A simulation-based explanation of consistency and asymptotic normality
Overview
In the frequentist approach to statistics, estimators are random variables because they are functions of random data. The finite-sample distributions of most of the estimators used in applied work are not known, because the estimators are complicated nonlinear functions of random data. These estimators have large-sample convergence properties that we use to approximate their behavior in finite samples.
Two key convergence properties are consistency and asymptotic normality. A consistent estimator gets arbitrarily close in probability to the true value. The distribution of an asymptotically normal estimator gets arbitrarily close to a normal distribution as the sample size increases. We use a recentered and rescaled version of this normal distribution to approximate the finite-sample distribution of our estimators.
I illustrate the meaning of consistency and asymptotic normality by Monte Carlo simulation (MCS). I use some of the Stata mechanics I discussed in Monte Carlo simulations using Stata.
Consistent estimator
A consistent estimator gets arbitrarily close in Read more…