MLE_Lecture 3 - Lecture 3 Maximum Likelihood Estimation for the Classical Normal Linear Regression Model Examination of MVU Efficiency of Estimators and

MLE_Lecture 3 - Lecture 3 Maximum Likelihood Estimation for...

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Lecture 3 Maximum Likelihood Estimation for the Classical Normal Linear Regression Model, Examination of MVU Efficiency of Estimators, and the “Super” Theorem of Maximum Likelihood Estimation ~ ~ ~ ε β + = X y ; ) , 0 ( ~ 2 ~ ~ I N σ ε ) 2 exp( * ) 2 ( 1 ) , , ( 2 ~ ~ ' 2 2 ~ 2 ~ σ ε ε πσ σ β = N X y L ) ( * ) ( 2 1 ) ln( 2 ) 2 ln( * 2 ) , , ( ln ~ ~ ' ~ ~ 2 2 ~ 2 β β σ σ π σ β X y X y N N X y L l = = = ) ( ) ( 2 1 ~ ~ ' ~ ~ 2 ~ ~ β β σ β β X y X y l + = ~ ' ' ~ ~ ' ' ~ ~ ' ' ~ ~ ' ~ ~ 2 * 2 1 β β β β β σ X X X y y X y y + = ~ ' ' ~ ~ ~ ' ~ ~ 2 2 * 2 1 β β β β β σ X X X y + = ~ ' ~ ' 2 2 2 * 2 1 β σ X X y X = ~ ' ~ ' 2 * 1 β σ X X y X F.O.C. with respect to ~ β : ~ ~ ' ~ ' 2 0 ˆ * ˆ 1 = β σ X X y X ~ ~ ' ~ ' 0 ˆ = β X X y X ~ ' ~ ' ˆ β X X y X = ;(Normal Equation of OLS) ( ) ~ ' 1 ' ~ ˆ y X X X = β + = ~ ~ ' ~ ~ 4 2 2 2 1 1 2 β β σ σ σ X y X y N l F.O.C. with respect to 2 σ : 1
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