Consider the linear regression model y = X3 + u with iid random variables and assume u ~ N(0,0²In). Conditional on X, show that the MLE estimator of 3 coincides with the OLS estimator of 3. Compare the Gauss-Markov Theorem to the Cramer-Rao Theorem in this case. Conditional on X, y ~ N(XB, o² In) and hence f(y|X, 3,0²) = = From here obtain the log-likelihood as n (2π0²)-¹/² exp i=1 (2πо²)-¹/² exp (Yi - X₁3)² 20² (y - XB)' (y - X³) 20² l(y|X, 3,0²) = − (n/2) ln(2ño²) – (y — Xß)' (y – Xß) 20² (2) Since the only term in (2) that depends on ß is the numerator of the second RHS term, maximizing l(y X, B, 0²) w.r.t 3 is equivalent to maximizing - (y - XB)' (y - XB) w.r.t. ß in that both yield the same argmax. At the same time, - (y - XB)' (y - X³) = -u'u n -Σu² i=1 (3) which is the negative of the sum of square residuals criterion function of OLS. Hence maximizing (y|X, B, 0²) in (2) to obtain BML yields the same result as minimizing ₁ u²in (3) to obtain BOLS. The GM and CR Theorems also coincide in this case, since we are applying MLE to a linear model.