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+ˉx2ni=1(xiˉx)2)ˆVar(b1)=s2(1n+¯x2ni=1(xi¯x)2)

^Var(b2)=s2ni=1(xiˉx)2ˆVar(b2)=s2ni=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) .