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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, 0² In) and hence f(y X, 3,0²) = From here obtain the log-likelihood as n Π(2πσ?)-1/2 exp (2πо²)-¹/2 exp i=1 - l(y|X, 3,0²) = (n/2) In (2πo²) - (y — Xß)' (y - XB) . 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, (Yi-x₁3)² 20² (y - XB)' (y - Xß) 20² - (y - XB)' (y - XB) = -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 Σuin (3) to obtain BOLS. The GM and CR Theorems also coincide in this case, since we are applying MLE to a linear model.

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, 0² In) and hence f(y X, 3,0²) = From here obtain the log-likelihood as n Π(2πσ?)-1/2 exp (2πо²)-¹/2 exp i=1 - l(y|X, 3,0²) = (n/2) In (2πo²) - (y — Xß)' (y - XB) . 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, (Yi-x₁3)² 20² (y - XB)' (y - Xß) 20² - (y - XB)' (y - XB) = -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 Σuin (3) to obtain BOLS. The GM and CR Theorems also coincide in this case, since we are applying MLE to a linear model.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.3: Least Squares Approximation
Problem 29EQ
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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.
Transcribed Image Text: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.
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