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

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1 Distribution of Estimates and Multivariate Regression Lecture XXVII I. Models and Distributional Assumptions A. Conditional Normal Model 1. The conditional normal model assumes that the observed random variables are distributed 2 ~ , i i y N x Thus, i i i E y x x and the variance of i y equals 2 . The conditional normal can be expressed as 2 ~ 0, i i i i y x N Further, the i 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: 2 2 2 1 1 , , exp 2 2 n i i i y x L x Taking the log of this likelihood function yields: 2 2 2 1 1 ln ln 2 ln 2 2 2 n i i i n n L y x As discussed in Lecture XVII, this likelihood function can be concentrated in such a way so that 2 2 2 1 ˆ ln ln 2 2 1 ˆ n i i i n n L y x n
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AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture XXIX Professor Charles Moss Fall 2007 2 So that the least squares estimator are also maximum likelihood estimators if the error terms are normal. 3. Proof of the variance of can be derived from the Gauss-Markov results. a) Note from last lecture: 1 1 1 1 1 ˆ n n i i i i i i i xx n n n i i i i i i i i x x d y x S d d x d (1)
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