Question

. Under the Gauss-Markov assumptions, we know that the Ordinary Least Squares (OLS) estimator βˆ is unbiased, efficient, and consistent. However, if the assumption that E[ϵ 2 i |X] = σ 2 i = σ 2 is violated while the assumption of E[ϵi , ϵj ] = 0 , ∀i ̸= j holds, that the least squares estimator is unbiased but is no longer efficient. The Generalized Least Squares (GLS) estimator, in this case, may be unbiased, consistent, and efficient relative to the least squares estimator.

Answer 1

It shall be noted that violation of classical-linear regression model assumption of E[ϵ 2 i |X] = σ 2 is called presence of heteroscedasticity.

The assumption of E[ϵi , ϵj ] = 0 , ∀i ̸= j holds, is called the assumption of no autocorrelation.

It is rightly said that when there is presence of heteroscedasticity, the OLS estimators is still unbiased but not efficient, that means, the OLS estimator in the presence of heteroscedasticity problem is not with minimum variance. However, the expected value of this OLS estimator is population estimator and hence, is unbiased.

To remove the problem of heteroscedasticity, the generalized least square (GLS) estimation technique is used. GLS is equivalent to applying ordinary least squares to a linearly transformed version of the data

This helps the OLS estimator unbiased, consistent and efficient in presence of heteroscedasticity.