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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 ˜˜˜˜™ ™ 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. ˜˜˜˜ ™ 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 ? ˜˜˜˜ ˜™ 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. ˜˜˜˜ ˜ 33/33...
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 Fall '10
 H.Bierens
 Econometrics, Least Squares, Regression Analysis, Stern School of Business, Loglinear

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