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Unformatted text preview: Applied Econometrics William Greene Department of Economics Stern School of Business Applied Econometrics 5. Regression Algebra and a Fit Measure 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 ′ e = y ′ y  y’Xb = y ′ y  b ′ X ′ y = e ′ y as e ′ X = (This is the F.O.C. for least squares.) 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 Dropping a Variable 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 31. 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 NeymanPearson class of tests of the regression model. The Fit of the Regression “Variation:” In the context of the “model” we speak of variation of a variable as movement of the variable, usually associated with (not necessarily caused by) movement of another variable. Total variation = = y ′ M y . M = I – i(i’i)1 i’ = the M matrix for X = a column of ones....
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 Spring '11
 PP
 Econometrics, Least Squares, Regression Analysis, Standard Error, ........., Stern School of Business

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