Lecture 29-2007 - Distribution of Estimates and...

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Distribution of Estimates and Distribution of Estimates and Multivariate Regression Multivariate Regression Lecture XXIX
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Models and Distributional Models and Distributional Assumptions Assumptions The conditional normal model assumes that the observed random variables are distributed Thus, E [ y i | x i ]= α + β x i and the variance of y i equals σ 2 . The conditional normal can be expressed as ( 29 2 , ~ σ β α i i x N y + ( 29 2 , 0 ~ ε N x y i i i i + + =
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Further, the ε i are independently and identically distributed (consistent with our BLUE proof). Given this formulation, the likelihood function for the simple linear model can be written: ( 29 ( 29 ( 29 = + - - = n i i i x y x L 1 2 2 2 2 exp 2 1 , , σ β α π
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Taking the log of this likelihood function yields: As discussed in Lecture XVII, this likelihood function can be concentrated in such a way so that ( 29 ( 29 ( 29 ( 29 = - - - - - = n i i i x y n n L 1 2 2 2 2 1 ln 2 2 ln 2 ln β α σ π ( 29 ( 29 ( 29 = - - = - - n i i i x y n n n L 1 2 2 2 1 ˆ 2 ˆ ln 2 ln
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So that the least squares estimator are also maximum likelihood estimators if the error terms are normal.
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Lecture 29-2007 - Distribution of Estimates and...

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