【Econometrics】Proof of Properties of Estimator

In the last post, we went through the properties of estimators. This post contains a mathematical proof of two properties: Unbiasedness and Consistency

 

 

【Econometrics】Estimator

 

I. Unbiasedness

 

Assumption:

• $X$ is a Random Variable ($RV$) that is $i.i.d$ (independent and identically distributed) with $E(X) = \mu$ and $V(X) = \sigma^2$.
  ($X\sim i.i.d.(\mu,\sigma^2)$)

 

Proof

 

$$
E(\bar{X}) = E(\frac{1}{N} \sum_{i=1}^N X_{i}) = \frac{1}{N} E(\sum_{i=1}^N X_{i}) =\frac{1}{N} \sum_{i=1}^N E(X_{i}) = \frac{1}{N} (N\mu) = \mu
$$

 

 

II. Consistency

 

Assumption:

Given $X\sim i.i.d. (\mu,\sigma^2)$ (unbiased), we need to show that $Var(\bar{X})\rightarrow 0$ as $N \rightarrow \infty$.

 

Proof

 

$Var(\bar{x}) = Var (\frac{1}{N} \sum_{i=1}^N X_{i})$

             $ = \frac{1}{N^2} Var(\sum_{i=1}^N X_{i})$

             $ = \frac{1}{N^2} [Var(X_1) + Var(X_2) + \cdots+Var(X_n) + 2Cov(X_1,X_2) + 2(Cov(X_1,X_3) + \cdots] $

             $ = \frac{1}{N^2}[Var(X_1) + Var(X_2) +\cdots+Var(X_n) + 0 + 0 +\cdots ] $  by independence 

             $ = \frac{1}{N^2}[Var(X_1) + Var(X_1) +\cdots+Var(X_1) + \cdots ] $  by identically distributed

             $ = \frac{1}{N^2} [NVar(X_1)] $

             $ = \frac{1}{N} [N\sigma^2] $

             $ = \frac{\sigma^2}{N}, $  which indeed converges to zero as $N$ goes to infinity