Why does xtreg with the mle option produce different results from xtreg with only the re option?

Question

A user asked about differing estimates and predictions from xtreg when fitting a random-effects model with and without the mle option:

I am getting inconsistent results when I try to use xtreg, re option. Initially I thought it was my lack of understanding of the options but I think there might be a problem. I have a continuous measure on 39 cases at 18 periods. The 39 cases are in 2 groups (fixed effect). I model a between group, linear time, squared time, and the interactions of group and time and squared time. The mle model for these data give a −1.9 coefficient for the group variable; the re model gives a 2.1 coefficient for the group variable. The time interaction terms are equivalent in both mle and re models.

Answer

The standard random-effects regression estimator, xtreg ..., re, is a generalized method of moments (GMM) estimator that is just a matrix-weighted average of the between and within estimators. The ML random-effects regression estimator, xtreg ..., mle, is an MLE that fully maximizes the likelihood of the random-effects model. They are different estimators of the same model that can and do produce different estimates.

You note that one coefficient is −1.9 under one estimator and 2.1 under another. This would surprise me only if both of these estimates were statistically significantly different from 0. Even then, I would be only surprised.

The predicted values for the re model are consistently higher (by 2–4 points on this scale) and are above any observed mean for the group indexed as 1. The predicted values for the mle model are close to the observed values.

The GMM estimator does not directly consider the group predicted values as one of its moments; therefore, its predictions need not match observed group means. Again, the GMM is a different estimator from the MLE, though both are consistent.