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I tried to prove Hausman test for two estimators efficient and inefficient one. Then I encountered with the statement for variance of two estimators saying that: Covariance between the efficient estimator and inefficient is equals to the variance of the efficient one. Thus, asymptotic variance of the difference is equal to the difference of variances of the efficient and ineffient estimator. Why is it so?

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Just a comment, I find a related result when reading Greene's textbook "Econometric Analysis".

Let us consider the GLS estimator (efficient) and the OLS estimator (inefficient) of $\beta$ in a linear regression model with heteroscedasticity. One can show that:

$\text{Cov}\left( \hat{\beta}_{GLS}\ ,\ \ \hat{\beta}_{GLS}- \hat{\beta}_{OLS}\right) = 0$

Note that $\hat{\beta}_{GLS}- \hat{\beta}_{OLS}$ is an unbiased linear estimator of 0.

Also, $\hat{\beta}_{GLS}$ is the BLUE of $\beta$ under the assumption of heteroscedasticity.

The intuition for this specific case is that one cannot improve a BLUE by adding or substracting any unbiased linear estimator of 0.

Hope this simple example could be helpful in someway.

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