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Unformatted text preview: Introduction to Econometrics 61 Technical Notes Introduction to Econometrics Lecture 6: Multiple Regression  Goodness of Fit Measuring the Goodness of Fit  R 2 For a multiple regression measures of goodness of fit based on the correlation coefficient between any individual regressor Z and Y Corr(Z,Y) = Cov(Z,Y)/ (Var(Z)Var(Y)) are not very useful. However we can still decompose Var(Y) into explained and unexplained components 1 . In the two variable case Y = e + e X + e Z + U e Var(Y) = Var( e + e X + e Z + U e ) = ( e ) 2 Var(X) + 2 e e Cov(X,Z) + 2 e Cov(X,U e ) + ( e ) 2 Var(Z) + 2 e Cov(Z,U e ) + Var(U e ) since e is a constant. In addition, Cov(X,U e ) and Cov(Z,U e ) are both zero (this was shown in Lecture 5), so this expression reduces to Var(Y) = {( e ) 2 Var(X) + 2 e e Cov(X,Z) + ( e ) 2 Var(Z)} + Var(U e ) The first three terms (in {}) are the Explained Sum of Squares SSE, the last the Residual Sum of Squares SSR (each divided by N1) 2 . So R 2 = (Var(Y)  Var(U e ))/Var(Y) = (SST  SSR)/SST = 1  SSR/SST This decomposition is exactly the same as in the case of simple regression: similarly the AN alysis O f VA riance table (set out in Lecture 4, repeated below) is equally useful for a multivariate regression. The value of R 2 depends on the precise form of Y (since Var(Y) is the variance around mean (Y)), so there is a vital rule to remember Never use R 2 to compare the fit of regressions with different dependent variables In addition the statistical significance associated with the value of R 2 is a function of the number of observations and the number of explanatory variables, so a second rule is Do not use R 2 (even informally) to compare regressions with a different number of observations R 2 always increases if more regressors are added, so if you want to compare two regressions with the same dependent variable you should use adjusted R 2 (Rbar squared) instead. Adjusted R 2 is defined as adjusted R 2 = R 2 {(K1)/(NK)}(1  R 2 ) so it can either increase or decrease if more regressors are added. Its often better to compare two regressions using the square root of the estimated residual variance s = ( (u e i ) 2 /(NK)) 1 The logic behind this decomposition for simple regression was discussed in Lecture 4. 2 Remember the warning (in Lecture 4) about the possible alternative meanings for the abbreviations SSE and SSR. Introduction to Econometrics 62 Technical Notes the standard error of estimate . This is because some transformations of the dependent variable affect R 2 , but leave s unchanged, so that a wider range of comparisons is possible using s. s has the same units of measurement as the dependent variable: if the dependent variable is a logarithm s has a natural interpretation as an average proportional error....
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This note was uploaded on 03/07/2012 for the course ECON 201 taught by Professor Cowell during the Spring '10 term at LSE.
 Spring '10
 Cowell
 Economics, Econometrics

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