On residuals from a regression of x 1 on x 2 this

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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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Fall 2008 under Econometrics Prof. Keunkwan Ryu 10 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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Fall 2008 under Econometrics Prof. Keunkwan Ryu 11 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 made being as n observatio each of can think We 2 2 2 + = - - + = i i i i i i u y y y y u y y
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 12 Goodness-of-Fit (continued) How do we think about how well our sample regression line fits our sample data? Can compute the fraction of the total sum of squares (SST) that is explained by the model, call this the R-squared of regression R 2 = SSE/SST = 1 – SSR/SST
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 13 Goodness-of-Fit (continued) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 - - - - = 2 2 2 2 2 ˆ ˆ ˆ ˆ ˆ values the and actual the between t coefficien n correlatio squared the to equal being as of think also can We y y y y y y y y R y y R i i i i i i
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 14 More about R -squared R 2 can never decrease when another independent variable is added to a regression, and usually will increase Because R 2 will usually increase with the number of independent variables, it is not a good way to compare models
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 15 3.3 The Expected Value of the OLS Estimators Assumption about unbiasedness Population model is linear in parameters: y = β 0 + β 1 x 1 + β 2 x 2 +…+ β k x k + u We can use a random sample of size n , {( x i1 , x i2 ,…, x ik , y i ): i =1, 2, …, n }, from the population model, so that the sample model is y i = β 0 + β 1 x i1 + β 2 x i2 +…+ β k x ik + u i E( u|x 1 , x 2 ,… x k ) = 0, implying that all of the explanatory variables are exogenous None of the x ’s is constant, and there are no exact linear relationships among them
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 16
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