Distribution is much easier to think about this

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distribution is Much easier to think about this variance under an additional assumption, so Assume Var( u|x 1 , x 2 ,…, x k ) = σ 2 (Homoskedasticity)
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 25 Variance of OLS (cont) Let x stand for ( x 1 , x 2 ,…x k ) Assuming that Var( u | x ) = σ 2 also implies that Var( y | x ) = σ 2 The 4 assumptions for unbiasedness, plus this homoskedasticity assumption are known as the Gauss-Markov assumptions
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 26 Variance of OLS (cont) ( 29 ( 29 ( 29 s ' other all on regressing from the is and where , 1 ˆ s Assumption Markov - Gauss Given the 2 2 2 2 2 x x R R x x SST R SST Var j j j ij j j j j - = - = σ β
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 27 Components of OLS Variances The error variance: a larger σ 2 implies a larger variance for the OLS estimators The total sample variation: a larger SST j implies a smaller variance for the estimators Linear relationships among the independent variables: a larger R j 2 implies a larger variance for the estimators
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 28
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 29 Misspecified Models ( 29 ( 29 ( 29 same the re ' then they ed, uncorrelat are and unless ˆ ~ Thus, ~ that so , ~ ~ ~ model ed misspecifi again the Consider 2 1 1 1 1 2 1 1 1 0 x x Var Var SST Var x y β β σ β β β < = + =
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 30 Misspecified Models (cont) While the variance of the estimator is smaller for the misspecified model, unless β 2 = 0 the misspecified model is biased As the sample size grows, the variance of each estimator shrinks to zero, making the variance difference less important
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 31 Estimating the Error Variance We don’t know what the error variance, σ 2 , is, because we don’t observe the errors, u i What we observe are the residuals, û i We can use the residuals to form an estimate of the error variance
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 32 Error Variance Estimate (cont) ( 29 ( 29 ( 29 ( 29 [ ] 2 1 2 2 2 1 ˆ ˆ thus, 1 ˆ ˆ j j j i R SST se df SSR k n u - = - - = σ β σ df = n – ( k + 1), or df = n k – 1 df (i.e. degrees of freedom) is the (number of observations) – (number of estimated parameters)
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 33 3.5The Gauss-Markov Theorem Given our 5 Gauss-Markov Assumptions it can be shown that OLS is “BLUE” Best Linear Unbiased Estimator Thus, if the assumptions hold, use OLS
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