Consider the simple regression with only a constant yi = + ui for i = 1, 2, . . ., n; where the uis

Consider the simple regression with only a constant yi = α + ui for i = 1, 2, . . ., n; where the ui’s are independent with mean zero and var(ui) = σ21 for i = 1, 2, . . . , n1; and var(ui) = σ22 for i = n1 + 1, . . . , n1 + n2 with n = n1 + n2.

(a) Derive the OLS estimator of α along with its mean and variance.

(b) Derive the GLS estimator of α along with its mean and variance.

(c) Obtain the relative efficiency of OLS with respect to GLS. Compute their relative efficiency for various values of σ22

/σ21

= 0.2, 0.4, 0.6, 0.8, 1, 1.25, 1.33, 2.5, 5; and n1/n = 0.2, 0.3, 0.4, . . . , 0.8. Plot this relative efficiency.

(d) Assume that ui is N(0, σ21

) for i = 1, 2, . . . , n1; and N(0, σ21

) for i = n1 +1, . . . , n1 +n2; with ui’s being independent. What is the maximum likelihood estimator of α, σ21 and σ22

?

(e) Derive the LR test for testing H0; σ21

= σ22 in part (d).