Consider the system y 1 1 x 1 , y 2 2 2 The disturbances are freely correlated Prove that GLS applied to the system leads to the OLS estimates of 1 and 2 but to a mixture of the least squares slopes in the regressions of y 1 and y 2 on x as the estimator of What is the mixture To simplify the algebra, assume (with no loss of generality) that x 0

Question: Consider the system y 1 = 1 + x + 1 , y 2 = 2 + 2 . The disturbances

Consider the system

y₁ = α₁ + βx + ε₁,
y₂ = α₂ + ε₂.

The disturbances are freely correlated. Prove that GLS applied to the system leads to the OLS estimates of α₁ and α₂ but to a mixture of the least squares slopes in the regressions of y₁ and y₂ on x as the estimator of β.What is the mixture? To simplify the algebra, assume (with no loss of generality) that x̅ = 0.

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