:23
• "i.i.d." is a sufficient condition, but not a necessary condition for consistency. Commented Jan 31, 2023 at 2:20
• @Zhanxiong But why is there consistency despite the lack of $iid?$
– Dave
Commented Jan 31, 2023 at 2:28
• @Dave Whereas a rigorous discussion requires a careful listing of assumptions on the parameter space, a quick answer is that: the MLE is a type of $M$-estimate, and the log-likelihood function of the logistic regression (under the usual regularity conditions) satisfies the condition of the Wald's consistency theorem (see Theorem 5.14 in Asymptotic Statistcs by A. W. van der Vaart), thus $\hat{\beta}$ is consistent. Commented Jan 31, 2023 at 2:52
• Dave, I added your other question's link in this post for the commentary there is definitely relevant here. Commented Jan 31, 2023 at 4:40