3133 part 5 regression algebra and fit aspects of

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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  0 .        What does  m  =  0  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
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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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