LinearRegression2

9252012 p kolm 9 proof recall n b1 n x i 1 i

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Unformatted text preview: P. KOLM 9 Proof: Recall n ˆ b1 = n å (x i =1 i - x ) yi SSTx where SSTx º å (x i - x ) ( SSTx = “total sum of squares of x”). Now, note that 2 i =1 n å (x i =1 n i - x ) yi =å (x i - x )(b0 + b1x i + ui ) i =1 n n n = å (x i - x )b0 + å (x i - x ) b1x i + å (x i - x ) ui i =1 i =1 n i =1 n i =1 n i =1 = b0 å (x i - x ) + b1 å (x i - x ) x i + å (x i - x ) ui i =1 Since n å (x n i =1 å (x i =1 i - x ) = 0, n i - x ) x i = å (x i - x ) 2 i =1 ˆ we can rewrite b1 as VER. 9/25/2012. © P. KOLM 10 n ˆ b1 = b1SSTx + å (x i - x ) ui i =1 n = b1 + SSTx å (x i =1 i - x ) ui SSTx Therefore, ˆ E (b1 ) = b1 + 1 SSTx n å (x i =1 i - x )E (ui ) = b1 ˆ The unbiasedness of b0 follows from ˆ ˆ E (b0 ) = E (y - b1x ) ˆ = E (b0 + b1x - b1x ) ˆx) = b + E (b x - b 0 1 1 = b0 We are done. VER. 9/25/2012. © P. KOLM 11 Sampling Variance of the OLS Estimators (1/2) Now we know that the sampling distribution of our estimator is centered around the true regression parameters, we ask: How spread out is the sample distribution? ˆ ˆ Or, in oth...
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