Probably a naive question. I am quoting this from Greene's econometrics book:
"The occasional statement that the properties of the MLE are only optimal in large samples is not true, however. It can be shown that when sampling is from an exponential family of distributions, there will exist sufficient statistics. If so, MLEs will be functions of them, which means that when minimum variance unbiased estimators exist, they will be MLEs. [See Stuart and Ord (1989).] Most applications in econometrics do not involve exponential families, so the appeal of the MLE remains primarily its asymptotic properties."
So why the variance estimate is downward biased in MLE for a normal distribution?