35 unbiasedness summary the ols estimates of β 1 and

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35 Unbiasedness Summary The OLS estimates of β 1 and β 0 are unbiased Proof of unbiasedness depends on our 4 assumptions – if any assumption fails, then OLS is not necessarily unbiased Remember unbiasedness is a description of the estimator – in a given sample we may be “near” or “far” from the true parameter Fall 2008 under Econometrics Prof. Keunkwan Ryu
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36 Variance of the OLS Estimators Now we know that the sampling distribution of our estimate is centered around the true parameter Want to think about how spread out this distribution is Much easier to think about this variance under an additional assumption, so Assume Var( u|x ) = σ 2 (Homoskedasticity) Fall 2008 under Econometrics Prof. Keunkwan Ryu
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37 Variance of OLS (cont) Var( u|x) = E( u 2 |x )-[E( u|x )] 2 E( u | x ) = 0, so σ 2 = E( u 2 |x ) = E( u 2 ) = Var( u ) Thus σ 2 is also the unconditional variance, called the error variance σ , the square root of the error variance is called the standard deviation of the error Can say: E( y|x )= β 0 + β 1 x and Var( y | x ) = σ 2 Fall 2008 under Econometrics Prof. Keunkwan Ryu
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38 Homoskedastic Case Fall 2008 under Econometrics Prof. Keunkwan Ryu
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39 Heteroskedastic Case Fall 2008 under Econometrics Prof. Keunkwan Ryu
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40 Variance of OLS (cont) ( 29 ( 29 ( 29 ( 29 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 ˆ 1 1 1 1 1 1 ˆ β σ σ σ σ β β Var s s s d s d s u Var d s u d Var s u d s Var Var x x x i x i x i i x i i x i i x = = = = = = = + = Fall 2008 under Econometrics Prof. Keunkwan Ryu
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41 Variance of OLS Summary The larger the error variance, σ 2 , the larger the variance of the slope estimate The larger the variability in the x i , the smaller the variance of the slope estimate As a result, a larger sample size should decrease the variance of the slope estimate Problem that the error variance is unknown Fall 2008 under Econometrics Prof. Keunkwan Ryu
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42 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 Fall 2008 under Econometrics Prof. Keunkwan Ryu
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43 Error Variance Estimate (cont) ( 29 ( 29 ( 29 ( 29 ( 29 2 / ˆ 2 1 ˆ is of estimator unbiased an Then, ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 2 2 1 1 0 0 1 0 1 0 1 0 - = - = - - - - = - - + + = - - = n SSR u n u x u x x y u i i i i i i i i σ σ β β β β β β β β β β Fall 2008 under Econometrics Prof. Keunkwan Ryu
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44 Error Variance Estimate (cont) ( 29 ( 29 ( 29 ( 29 2 1 2 1 1 2 / ˆ ˆ se , ˆ of error standard the have then we for ˆ substitute we if ˆ sd that recall regression the of error Standard ˆ ˆ - = = = = x x s i x σ β β σ σ σ β
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