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Unformatted text preview: Part 5: Regression Algebra and Fit Linear Least Squares Subject to Restrictions Restrictions : Theory imposes certain restrictions on parameters. Some common applications Dropping variables from the equation = certain coefficients in b forced to equal 0. (Probably the most common testing situation. “Is a certain variable significant?”) • Adding up conditions: Sums of certain coefficients must equal fixed values. Adding up conditions in demand systems. Constant returns to scale in production functions. • Equality restrictions: Certain coefficients must equal other coefficients. Using real vs. nominal variables in equations. General formulation for linear restrictions: Minimize the sum of squares, ee, subject to the linear constraint Rb = q . ˜˜˜™ ™ 26/33 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 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 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 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 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 β β + ∑ ∑ ˜˜˜˜™ ™ 27/33 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 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 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...
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 Fall '10
 H.Bierens
 Econometrics, Least Squares, Regression Analysis, Stern School of Business, Loglinear

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