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Lecture 6 - Multiple Regression Goodness of Fit We can...

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1 Multiple Regression: Goodness of Fit We can decompose Var(Y) using a multiple regression y i = α + β x i + θ z i + u i Var(Y) = {( β e ) 2 Var(X) + 2 β e θ e Cov(X,Z) + ( θ e ) 2 Var(Z)} + Var(U e ) (Other terms such as Cov( α e , X) and Cov(X,U e are 0) As in simple regression, this decomposition can be set out in an Analysis of Variance (ANOVA) table The ANOVA table Source of variance Sum of squares Degrees of freedom Mean square Total SST N-1 SST/N-1 (Var Y) Explained SSE K-1 SSE/K-1 ( F A ) Residual SSR N-K SSR/N-K ( F B ) SST has N-1 degrees of freedom because it is the variance around Mean(Y) R 2 and ‘adjusted’ R 2 R 2 = SSE/SST = (SST - SSR)/SST The value of R 2 depends on the form of Y: never use R 2 to compare regressions with different dependent variables R 2 must increase as more regressors are added Use ‘adjusted R 2 ’ to compare two regressions with same dependent variable and number of observations adjusted R 2 = R 2 - {(K-1)/(N-K)}(1 - R 2 ) Alternatively use the estimated value of σ (the residual variance) s = { Σ (u e i ) 2 /(N-K)}
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