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Slides29-2010 - Distribution of Estimates and Multivariate...

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Distribution of Estimates and Multivariate Regression: Lecture XXIX Charles B. Moss November 16, 2010 Charles B. Moss () Distribution and Multivariate Regression November 16, 2010 1 / 17
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1 Models and Distributional Assumptions 2 Multivariate Regression Models Gauss-Markov Theorem Charles B. Moss () Distribution and Multivariate Regression November 16, 2010 2 / 17
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Models and Distributional Assumptions Conditional Normal Model The conditional normal model assumes that the observed random variables are distributed y i N ( α + β x i , σ 2 ) (1) Thus, E [ y i | x i ] = α + β x i (2) Charles B. Moss () Distribution and Multivariate Regression November 16, 2010 3 / 17
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Continued and the variance of y i equals σ 2 . The conditional normal can be expressed as y i = α + β x i + i i N ( 0 , σ 2 ) (3) Further, the are independently and identically distributed (consistent with our BLUE proof). 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 + β x i )) 2 2 σ 2 (4) Taking the log of this likelihood function yields ln ( L ) = n 2 ln (2 π ) n 2 ln ( σ 2 ) 1 2 σ 2 n i =1 ( y i α β x i ) 2 (5) Charles B. Moss () Distribution and Multivariate Regression November 16, 2010 4 / 17
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Continued 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 β x i ) 2 (6) So that the least squares estimator are also maximum likelihood estimators if the error terms are normal. Proof of the variance of β can be derived from the Gauss-Markov results.
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