Consistency

Econometrics

In a general notation, we suppose $\hat{\theta }$ is an estimator of $\theta$, so we call $\hat{\theta }$ consistent if it converge in probability to $\theta$, that is,

$$pli{m}_{n\to \infty }\hat{\theta }=\theta$$

To work out the probability limit of $\hat{\theta }$, we should use some tools such as the Law of Large Numbers

Some sufficient conditions for Consistency:

1. ${\lim }_{n\to \infty }E(\hat{\theta })=\theta$
2. ${\lim }_{n\to \infty }Var(\hat{\theta })=0$

If both conditions are satisfied, we could ensure that $\hat{\theta }$ is consistent.