General formulation for linear restrictions minimize

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General formulation for linear restrictions: Minimize the sum of squares, ee, subject to the linear constraint Rb = q . ™    26/33
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Part 5: Regression Algebra and Fit Restricted Least Squares 1 2 3 i 1 i1 2 i2 3 i Force a coefficient to equal In practice, restrictions can usually be imposed by solving them out. 1. Drop the variable from the equation Problem: Minimize for , , (y x zero x x . β β β n 2 3 3 i 1 n 2 1 2 i 1 i1 2 i2 i 1 1 2 3 3 1 2 1 2 i 1 i1 Adding up restri ) subject to 0 Solution: Minimize for , (y x x ) 2. Impose + + = 1. Strategy: =1 . Solution: Minimize for , ( ct x . y ion = = β = β β β β β β -β -β β β - n 2 2 i2 1 2 i3 i 1 n 2 i i3 1 i1 i3 2 i2 i3 i 1 3 2 1 2 3 i 1 i1 2 i2 3 x (1 )x ) = [(y x ) (x x ) (x x )] 3. Impose Minimize for , , (y x x Equality restriction. = = β - -β -β - - - β = β β β β n 2 i3 3 2 i 1 n 2 1 2 i 1 i1 2 i2 i3 i 1 x ) subject to Solution: Minimize for , [y x (x x )] In each case, least squares using transformations of the data. = = β = β β β + ™    27/33
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Part 5: Regression Algebra and Fit Restricted Least Squares Solution p General Approach: Programming Problem Minimize for L = ( y - Xb ) ¢ ( y  -  Xb subject to      Rb  =  q Each row of  R  is the K coefficients in a restriction. There are J restrictions:  J rows p b 3 = 0:     R  = [0,0,1,0,…]  q  = (0). p b 2 =  b 3:    R  = [0,1,-1,0,…] q  = (0) p b 2 = 0,  b 3 = 0:   R  = 0,1,0,0,… q  =  0                                0,0,1,0,…       0 ™    28/33
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Part 5: Regression Algebra and Fit Solution Strategy p Quadratic program: Minimize quadratic criterion  subject to linear restrictions p All restrictions are binding p Solve using Lagrangean formulation p Minimize over ( b , l           L* = ( y  -  Xb ) ¢ ( y  -  Xb ) + 2 l ¢ ( Rb -q ) (The 2 is for convenience – see below.) ™    29/33
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Part 5: Regression Algebra and Fit Restricted LS Solution 1 Necessary Conditions L* 2 ( ) 2 L* 2( ) Divide everything by 2. Collect in a matrix form ˆ or = Solution = Does no - = - - + = = β - =  = θ ÷ ÷  X y X R 0 R q 0 X X R X y A w. A w R 0 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
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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.
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