Lecture 4

Lecture 4 - Properties of OLS residuals The aim of...

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1 Properties of OLS residuals The aim of regression analysis is to partition y into systematic and nonsystematic components. A desirable estimator generates u i which are not systematic For OLS • the mean value of u e (the regression residuals) is zero if the regression contains a constant • the correlation of u e with any x variable included in the regression is zero: in simple regression Cov(u e ,x) = Cov(y - α e - β e x, x) = Cov(y,x) - β e Cov(x, x) = Cov(y,x) - {Cov(x,y)/V(x)}V(x) = 0 Analysing the variance of Y The within-sample variance of Y is Var(Y) = Var( α e + β e X+ U e ) = ( β e ) 2 Var(X) + 2 β e Cov(X,U e ) + Var(U e ) which can be decomposed into two components Var(Y) = ( β e ) 2 Var(X) + Var(U e )C o v ( X , U e ) = 0 Total Sum of Squares (SST) = Explained Sum of Squares (SSE) + Residual Sum of Squares (SSR) R 2 = ( β e ) 2 Var(X)/Var(Y) = SSE/SST R 2 = (SST - SSR)/SST = 1 - SSR/SST This decomposition works for all OLS regressions The ANOVA table Source Sum of squares (around mean) Degrees of freedom Mean square Total SST (N-1) SST/(N-1) {Var(Y)} Explained SSE {(N-1)* β 2 *Var(X)} 1S S E Residual SSR {SST - SSE} (N-2) SSR/(N-2) {s 2 }
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2 Regression and causality Correlation is symmetric
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Lecture 4 - Properties of OLS residuals The aim of...

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