Econometrics-I-5 - Applied Econometrics William Greene Department of Economics Stern School of Business Applied Econometrics 5 Regression Algebra

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Applied Econometrics William Greene Department of Economics Stern School of Business
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Applied Econometrics 5. Regression Algebra and a Fit Measure
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The Sum of Squared Residuals       b  minimizes  e e  = ( y  -  Xb ) ( y  -  Xb ).        Algebraic equivalences, at the solution         b  = ( X X ) -1 X y       e’e   =    y e  (why?   e’  =  y’  –  b’X’ )       e  =   y y  -  y’Xb   =   y y  -  b X y               =   e y  as  e 0          (This is the F.O.C. for least squares.)
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Minimizing e e Any other coefficient vector has a larger sum of squares.  A  quick proof:      d  = the vector, not  b      u  =  y  -  Xd .    Then,  u u  = ( y  -  Xd ) ( y - Xd )                = [ y  -  Xb  -  X ( d  -  b )] [ y  -  Xb  -  X ( d  -  b )]               = [ e   -  X ( d  -  b )]  [ e   -  X ( d  -  b )] Expand to find  u u  =  e e  + ( d - b ) X X ( d - b )  >    e e  
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An important special case.  Suppose [ b ,c]=the regression coefficients in a  regression of  y  on [ X , z ]and  d  is the same, but computed to force the  coefficient on  z  to be 0.  This removes  z  from the regression.  (We’ll  discuss how this is done shortly.)  So, we are comparing the results that  we get with and without the variable  z  in the equation.   Results which we  can show: Dropping a variable(s) cannot improve the fit - that is, reduce the sum of  squares. Adding a variable(s) cannot degrade the fit - that is, increase the sum of  squares. The algebraic result is on text page 34.  Where  u  = the residual in the  regression of  y  on [ X,z ] and  e  = the residual in the regression of  y  on  X   alone,          u’u  =  e e  –  c 2   ( z * z *)      e e  where  z * =  M X z . This result forms the basis of the Neyman-Pearson class of tests of the 
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This note was uploaded on 09/27/2008 for the course FM 101 taught by Professor Greece during the Spring '08 term at New York College of Podiatric Medicine.

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Econometrics-I-5 - Applied Econometrics William Greene Department of Economics Stern School of Business Applied Econometrics 5 Regression Algebra

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