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Unformatted text preview: 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 1 Models and Distributional Assumptions 2 Multivariate Regression Models GaussMarkov Theorem Charles B. Moss () Distribution and Multivariate Regression November 16, 2010 2 / 17 Models and Distributional Assumptions Conditional Normal Model I 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 Continued I 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 ( , σ 2 ) (3) Further, the are independently and identically distributed (consistent with our BLUE proof). I Given this formulation, the likelihood function for the simple linear model can be written L ( α, β, σ 2  x ) = n Y 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 X i =1 ( y i − α − β x i ) 2 (5) Charles B. Moss () Distribution and Multivariate Regression November 16, 2010 4 / 17 Continued I 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 X 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....
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This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.
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