02_09_2016 - Stat 151A Notes Lecture 7 Jeff Weng Why OLS is...

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Stat 151A Notes Lecture 7 (02/9/16) Jeff Weng Why OLS is BLUE (Proof + Interpretation) Gauss-Markov theorem states that in a linear regression model in which errors have expec- tation zero and are uncorrelated and have equal variance, the best linear unbiased estimator (BLUE) of the coefficients is given by OLS (best means lowest variance of estimate). Assumptions: E ( i ) = 0 V ar ( i ) = σ 2 < Cov ( i , j ) = 0 , i 6 = j Model: y = + , ( y, R n , β R k , X R n × k ) X is not random, but is observable. β is not random and not observable. is random, y is random. BLUE stands for Best Linear Unbiased Estimator. Show that OLS is BLUE: E ( ˆ β ) = β which holds for OLS. ˆ β = ( X T X ) - 1 X T y, ˆ β is not linear in x, but in y. Also y is random variable so ˆ β and thus ˆ β is linear in Y. Let ˜ B = Cy, C = ( X T X ) - 1 X T + D, where D is k × n non-zero matrix. E [ Cy ] = E [((( X T X ) - 1 X T + D )( + )] = (( X T X ) - 1 X T + D ) + (( X ( X T ) - 1 X T + D ) E [ ] = β + DXβ = 0 V ar [ ˜ B ] = V ar [ Cy ] = CV ar [ y ] C T = σ 2 CC T = σ 2 (( X T X ) - 1 + D )( X ( X T X ) - 1 + D T ) = σ 2 (( X T X ) - 1 X T X ( X T X ) - 1 ) + ( X T X ) - 1 X T D T + DX ( X
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