Econometrics-I-5 - Econometrics I Professor William Greene...

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Part 5: Regression Algebra and Fit Econometrics I Professor William Greene Stern School of Business Department of Economics
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Part 5: Regression Algebra and Fit Econometrics I Part 5 – Regression               Algebra and Fit ™    1/33
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Part 5: Regression Algebra and Fit 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’ and X’e=0  ) (This is the F.O.C. for least squares.) e¢ e   =   y¢ y  -  y’Xb   =   y¢ y  -  b¢ X¢ y         =   e¢ y  as  e¢ X  0   (or  e’y = y’e ) ™    2/33
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Part 5: Regression Algebra and Fit Minimizing ee Any other coefficient vector has a larger sum of squares.  A  quick proof:     d  = the vector, not equal to  b     u  =  y  –  Xd = y – Xb + Xb – Xd     = e  -  X ( d  -  b ).    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   ™    3/33
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Part 5: Regression Algebra and Fit Dropping a Variable An important special case.  Suppose          bX,z  = [ b ,c]               = the regression coefficients in a regression of  y  on [ X , z ]         bX    = [ d ,0]              = is the same, but computed to force the coefficient on  z            to equal 0.  This removes  z  from the regression.  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, it cannot reduce the  sum of squared residuals.   Adding a variable(s) cannot degrade the fit - that is, it cannot increase the  sum of squared residuals. ™    4/33
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Part 5: Regression Algebra and Fit Adding a Variable Reduces the Sum of Squares Theorem 3.5 on text page 38. u = the residual in the regression of y on [ X,z ] e = the residual in the regression of y on X alone, u ¢ u = ee – c 2 ( z * z *) ee where z * = MXz . ™    5/33
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Part 5: Regression Algebra and Fit The Fit of the Regression p “Variation:” In the context of the “model” we speak of covariation of a variable as movement of the variable, usually associated with (not necessarily caused by) movement of another variable.
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