04ProofOfTheGauss-MarkovTheorem(1)

04ProofOfTheGauss-Ma - Proof of the Gauss-Markov Theorem Suppose dy is any linear unbiased estimator other than the OLS ^ estimator C ^ Need to

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Proof of the Gauss-Markov Theorem Suppose dy is any linear unbiased estimator other than the OLS ^ estimator C. ^ Need to show Var(dy) > Var(C). ^ ^ Can relate the two Var by writing Var(dy) = Var(dy - C + C) ^ ^ Var(dy) = Var(dy - C + C) ^ ^ ^ ^ = Var(dy - C) + Var(C) + 2 Cov(dy - C, C). Need to do two things: ^ ^ 1) show that Var(dy - C) > 0 unless dy = C ^ ^ 2) show that Cov(dy - C, C) = 0 Proof of the Gauss-Markov Theorem Gauss-Markov Th'm: ^ The OLS estimator, C, is the unique BLUE of C in GM model: y = X + , N(0, 2 I) ^ Need to show Var(C) is strictly less than the variance of any other linear unbiased estimator of C for all IRp and 2 IR+ . Outline of the proof: Consider some other linear unbiased estimator, dy. ^ ^ Write Var(dy) as Var[(dy - C) + C] ^ ^ ^ ^ Show Cov(dy - C, C) = 0 and Var(dy - C) > 0 unless dy = C ^ ^ unless dy = C. Hence, Var(dy) > Var(C) Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 1/4 Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 2/4 GM Proof: item 1 GM proof: item 2 ^ ^ Need to show Cov(dy - C, C) = 0. ^ ^ Cov(dy - C, C) = Cov(dy - y, y) = Cov[(d - )y, y] = (d - )Var(y) = 2 (d - ) = 2 (d - )X[(X X)- ] C The key is (d - )X in the middle of the above equ. Remember any linear unbiased estimator ky of C satisfies E(ky) = C IRp kX = C IRp kX = C. both dy and y are linear unbiased est. of C, so (d - )X = dX - X = C-C=0 ^ ^ Hence, Cov(dy - C, C) = 0. Statistics 511 3/4 Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 4/4 ^ and (X X)- X y are both solutions to the normal equations Because C is estimable, ^ C is the same for all solutions to the normal equations. ^ Thus, C = y where C(X X)- X , and = Var[(d - )y] = (d - )Var(y)(d - ) = (d - ) I(d - ) = 2 ||d - ||2 > 0. unless d = 2 ^ Var(dy - C) = Var(dy - y) Copyright c 2011 Dept. of Statistics (Iowa State University) ...
View Full Document

This note was uploaded on 02/11/2012 for the course STAT 511 taught by Professor Staff during the Spring '08 term at Iowa State.

Ask a homework question - tutors are online