Addressing Alternative Explanations: Multiple Regression17.871Did Clinton hurt Gore example Did Clinton hurt Gore in the 2000 election? Treatment is not liking Bill Clinton How would you test this?Bivariate regression of Gore thermometer on Clinton thermometerClinton thermometerDid Clinton hurt Gore example What alternative explanations would you need to address? Nonrandom selection into the treatment group (disliking Clinton) from many sources Let’s address one source: party identification How could we do this? Matching: compare Democrats who like or don’t like Clinton; do the same for Republicans and independents Multivariate regression: control for partisanship statisticallyDemocratic pictureClinton thermometerIndependent pictureClinton thermometerRepublican pictureClinton thermometerCombined data pictureClinton thermometerCombined data picture with regression: bias!Clinton thermometerCombined data picture with “true” regression lines overlaidClinton thermometerTempting yet wrong normalizationsClinton thermometerClinton thermometerSubtract the Goretherm. from theavg. Gore therm. scoreSubtract the Clintontherm. from theavg. Clinton therm. score3D RelationshipThe Linear Relationship between Three VariablesiiiiXXY,22,110Clinton thermometerGorethermometerParty IDMultivariate slope coefficients)var(),cov(ˆ- )var(),cov(ˆ vs.)var(),cov(ˆ1212111111XXXXYXXYXMMBWhen does ? Obviously, when 0)var(),cov(ˆ1212XXXMMB11ˆˆClinton effect (on Gore) in bivariate (B) regressionClinton effect (on Gore) in multivariate (M) regressionParty ID effect (on Gore) in multivariate (M) regressionBivariate estimate:Multivariate estimate:Clinton effect on Party ID in bivariate regressionThe Slope CoefficientsniiniiiniiniiiniiniiiniiniiiXXXXXXXXXXYYXXXXXXXXXXYY12,221,22,11112,221,12212,111,22,11212,111,111)())((ˆ- )())((ˆand )())((ˆ- )())((ˆX1is Clinton thermometer, X2is PID, and Y is Gore thermometerThe Slope Coefficients More Simply)var(),cov(ˆ- )var(),cov(ˆand)var(),cov(ˆ- )var(),cov(ˆ22112221212111XXXXYXXXXXYXX1is Clinton thermometer, X2is PID, and Y is Gore thermometerThe Matrix formy1y2…yn1 x1,1x2,1… xk,11 x1,2x2,2… xk,21 … … … …1 x1,nx2,n… xk,n ( )X X X y13D Linear RelationshipThe Output. reg gore clinton party3Source | SS df MS Number of obs = 1745-------------+------------------------------ F( 2, 1742) = 1048.04Model | 629261.91 2 314630.955 Prob > F = 0.0000Residual | 522964.934 1742 300.209492 R-squared = 0.5461-------------+------------------------------ Adj R-squared = 0.5456Total | 1152226.84 1744 660.68053 Root MSE = 17.327------------------------------------------------------------------------------gore | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------clinton | .5122875 .0175952 29.12 0.000 .4777776 .5467975party3 | 5.770523 .5594846 10.31 0.000 4.673191 6.867856_cons | 28.6299 1.025472 27.92 0.000 26.61862 30.64119------------------------------------------------------------------------------Interpretation of clinton effect: Holding constant party identification, a one-point increase in the Clinton feeling thermometer is associated with a .51 increase in the Gore thermometer.Separate regressions(1) (2) (3)Intercept 23.1 55.9 28.6Clinton 0.62 -- 0.51Party -- 15.7 5.8Is the Clinton effect causal? That is, should we be convinced that negative feelings about Clinton really hurt Gore? No! The regression analysis has only ruled out nonrandom selection on party ID. Nonrandom selection into the treatment could occur from Variables other than party ID, or Reverse causation, which is feelings about Gore influencing feelings about Clinton. Additionally, the regression analysis may not have entirely ruled out nonrandom selection on party ID because it may have assumed he wrong functional form. E.g., what if nonrandom selection on strong Republican/strong DemocratSummary: Why we control Address alternative explanations by removing confounding effects Improve efficiencyWhy did the Clinton Coefficient change from 0.62 to 0.51. corr gore clinton party, cov(obs=1745)| gore clinton party3-------------+---------------------------gore | 660.681clinton | 549.993 883.182party3 | 13.7008 16.905 .8735The Calculations5122.01105.06227.0182.883905.167705.5182.883993.549)var(),cov(ˆ)var(),cov(ˆ6227.0182.883993.549)var(),cov(ˆ211clintonpartyclintonclintonclintongoreclintonclintongoreMMB. corr gore clinton party,cov(obs=1745)| gore clinton party3-------------+---------------------------gore | 660.681clinton | 549.993 883.182party3 | 13.7008 16.905 .8735Accounting for total effectsMMMBMMBMMXXXXYX21211212111212111ˆˆ ˆˆ- ˆˆ)var(),cov(ˆ- )var(),cov(ˆAccounting for the total effect21211ˆˆ ˆMMB21Total effect = Direct effect + indirect effectYX1X2M2ˆM1ˆ Accounting for the total effects in the Gore thermometer exampleEffect Total Direct IndirectClinton 0.62 0.51 0.11Party 15.7 5.8 9.9Other approaches to addressing confounding effects? Experiments Difference-in-differences designs Others? Is regression the best approach to addressing confounding effects? ProblemsDrinking and Greek Life Example Why is there a correlation between living in a fraternity/sorority house and drinking? Greek organizations often emphasize social gatherings that have alcohol. The effect is being in the Greek organization itself, not the house. There’s something about the House environment itself.Dependent variable: Times Drinking in Past 30 Days. infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493weight99 494-512 using da3818.dat,clear(14138 observations read). recode timespast30 timeshs (1=0) (2=1.5) (3=4) (4=7.5) (5=14.5) (6=29.5) (7=45)(timespast30: 6571 changes made)(timeshs: 10272 changes made). replace timespast30=0 if
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