Addressing Alternative Explanations:Explanations: Multiple Regressiong17.871Spring 2012Did Cli t h t G lDid Clinton hurt Gore example Did Clinton hurt Gore in the 2000 election?Treatment is not liking Bill ClintonTreatment is not liking Bill Clinton How would you test this?Bivariate regression of Gore thermometer onBivariate regression of Gore thermometer on Clinton thermometerClinton thermometerDid Cli t h t G lDid Clinton hurt Gore example What alternative explanations would you need to address?Nonrandom selection into the treatment group (dislikingNonrandom 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 statistically Also called multiple regression, Ordinary Least Squares (OLS) Presentation below is intuitiveDtiitDemocratic pictureClinton thermometerId d t itIndependent pictureClinton thermometerRbli itRepublican pictureClinton thermometerCbiddtitCombined data pictureClinton thermometerCombined data picture withCombined data picture with regression: bias!Clinton thermometerCombined data picture withCombined data picture with “true” regression lines overlaidClinton thermometerTempting yet wrong pgy gnormalizationsSubtract the Goretherm. from theavg. Gore therm.Clinton thermometeravg. Gore therm. scoreSubtract the ClintonSubtract the Clintontherm. from theavg. Clinton therm. Clinton thermometerscore3D R l ti hi3D Relationship3D Linear Relationship3D Linear Relationship3D R l ti hi Cli t3D Relationship: Clinton0501003D R l ti hi t3D Relationship: partyRepIndDemThe Linear Relationship between ThreeThe Linear Relationship between Three VariablesClintonGoreXXYClinton thermometerGorethermometerParty IDiiiiXXY,22,110STATA: 12reg y x1 x2reg gore clinton party3Multivariate slope coefficientsMultivariate slope coefficientsClinton effect (on Gore) in bivariate (B)vs),cov(ˆ1YXBbivariate (B) regressionAre Gore and Party ID related?Bivariate estimate:),cov(ˆ-),cov(ˆvs.)var()(21211111XXYXXMMBivariate estimate:Multivariate estimate:)var()var(1211XXClinton effect (G)iMultivariate estimate:Are Clinton and Party ID )cov(ˆXXMMBˆˆ(on Gore) in multivariate (M) regressionyrelated?When does ? Obviously, when 0)var(),cov(ˆ1212XXXMMB11X1is Clinton thermometer, X2is PID, and Y is Gore thermometerTh Sl C ffi i tThe Slope CoefficientsnniiinniiiXXXXXXYY1,22,1121,111and))((ˆ-))((ˆnnniiniiXXXX12,11212,111)()(nniiinniiiXXXXXXXXXXYY21,22,11121,122)())((ˆ- )())((ˆiiiiXXXX12,2212,22)()(X1is Clinton thermometer, X2is PID, and Y is Gore thermometerThe Slope Coefficients MoreThe Slope Coefficients More Simplyand),cov(ˆ),cov(ˆ211XXYXand)var(-)var(1211XX)var(),cov(ˆ- )var(),cov(ˆ21122XXXXYX)var()var(22XXX1is Clinton thermometer, X2is PID, and Y is Gore thermometerTh M t i fThe Matrix formy1y21x1,1x2,1…xk,11x1,2x2,2…xk,2…y1…………1xxxyn1x1,nx2,n…xk,n()XXXy1()yTh O t tThe Output. reg gore clinton party3Source | SS df MS Number of obs = 1745-------------+------------------------------ F( 2, 1742) = 1048.04Model | 629261.91 2 314630.955 Prob > F=0.0000Model | 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 6.867856py|_cons | 28.6299 1.025472 27.92 0.000 26.61862 30.64119------------------------------------------------------------------------------Interpretation ofclintoneffect:Holding constant party identificationa one-Interpretation of clintoneffect: Holding constant party identification, a one-point increase in the Clinton feeling thermometer is associated with a .51 increase in the Gore thermometer.St iSeparate regressions(1) (2) (3)Intercept 23.1 55.9 28.6pClinton 0.62 -- 0.51Party--15.75.8Party15.75.8Is the Clinton effect causal?Is the Clinton effect causal? That is, should we be convinced that negative feelings about Clinton really hurt Gore?feelings about Clinton really hurt Gore? No! The regression analysis has only ruled out linear dlti tIDnonrandom selection on party ID. Nonrandom selection into the treatment could occur fromV i bl th th t IDVariables other than party ID, or Reverse causation, that is, feelings about Gore influencing feelings about Clinton.Additionally the regression analysis may not haveAdditionally, the regression analysis may not have entirely ruled out nonrandom selection even on party ID because it may have assumed the wrong functional form. E.g., what if nonrandom selection on strong Republican/strong Democrat, but not on weak partisansOther approaches to addressingOther approaches to addressing confounding effects? ExperimentsDifference-in-differences designsDifferenceindifferences designs Others?Summary: Why we controlSummary: Why we control Address alternative explanations by removing fdi fftconfounding effects Improve efficiencyWhy did the Clinton CoefficientWhy did the Clinton Coefficient change from 0.62 to 0.51. corr gore clinton party, cov(obs=1745)| gore clinton party3-------------+---------------------------+gore | 660.681clinton | 549.993 883.182t3| 13 7008 16 905 8735party3| 13.7008 16.905 .8735Th C l l tiThe Calculations993549)cov(ˆclintongoreB6227.0182.883993.549)var(),cov(ˆ1clintonclintongoreB)(),cov(ˆ)(),cov(ˆ21li tpartyclintonli tclintongoreMM905.167705.5993.549)var()var(21clintonclinton1105.06227.0182.883182.883. corr gore clinton party,cov(obs=1745)| gore clinton
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