Econometrics-I-5

∂ ′ ′ = = ∂ ∂ = β = ∂ ′ ′ ′ = θ ?

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Unformatted text preview: - ∂ ′ ′ = -- + = ∂ ∂ = β - = ∂ ′ ′ ′ = θ ÷ ÷ X y X R R q X X R X y A w. A w R q β λ β λ β θ λ t rely on full rank of . Relies on column rank of = K J. + X A &#152;&#152;&#152;&#152;™ ™ 30/33 Part 5: Regression Algebra and Fit Restricted Least Squares 1 1 1 1 If has full rank, there is a partitioned solution for * and * = - ( ) [ ( ) ]( ) * [ ( ) ]( ) where the simple least squares coefficients, = ( ) . There are cas---- β λ ′ ′ ′ ′- ′ ′ =- ′ ′ = X β* b X X R R X X R Rb q R X X R Rb q b b X X X y λ 1 2 1 2 3 4 1 2 3 4 1 2 3 4 es in which does not have full rank. E.g., = [1, , , , , , ] where , , , are a complete set of dummy variables with coefficients a ,a ,a ,a . Unrestricted cannot be computed. Restri X X x x d d d d d d d d b 1 2 3 4 cted LS with a +a +a +a = 0 can be computed. &#152;&#152;&#152;&#152; ™ 31/33 Part 5: Regression Algebra and Fit Aspects of Restricted LS 1. b * = b - Cm where m = the “discrepancy vector” Rb - q . Note what happens if m = . What does m = mean? 2. l = [ R ( X¢ X )-1 R ¢ ]-1( Rb - q ) = [ R ( X¢ X )-1 R ¢ ]- 1 m. When does l = 0. What does this mean? 3. Combining results: b * = b - ( X¢ X )-1 R ¢ l . How could b * = b ? &#152;&#152;&#152;&#152; &#152;™ 32/33 Part 5: Regression Algebra and Fit 1 Restrictions and the Criterion Function Assume full rank case. (The usual case.) = ( ) uniquely minimizes ( - ) (y- ) = . ( - ) ( - ) < ( - *) ( - b*) for any * . Imposing restri- ′ ′ ′ ′ β ′ ′ ≠ X b X X X y y X X y Xb y Xb y Xb y X b b β ε ε 2 2 ctions cannot improve the criterion value. It follows that R * < R . Restrictions must degrade the fit. &#152;&#152;&#152;&#152; &#152; 33/33...
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∂ ′ ′ = = ∂ ∂ = β = ∂ ′ ′ ′ = θ X y...

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