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

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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 Gauss-Markov 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 ( + x i , 2 ) (1) Thus, E [ y i | x i ] = + x i (2) Charles B. Moss () Distribution and Multivariate Regression November 16, 2010 3 / 17 Continued I and the variance of y i equals 2 . The conditional normal can be expressed as y i = + x i + i i N ( , 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 ( , , 2 | x ) = n Y i =1 1 2 exp " ( y i ( + x i )) 2 2 2 # (4) Taking the log of this likelihood function yields ln ( L ) = n 2 ln (2 ) n 2 ln ( 2 ) 1 2 2 n X i =1 ( y i 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 ( 2 ) n 2 2 = 1 n n X i =1 ( y i 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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Slides29-2010 - Distribution of Estimates and Multivariate...

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