321w4p2 - Explaining Differences in Y We can use regression...

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Explaining Differences in Y We can use regression analysis to explain why some observations have high/low values of Y n SST = ∑ i=1 (Y i -Y) 2 = Sum of Total Deviations Sq. n SSE = ∑ i=1 (Y i -Y) 2 = Sum of Explained Dev's Sq. n SSR = ∑ i=1 u i 2 = Sum of Unexplained Dev's Sq. ^
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SST=SSE+SSR Proof: n n SST = ∑ i=1 (Y i -Y) 2 = ∑ i=1 ((Y i -Y i ) + (Y i -Y)) 2 n = ∑ i=1 ((u i ) + (Y i -Y)) 2 n n n = ∑ i=1 (u i ) 2 + ∑ i=1 2u (Y i -Y) + ∑ i=1 (Y i -Y) 2 = SSR + 0 + SSE n How do we know that i=1 2u (Y i -Y)=0? Because we have already claimed that the sample covariance between the residuals and fitted values is zero. ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
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Sampling Distribution of OLS Estimators: 1. OLS Estimators are unbiased, E[ β 1 ]= β 1 Proof: n n i=1 (X i -X)(Y i -Y) = ∑ i=1 (X i Y i -X i Y -XY i +XY ) n = ∑ i=1 [(X i -X)Y i – X i Y +XY ] n n n = ∑ i=1 (X i -X)Y i – ∑ i=1 X i Y +∑ i=1 XY n = ∑ i=1 (X i -X)Y i – nXY + nXY ^ numerator:
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so we can write our OLS estimate of β 1 as: n i=1 (X i -X)Y i β 1 = n i=1 (X i -X) 2 n i=1 (X i -X)( β 0 + β 1 X i +u i ) β 1 = n i=1 (X i -X) 2 n n n n i=1 (X i -X)(
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