Addressing Alternative. Multiple Regression Spring 2012
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1 Addressng Alternatve Explanatons: Multple Regresson 7.87 Sprng 0
2 Dd Clnton hurt Gore example Dd Clnton hurt Gore n the 000 electon? Treatment s not lkng Bll Clnton How would you test ths?
3 Bvarate regresson of Gore thermometer on Clnton thermometer Clnton thermometer
4 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
5 Democratc pcture Clnton thermometer
6 Independent d pcture Clnton thermometer
7 Republcan pcture Clnton thermometer
8 Combned data pcture Clnton thermometer
9 Combned data pcture wth regresson: bas! Clnton thermometer
10 Combned data pcture wth true regresson lnes overlad Clnton thermometer
11 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
12 3D Relatonshp
13 3D Lnear Relatonshp
14 3D Relatonshp: Clnton
15 3D Relatonshp: party Rep Ind Dem
16 The Lnear Relatonshp between Three Varables Gore thermometer Clnton thermometer Party ID Y 0,, STATA: reg y x x reg gore clnton party3
17 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
18 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
19 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
20 The Matrx form y x, x, x k, y x, x, x k, y n x,n x,n x k,n ( y
21 The Output t. reg gore clnton party3 Source SS df MS Number of obs = F(, 74 = Model Prob > F = Resdual R-squared = Adj R-squared = Total Root MSE = gore Coef. Std. Err. t P> t [95% Conf. Interval] clnton party _cons 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.
22 Separate regressons ( ( (3 Intercept Clnton Party
23 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
24 Other approaches to addressng confoundng effects? Experments Dfference-n-dfferences n desgns Others?
25 Summary: Why we control Address alternatve explanatons by removng confoundng effects Improve effcency
26 Why dd the Clnton Coeffcent change from 0.6 to 0.5. corr gore clnton party, cov (obs=745 gore clnton party gore clnton party
27 The Calculatons l ˆ B ˆ cov( gore, clnton var( clnton ˆ cov( gore, clnton var( clnton M ˆ M cov( clnton, party var( clnton. corr gore clnton party,cov (obs=745 gore clnton party gore clnton party
28 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.
29 Dependent varable: Tmes Drnkng n Past 30 Days
30 . nfx age 0- resdence 6 greek 4 screen 0 tmespast30 03 howmuchpast30 04 gpa studyng 8 tmeshs 35 howmuchhs 36 socalzng 83 stwgt weght 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
31 . tab tmespast30 tmespast30 Freq. Percent Cum , , , , , Total 3,
32 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
33 Three Regressons Dependent varable: number of tmes drnkng n past 30 days Lve n frat/sor house (ndcator varable (0.35 (0.38 Member of frat/sor (ndcator varable ( (0.8 Intercept (0.56 (0.059 (0.059 R 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
34 The Pcture Lvng n frat house =0.9 M ˆ =.6 Drnks k per 30 days Y Member of fraternty y M ˆ =.44
35 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
36 Accountng for the effects of frat house lvng and Greek membershp on drnkng Effect Total Drect Indrect Member of Greek org. (85% (5% Lve n frat/ sor. house (5% (49%
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