# Properties of OLS Estimators

##### The properties of the OLS Estimators are:
1. ${\displaystyle E(Y_{i}|X=x_{i})=\beta _{1}+\beta _{2}x_{i}}$
2. ${\displaystyle var(Y_{i}|X=x_{i})=\sigma ^{2}}$
3. ${\displaystyle Y_{1}|X=x_{1};Y_{2}|X=x_{2}...Y_{n}|X=x_{n}}$
##### The expectation of estimators ${\displaystyle B_{1}}$ and ${\displaystyle B_{2}}$ are:

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

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

##### The variance and covariance of estimators ${\displaystyle B_{1}}$ and ${\displaystyle B_{2}}$ are:

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

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

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