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 statistically Also called multiple regression, Ordinary Least Squares (OLS) Presentation below is intuitiveDemocratic 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 Relationship3D Linear RelationshipThe Linear Relationship between Three VariablesiiiiXXYεβββ+++=,22,110Clinton thermometerGorethermometerParty IDSTATA: reg y x1 x2reg gore clinton party3Multivariate slope coefficients)var(),cov(ˆ- )var(),cov(ˆ vs.)var(),cov(ˆ1212111111XXXXYXXYXMMBβββ==When does ? Obviously, when 0)var(),cov(ˆ1212=XXXMβMB11ˆˆββ=Clinton 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 regressionX1is Clinton thermometer, X2is PID, and Y is Gore thermometerThe Slope Coefficients∑∑∑∑∑∑∑∑========−−−−−−=−−−−−−=niiniiiniiniiiniiniiiniiniiiXXXXXXXXXXYYXXXXXXXXXXYY12,221,22,11112,221,12212,111,22,11212,111,111)())((ˆ- )())((ˆand )())((ˆ- )())((ˆββββX1is Clinton thermometer, X2is PID, and Y is Gore thermometerThe Slope Coefficients More Simply)var(),cov(ˆ- )var(),cov(ˆand)var(),cov(ˆ- )var(),cov(ˆ22112221212111XXXXYXXXXXYXββββ==X1is Clinton thermometer, X2is PID, and Y is Gore thermometerThe Matrix formy1y2…yn1x1,1x2,1…xk,11x1,2x2,2…xk,21…………1x1,nx2,n…xk,nβ=′′−()XX Xy1The 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 linearnonrandom selection on party ID. Nonrandom selection into the treatment could occur from Variables other than party ID, or Reverse causation, that is, feelings about Gore influencing feelings about Clinton. Additionally, 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 addressing confounding effects? Experiments Difference-in-differences designs Others?Summary: 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(ˆ211=−=−=−====clintonpartyclintonclintonclintongoreclintonclintongoreMMBβββ. corr gore clinton party,cov(obs=1745)| gore clinton party3-------------+---------------------------gore | 660.681clinton | 549.993 883.182party3 | 13.7008 16.905 .8735Drinking 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 screen<=3(4631 real changes made). tab timespast30timespast30 | Freq. Percent Cum.------------+-----------------------------------0 | 4,652 33.37 33.371.5 | 2,737 19.64 53.014 | 2,653 19.03 72.047.5 | 1,854 13.30 85.3414.5 | 1,648 11.82
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