Weighted least squares estimator

[Kanyarabi]

weighted least squares (WLS) estimator: If the errors are heteroskedastic, then OLS is no longer BLUE. If the heteroskedasticity is known (i.e., if the conditional variance of given is known up to a constant factor of proportionality) then an alternate estimator exists with a smaller variance than the OLS estimator. This method, weighted least squares (WLS), weights the i th observation by the inverse of the square root of the conditional variance of given . Because of this weighting, the errors in this weighted regression are homoskedastic, so OLS, when applied to the weighted data, is BLUE.

This is what i read in your notes and also the Miller textbook. As it says if the error terms are heteroskedastic then there are other estimator apart of OLS estimator, whereas in the last line it says the errors are homeskedastic

 

[David Harper CFA FRM]

@Kanyarabi OLS assumes homoskedastic (constant variance) error terms. Stock & Watson (not Miller) is saying that when the errors are hetero, the WLS procedure produces better regression coefficients (actually, in the strict sense, it's not a messy heteroskedasticity but a variance that increases as a function of the independent variable, so it's an increasing but conveniently well-behaved variance). What I am calling the WLS procedure is much like the OLS (that minimizes the residual sum of squares, RSS) except that it minimizes weighted RSS. So we can perform the WLS can get better estimates. Additionally, we can then weight the data (in a consistent model) and then perform OLS on the weighted data such that these estimators (i.e., OLS estimators on weighted data) are BLUE because these errors will be homeskedastic. Thanks,

 

[Kanyarabi]

Thanks David. I feel my concepts are not very clear here which is why i am unable to connect the dots. According to my understanding errors are nothing but the difference between actual (Y) and predicted (Y^) independent variable.
Homosked - is when errors have a constant variance so are we saying this difference of actual and predicted Y is constant or is it E(ui|X)=0
Thank you.