Lecture 5 Guass-Markov Theorem

Lecture 5 Guass-Markov Theorem - Economics 326 Methods of...

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Unformatted text preview: Economics 326 Methods of Empirical Research in Economics Lecture 5: Gauss-Markov Theorem Vadim Marmer University of British Columbia January 27, 2011 There are many alternatives estimators I The OLS estimator is not the only estimator we can construct. There are alternative estimators with some desirable properties. I Example: Using only the &rst two observations, suppose that X 2 6 = X 1 . ˜ β = Y 2 & Y 1 X 2 & X 1 . I ˜ β is linear: ˜ β = c 1 Y 1 + c 2 Y 2 , where c 1 = & 1 X 2 & X 1 and c 2 = 1 X 2 & X 1 . 1/13 Unbiasedness of ˜ β I If Y i = α + β X i + U i and E ( U i j X 1 , . . . , X n ) = , then ˜ β is unbiased: ˜ β = Y 2 & Y 1 X 2 & X 1 = ( α + β X 2 + U 2 ) & ( α + β X 1 + U 1 ) X 2 & X 1 = β ( X 2 & X 1 ) X 2 & X 1 + U 2 & U 1 X 2 & X 1 = β + U 2 & U 1 X 2 & X 1 , and E & ˜ β j X 1 , X 2 ¡ = β + E ¢ U 2 & U 1 X 2 & X 1 j X 1 , X 2 £ = β + E ( U 2 j X 1 , X 2 ) & E ( U 1 j X 1 , X 2 ) X 2 & X 1 = β . 2/13 An optimality criterion I Among all linear and unbiased estimators, an estimator with the smallest variance is called the Best Linear Unbiased Estimator(BLUE)....
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This note was uploaded on 03/24/2012 for the course ECON 326 taught by Professor Whisler during the Spring '10 term at UBC.

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Lecture 5 Guass-Markov Theorem - Economics 326 Methods of...

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