Addressng Alternatve Explanatons: Multple Regresson 7.87 Sprng 0
Dd Clnton hurt Gore example Dd Clnton hurt Gore n the 000 electon? Treatment s not lkng Bll Clnton How would you test ths?
Bvarate regresson of Gore thermometer on Clnton thermometer Clnton thermometer
Dd Clnton hurt Gore example What alternatve explanatons would you need to address? Nonrandom selecton nto the treatment group (dslkng Clnton from many sources Let s address one source: party dentfcaton How could we do ths? Matchng: compare Democrats who lke or don t lke Clnton; do the same for Republcans and ndependents Multvarate regresson: control for partsanshp statstcally Also called multple regresson, Ordnary Least Squares (OLS Presentaton below s ntutve
Democratc pcture Clnton thermometer
Independent d pcture Clnton thermometer
Republcan pcture Clnton thermometer
Combned data pcture Clnton thermometer
Combned data pcture wth regresson: bas! Clnton thermometer
Combned data pcture wth true regresson lnes overlad Clnton thermometer
Temptng yet wrong normalzatons Subtract the Gore therm. from the avg. Gore therm. score Clnton thermometer Subtract the Clnton therm. from the avg. Clnton therm. score Clnton thermometer
3D Relatonshp
3D Lnear Relatonshp
3D Relatonshp: Clnton 00 50 0
3D Relatonshp: party Rep Ind Dem
The Lnear Relatonshp between Three Varables Gore thermometer Clnton thermometer Party ID Y 0,, STATA: reg y x x reg gore clnton party3
Multvarate slope coeffcents Bvarate estmate: Multvarate estmate: ˆ ˆ B M Clnton effect (on Gore n bvarate (B regresson cov(, Y var( vs. cov(, Y ˆ M cov(, - var( var( Are Gore and Party ID related? Clnton effect (on Gore n multvarate (M regresson ˆ ˆ B M ˆ cov(, var( Are Clnton and Party ID related? M When does? Obvously, when 0 s Clnton thermometer, s PID, and Y s Gore thermometer
Th Sl C ff t The Slope Coeffcents n n n n Y Y,,, and ( ( ˆ - ( ( ˆ n n n n,, ( ( n n n n Y Y,,, ( ( ( ˆ - ( ( ( ˆ,, ( ( s Clnton thermometer, s PID, and Y s Gore thermometer
The Slope Coeffcents More Smply cov(, Y ˆ cov(, - var( var( ˆ cov(, Y ˆ cov(, - var( var( ˆ and s Clnton thermometer, s PID, and Y s Gore thermometer
The Matrx form y x, x, x k, y x, x, x k, y n x,n x,n x k,n ( y
The Output t. reg gore clnton party3 Source SS df MS Number of obs = 745 -------------+------------------------------ F(, 74 = 048.04 Model 696.9 34630.955 Prob > F = 0.0000 Resdual 5964.934 74 300.0949 R-squared = 0.546 -------------+------------------------------ Adj R-squared = 0.5456 Total 56.84 744 660.68053 Root MSE = 7.37 ------------------------------------------------------------------------------ gore Coef. Std. Err. t P> t [95% Conf. Interval] -------------+---------------------------------------------------------------- clnton.5875.07595 9. 0.000.4777776.5467975 party3 5.77053.5594846 0.3 0.000 4.6739 6.867856 _cons 8.699.0547 7.9 0.000 6.686 30.649 ------------------------------------------------------------------------------ Interpretaton of clnton effect: Holdng constant party dentfcaton a one- Interpretaton of clnton effect: Holdng constant party dentfcaton, a onepont ncrease n the Clnton feelng thermometer s assocated wth a.5 ncrease n the Gore thermometer.
Separate regressons ( ( (3 Intercept 3. 55.9 8.6 Clnton 0.6 -- 0.5 Party -- 5.7 5.8
Is the Clnton effect causal? That s, should we be convnced that negatve feelngs about Clnton really hurt Gore? No! The regresson analyss has only ruled out lnear nonrandom selecton on party ID. Nonrandom selecton nto the treatment could occur from Varables other than party ID, or Reverse causaton, that s, feelngs about Gore nfluencng feelngs about Clnton. Addtonally, the regresson analyss may not have entrely ruled out nonrandom selecton even on party ID because t may have assumed the wrong functonal form. E.g., what f nonrandom selecton on strong Republcan/strong Democrat, but not on weak partsans
Other approaches to addressng confoundng effects? Experments Dfference-n-dfferences n desgns Others?
Summary: Why we control Address alternatve explanatons by removng confoundng effects Improve effcency
Why dd the Clnton Coeffcent change from 0.6 to 0.5. corr gore clnton party, cov (obs=745 gore clnton party3 -------------+--------------------------- gore 660.68 clnton 549.993 883.8 party3 3.7008 6.905.8735
The Calculatons l ˆ B ˆ cov( gore, clnton var( clnton 549.993993 883.8 0.67 ˆ cov( gore, clnton var( clnton M ˆ M 549.993 6.905 5.7705 883.8 883.8 0.67 0.05 0.5 cov( clnton, party var( clnton. corr gore clnton party,cov (obs=745 gore clnton party3 -------------+--------------------------- gore 660.68 clnton 549.993 883.8 party3 3.7008 6.905.8735
D Drnkng and dgreek klfe Example Why s there a correlaton between lvng n a fraternty/sororty y house and drnkng? Greek organzatons often emphasze socal gatherngs g that have alcohol. The effect s beng n the Greek organzaton tself, not the house. There s somethng about the House envronment tself.
Dependent varable: Tmes Drnkng n Past 30 Days
. nfx age 0- resdence 6 greek 4 screen 0 tmespast30 03 howmuchpast30 04 gpa 78-79 studyng 8 tmeshs 35 howmuchhs 36 socalzng 83 stwgt 99 475-493 weght99 494-5 usng da388.dat,clear (438 observatons read. recode tmespast30 tmeshs (=0 (=.5 (3=4 (4=7.5 (5=4.5 (6=9.5 (7=45 (tmespast30: 657 changes made (tmeshs: 07 changes made. replace tmespast30=0 f screen<=3 (463 real changes made
. tab tmespast30 tmespast30 Freq. Percent Cum. ------------+----------------------------------- 0 4,65 33.37 33.37.5,737 9.64 53.0 4,653 9.03 7.04 7.5,854 3.30 85.34 4.5,648.8 97.7 9.5 350.5 99.68 45 45 0.3 00.00 ------------+----------------------------------- Total 3,939 00.00
Key explanatory varables Lve n fraternty/sororty house Indcator varable (dummy varable Coded f lve n, 0 otherwse Member of fraternty/sororty Indcator varable (dummy varable Coded f member, 0 otherwse
Three Regressons Dependent varable: number of tmes drnkng n past 30 days Lve n frat/sor house (ndcator varable 4.44 ---.6 (0.35 (0.38 Member of frat/sor (ndcator varable ---.88 (0.6.44 (0.8 Intercept 454 4.54 47 4.7 47 4.7 (0.56 (0.059 (0.059 R.0.03.05 N 3,876 3,876 3,876 N t St d d th C B t l What s the substantve nterpretaton of the coeffcents? Note: Standard errors n parentheses. Corr. Between lvng n frat/sor house and beng a member of a Greek organzaton s.4
The Pcture Lvng n frat house =0.9 M ˆ =.6 Drnks k per 30 days Y Member of fraternty y M ˆ =.44
Accountng for the total t effect ˆ B ˆ M ˆ M Total effect = Drect effect + ndrect effect Lvng n frat house =0.9 M ˆ =.6 Drnks per 30 days Y Member of fraternty M ˆ =.44
Accountng for the effects of frat house lvng and Greek membershp on drnkng Effect Total Drect Indrect Member of.88.44 0.44 Greek org. (85% (5% Lve n frat/ 4.44.6.8 sor. house (5% (49%