LinearRegression2

Or in other words what is var b and var b 0 1 to

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Unformatted text preview: er words, what is Var (b ) and Var (b ) ? 0 1 To answer this question, we need an additional assumption:2 Homoskedasticity: Assume Var (u | x ) = s 2 [SLR.5] VER. 9/25/2012. © P. KOLM 12 What Does Homoskedasticity Mean? VER. 9/25/2012. © P. KOLM 13 What Does Heteroskedasticity Mean? VER. 9/25/2012. © P. KOLM 14 Sampling Variance of the OLS Estimators (2/2) It can be shown that: ˆ Var (b1 ) = s2 n å (x i =1 2 ˆ Var (b0 ) = sn i -1 - x )2 n åx i =1 n 2 i å (x i - x )2 s2 = SSTx 2 sn = -1 n åx i =1 2 i SSTx i =1 VER. 9/25/2012. © P. KOLM 15 n ˆ Proof: Recall: b1 = b1 + å (x i =1 i - x ) ui SSTx Therefore, n æ ö ÷ çb + 1 ˆ ) = Var ç Var (b1 å (x i - x )ui ÷ ÷ ç 1 SST ÷ ç è ø x i =1 æn ö 1 ç (x - x )u ÷ ÷ = Var çå i i÷ ç i =1 ÷ SSTx2 è ø = = = = n 1 Var ((x i - x )ui ) 2å SSTx i =1 n 1 å (x i - x )2Var (ui ) SSTx2 i =1 s2 n å (x i - x )2 SSTx2 i =1 s2 s2 SSTx = 2 SSTx SSTx ˆ Var (b0 ) follows in a similar way (For you: Do the calculation!) VER. 9/25/2012. © P. KOLM 16 s2 ˆ Var (b1 ) = SSTx 2 ˆ Var (b0 ) = sn -1 n åx i =1 2 i SSTx Remarks3 Let’s make sure the following make sense: The larger the error variance, s2 , the larger the variance of the intercept and slope estimates The larger the variability in the x i , the smaller the variance of the slope estimate A larger sample size typically decreases the variance of the intercept and slope estimates...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.

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