Consistency

An estimator $\hat\beta$ is consistent if it converges in probability to the true parameter $\beta$ as the sample size grows: $\hat\beta \xrightarrow{p} \beta$ as $n \to \infty$. Consistency is a large-sample property: a consistent estimator may be biased in finite samples, but its bias shrinks with $n$. Contrast with unbiased: $\mathbb E[\hat\beta] = \beta$ holds at every sample size, not just asymptotically.

When to use

The distinction matters because OLS under endogeneity is both biased AND inconsistent — it converges to the wrong target no matter how much data you collect. 2SLS is biased in small samples (first-stage estimation uncertainty) but consistent, so it improves with $n$. MLE for logit / probit is similarly consistent but not unbiased. When choosing estimators, consistency is the minimum bar — without it more data is no help.