Lecture29-2010 - Distribution of Estimates and Multivariate...

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Distribution of Estimates and Multivariate Regression: Lecture XXIX Charles B. Moss November 16, 2010 I. Models and Distributional Assumptions A. Conditional Normal Model 1. The conditional normal model assumes that the observed ran- dom variables are distributed y i N α + βx i , σ 2 (1) Thus, E [ y i | x i ] = α + βx i (2) and the variance of y i equals σ 2 . The conditional normal can be expressed as y i = α + βx i + i i N 0 , σ 2 (3) Further, the are independently and identically distributed (consistent with our BLUE proof). 2. Given this formulation, the likelihood function for the simple linear model can be written L α, β, σ 2 | x = n i =1 1 2 πσ exp ( y i ( α + βx i )) 2 2 σ 2 (4) Taking the log of this likelihood function yields 1
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XXIX Fall 2010 ln ( L ) = n 2 ln (2 π ) n 2 ln σ 2 1 2 σ 2 n i =1 ( y i α βx i ) 2 (5) As discussed in Lecture XVII, this likelihood function can be concentrated in such a way so that ln ( L ) ∝ − n 2 ln ˆ σ 2 n 2 ˆ σ 2 = 1 n n i =1 ( y i α βx i ) 2 (6) So that the least squares estimator are also maximum likeli- hood estimators if the error terms are normal. 3. Proof of the variance of β can be derived from the Gauss- Markov results. a) Note from last lecture ˆ β = n i =1 d i y i = n i =1 ( x i ¯ x ) S xx ( α + βx i + i ) = n i =1 d i α + n i =1 d i βx i + n i =1 d i i (7) b) Under our standard assumptions about the error term, we have E n i =1 d i i = n i =1 d i E ( i ) = 0 (8)
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