gauss_markov_linear_2010

# gauss_markov_linear_2010 - EC 531 ECONOMETRICS by Prof...

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Unformatted text preview: EC 531 ECONOMETRICS by Prof. Burak Saltoglu Lecture 4 Topics to be discussed • Properties of OLS Estimators • GAUSS Markov Theory 27.10.2009 2 Gauss-Markov Theorem 1 1 ˆ C la im : O L S e s tim a to rs (i.e ) a re lin e a r u n b ia s e d a n d m o s t e ffic ie n t e s tim a to rs a m o n g th e o th e r lin e a r c la s s o f e s tim a to If Y = is th e tru e m o d rs . e l i i i X u β β β + + i i i i 1 2 2 i i y ˆ β = x xY x x = ∑ ∑ ∑ ∑ 1 i 1 i i 2 i ˆ ˆ β = Y β = w Y i x x ⇒ ÷ ÷ ∑ ∑ ∑ w i 1 1 ˆ β is a linear estimator of so β 27.10.2009 3 PROPERTIES OF OLS ESTIMATORS: GAUSS MARKOV THEOREM The OLS Estimators are, among the linear class of estimators, unbiased and efficient (i.e. Has the least variance). First, OLS estimators are linear Since, 1 2 ˆ where the weights which are fixed in every samples are given by i i i i i N i i w y w x w x β = Σ = ∑ 27.10.2009 4 GAUSS MAKROV: UNBIASEDNESS Claim II: OLS Estimators are unbiased: ˆ ) 1 1 E(β = β 27.10.2009 5 Gauss-Markov 1 Y = true model i i i X u β β + + i i i i 1 2 2 i i y ˆ β = x xY x x = ∑ ∑ ∑ ∑ 1 i 2 i ˆ β = Y i x x ÷ ÷ ∑ ∑ w i 1 i i ˆ β = w Y ∑ 27.10.2009 6 2 2 2 1 1 2 2 2 2 2 2 2 : 1 sin ( ) ( ) ( ) ( ) ..( ) 1 i i i i i i i i i i n n i i i i i i i note also w X ce x x X X x x X X X X X X X X X X X X x x X X X x x x = =-- +- +- ⇒- = → ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ 0 since i i note w x = = ∑ ∑ 27.10.2009 7 Gauss-Markov 1 Y = true model i i i X u β β + + i i i i 1 2 2 i i y ˆ β = x xY x x = ∑ ∑ ∑ ∑ 1 i 2 i ˆ β = Y i x x ÷ ÷ ∑ ∑ w i 1 i i ˆ β = w Y ∑ 27.10.2009 8 GAUSS MAKROV: UNBIASEDNESS Claim OLS Estimators are unbiased i.e the following equality should hold : ˆ ) 1 1 E(β = β 2 i i i x w x = ∑ 1 i i ˆ β = w Y ∑ Proof: 27.10.2009 9 Gauss-Markov...
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## This note was uploaded on 10/25/2011 for the course ECON 501 taught by Professor Zobuz during the Spring '11 term at Istanbul Technical University.

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gauss_markov_linear_2010 - EC 531 ECONOMETRICS by Prof...

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