Properties of OLS Estimators

The properties of the OLS Estimators are:

1. $E(Y_{i}|X=x_{i})=\beta _{1}+\beta _{2}x_{i}$$E(Y_{i}|X=x_{i})=\beta _{1}+\beta _{2}x_{i}$
2. $var(Y_{i}|X=x_{i})=\sigma ^{2}$$var(Y_{i}|X=x_{i})=\sigma ^{2}$
3. $Y_{1}|X=x_{1};Y_{2}|X=x_{2}...Y_{n}|X=x_{n}$$Y_{1}|X=x_{1};Y_{2}|X=x_{2}...Y_{n}|X=x_{n}$

The expectation of estimators $B_{1}$$B_{1}$ and $B_{2}$$B_2$ are:

$E(B_{2})=\beta _{2}$$E(B_{2})=\beta _{2}$

$E(B_{1})=\beta _{1}$$E(B_{1})=\beta _{1}$

The variance and covariance of estimators $B_{1}$$B_{1}$ and $B_{2}$$B_2$ are:

$var(B_{2})={{\sigma ^{2}} \over {\sum {i=1}^{n}(x_{i}-{\bar {x}})^{2}}}$$var(B_{2})={{\sigma ^{2}} \over {\sum {i=1}^{n}(x_{i}-{\bar {x}})^{2}}}$

$var(B_{1})=\sigma ^{2}[{{1} \over {n}}+{{{\bar {x}}^{2}} \over {\sum {i=1}^{n}(x_{i}-{\bar {x}})^{2}}}$$var(B_{1})=\sigma ^{2}[{{1} \over {n}}+{{{\bar {x}}^{2}} \over {\sum {i=1}^{n}(x_{i}-{\bar {x}})^{2}}}$

$cov(B_{1},B_{2})=-{{\sigma ^{2}{\bar {x}}} \over {\sum {i=1}^{n})x_{i}-{\bar {x}}^{2}}}$$cov(B_{1},B_{2})=-{{\sigma ^{2}{\bar {x}}} \over {\sum {i=1}^{n})x_{i}-{\bar {x}}^{2}}}$