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ch03 - Multiple Regression Analysis y = 0 1x1 2x2 kxk u 1...

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Economics 20 - Prof. Anderson 1 Multiple Regression Analysis y = β 0 + β 1 x 1 + β 2 x 2 + . . . β k x k + u 1. Estimation
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Economics 20 - Prof. Anderson 2 Parallels with Simple Regression β 0 is still the intercept β 1 to β k all called slope parameters u is still the error term (or disturbance) Still need to make a zero conditional mean assumption, so now assume that E( u|x 1 ,x 2 , …,x k ) = 0 Still minimizing the sum of squared residuals, so have k+1 first order conditions
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Economics 20 - Prof. Anderson 3 Interpreting Multiple Regression tion interpreta a has each is that , ˆ ˆ that implies fixed ,..., holding so , ˆ ... ˆ ˆ ˆ so , ˆ ... ˆ ˆ ˆ ˆ 1 1 2 2 2 1 1 2 2 1 1 0 ribus ceteris pa x y x x x x x y x x x y k k k k k β β β β β β β β β = + + + = + + + + =
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Economics 20 - Prof. Anderson 4 A “Partialling Out” Interpretation ( 29 2 2 0 1 1 2 1 1 1 2 2 1 1 0 ˆ ˆ ˆ ˆ regression estimated the from residuals the are ˆ where , ˆ ˆ ˆ then , ˆ ˆ ˆ ˆ i.e. , 2 where case he Consider t x x r r y r x x y k i i i i γ γ β β β β + = = + + = =
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Economics 20 - Prof. Anderson 5 “Partialling Out” continued Previous equation implies that regressing y on x 1 and x 2 gives same effect of x 1 as regressing y on residuals from a regression of x 1 on x 2 This means only the part of x i1 that is uncorrelated with x i2 are being related to y i so we’re estimating the effect of x 1 on y after x 2 has been “partialled out”
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Economics 20 - Prof. Anderson 6 Simple vs Multiple Reg Estimate sample in the ed uncorrelat are and OR ) of effect partial no (i.e. 0 ˆ : unless ˆ ~ Generally, ˆ ˆ ˆ ˆ regression multiple with the ~ ~ ~ regression simple the Compare 2 1 2 2 1 1 2 2 1 1 0 1 1 0 x x x x x y x y = + + = + = β β β β β β β β
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Economics 20 - Prof. Anderson 7 Goodness-of-Fit ( 29 ( 29 SSR SSE SST Then (SSR) squares of sum residual the is ˆ (SSE) squares of sum explained the is ˆ (SST) squares of sum total the is : following the define then We ˆ ˆ part, d unexplaine an and part, explained an of up
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