Estimating the variance of the OLS estimators
Summary
Setup
The LRM with random sampling
yi=β1+β2xi+εii=1,…,nyi=β1+β2xi+εii=1,…,n
Estimated variance of b1b1 and b2b2
If we replace σ2σ2 with s2s2 in the OLS GM variance formulas, we get the estimated variance of b1b1 and b2b2 :
^Var(b1)=s2(1n+ˉx2∑ni=1(xi−ˉx)2)ˆVar(b1)=s2(1n+¯x2∑ni=1(xi−¯x)2)
^Var(b2)=s2∑ni=1(xi−ˉx)2ˆVar(b2)=s2∑ni=1(xi−¯x)2
Standard errors
- Under GM, the estimated variances are unbiased and consistent estimators of the true variances.
- The square root of the estimated variances are called the OLS standard errors of the regression parameters and they are denoted by SE(b1)SE(b1) and SE(b2)SE(b2) .