Properties of OLS Estimators The properties of the OLS Estimators are:[edit | edit source] E ( Y i | X = x i ) = β 1 + β 2 x i {\displaystyle E(Y_{i}|X=x_{i})=\beta _{1}+\beta _{2}x_{i}} v a r ( Y i | X = x i ) = σ 2 {\displaystyle var(Y_{i}|X=x_{i})=\sigma ^{2}} Y 1 | X = x 1 ; Y 2 | X = x 2 . . . Y n | X = x n {\displaystyle Y_{1}|X=x_{1};Y_{2}|X=x_{2}...Y_{n}|X=x_{n}} The expectation of estimators B 1 {\displaystyle B_{1}} and B 2 {\displaystyle B_{2}} are:[edit | edit source] E ( B 2 ) = β 2 {\displaystyle E(B_{2})=\beta _{2}} E ( B 1 ) = β 1 {\displaystyle E(B_{1})=\beta _{1}} The variance and covariance of estimators B 1 {\displaystyle B_{1}} and B 2 {\displaystyle B_{2}} are:[edit | edit source] v a r ( B 2 ) = σ 2 ∑ i = 1 n ( x i − x ¯ ) 2 {\displaystyle var(B_{2})={{\sigma ^{2}} \over {\sum {i=1}^{n}(x_{i}-{\bar {x}})^{2}}}} v a r ( B 1 ) = σ 2 [ 1 n + x ¯ 2 ∑ i = 1 n ( x i − x ¯ ) 2 {\displaystyle var(B_{1})=\sigma ^{2}[{{1} \over {n}}+{{{\bar {x}}^{2}} \over {\sum {i=1}^{n}(x_{i}-{\bar {x}})^{2}}}} c o v ( B 1 , B 2 ) = − σ 2 x ¯ ∑ i = 1 n ) x i − x ¯ 2 {\displaystyle cov(B_{1},B_{2})=-{{\sigma ^{2}{\bar {x}}} \over {\sum {i=1}^{n})x_{i}-{\bar {x}}^{2}}}}