Addressing Alternative. Multiple Regression Spring 2012

Transcription

1 Addressing Alternative Explanations: Multiple Regression Spring

2 Did Clinton hurt Gore example Did Clinton hurt Gore in the 2000 election? Treatment is not liking Bill Clinton 2

3 Bivariate regression of Gore thermometer on Clinton thermometer Clinton thermometer 3

4 Did 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 intuitive 4

5 Democratic picture Clinton thermometer 5

6 Independent picture Clinton thermometer 6

7 Republican picture Clinton thermometer 7

8 Combined data picture Clinton thermometer 8

9 Combined data picture with regression: bias! Clinton thermometer 9

10 Combined data picture with true regression lines overlaid Clinton thermometer 10

11 Tempting yet wrong normalizations Subtract the Gore therm. from the avg. Gore therm. score Clinton thermometer Subtract the Clinton therm. from the avg. Clinton therm. score Clinton thermometer 11

12 3D Relati tionship 12

13 3D Linear Relationship 13

14 3D Relationshi hip: Cli Clinton

15 3D Relati tionship: party Rep Ind Dem 15

16 The Linear Relationship between Three Variables Gore thermometer Clinton thermometer Party ID Y i 0 1 X 1, i 2 X 2, i i 16

17 The method of least squares (again) Pick 0, 1, and 2 to minimize n i i 1 (Y Y ˆ ) 2 or i n (Y X X ) 2 i 0 1 i 2 2 i 1 17

18 The Slope Coefficients n (Y Y )(X X ) (X X )(X X ) i 1 1,i 1 1,i 2 2, i ˆ i 1 ˆ i and n 2 n (X X ) 2 1 1,i (X 1 X1, ) 2 i n i 1 (Y Y )(X X ) (X X )(X X ˆ i 1 ˆ i n 1 n (X 2 2 X 2,i ) (X 2 2 X 2,i ) i 1 n n i 1 i 2 1,i 1 1,i 2 2, i ) X 1 is Clinton thermometer, X 2 is PID, and Y is Gore thermometer i 1 0 Y 1 X 1 2 X 2 18

19 The Slope Coefficients More Simply ˆ cov(x,y ) 1 ˆ cov(x 1, X 2 ) 1 2 var(x ) var(x ) and ˆ cov(x,y ) 2 - ˆ cov(x 1, X 2 ) 2 var(x ) 1 var(x ) 2 2 X 1 is Clinton thermometer, X 2 is PID, and Y is Gore thermometer 19

20 The Matrix form y 1 y 2 1 x 1,1 x 2,1 x k,1 1 x 1,2 x 2,2 x k,2 1 y n 1 x 1,n x 2,n x k,n ( X X ) 1 X y 20

21 Multivariate slope coefficients cov(x,y ) Bivariate estimate: ˆ B 1 1 vs. var( X 1 ) Are Gore and Party ID related? ˆ cov(x M 1,Y ) ˆ cov(x M 1, X 2 ) Multivariate estimate: 1-2 var( X 1 ) var( X 1 ) Clinton effect (on Gore) in multivariate (M) regression Clinton effect (on Gore) in bivariate (B) regression cov( X, X ) Are Clinton and Party ID related? B M ˆ 1 2 When does ˆ ˆ? Obviously, when M var( X 1 ) X 1 is Clinton thermometer, X 2 is PID, and Y is Gore thermometer 21

22 The Output t. reg gore clinton party3 Source SS df MS Number of obs = F( 2, 1742) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = Prob gore Coef. Std. Err. t P> t [95% Conf. Interval] clinton party _cons 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. 22

23 Separate regressions (1) (2) (3) Intercept Clinton Party ˆ cov(x,y ) 1 ˆ cov(x, X ) and var(x 1 ) var(x 1 ) ˆ cov(x,y ˆ, 2 Y ) cov(x 1 X 2 ) 2-1 var(x 2 ) var(x 2 ) 23

24 Why did the Clinton Coefficient change from 0.62 to corr gore clinton party, cov (obs=1745) gore clinton party gore clinton party

25 The Calculations cov(gore, clinton) ) ˆ B var(clinton) ˆ M cov(gore, clinton) ˆ M cov(clinton, party) 1 2 var(clinton) var(clinton) corr gore clinton party,cov (obs=1745) gore clinton party gore clinton party

26 Another way of thinking about this Rewrite ˆ cov(gore, clinton) M 1 2 var(clinton) as cov(gore, clinton) var(clinton) ˆ M M ˆ ˆ1 M 2 cov(clinton, party) var(clinton) cov(clinton, party) var(clinton) Total effect = Direct effect + indirect effect The Total Effect of the Clinton thermometer on the Gore thermometer (.61) can be Broken down into a direct effect of.51, plus an indirect effect (though party) of.11 26

27 Di Drinking and dg Greek klif 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. 27

28 Dependent variable: Times Drinking in Past 30 Days :HFKVOHU+HQU\&ROOHJH$OFRKRO6WXG\+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXU&UHDWLYH&RPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH 28

29 . infix age residence 16 greek 24 screen 102 timespast howmuchpast gpa studying 281 timeshs 325 howmuchhs hh 326 socializing i 283 stwgt_ weight 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: changes made). replace timespast30=0 if screen<=3 (4631 real changes made) 29

30 . tab timespast30 timespast30 Freq. Percent Cum , , , , , Total 13,

31 Key explanatory variables Live in fraternity/sorority house Indicator variable (dummy variable) Coded 1 if live in, 0 otherwise Member of fraternity/sorority Indicator variable (dummy variable) Coded 1 if member, 0 otherwise 31

32 Three Regressions Dependent variable: number of times drinking in past 30 days Live in frat/sor house (indicator variable) Member of frat/sor (indicator variable) Intercept S.E.R. R (0.35) N 13,876 13,876 What is the substantive interpretation of the coefficients? (0.56) (0.16) 4.27 (0.059) (0.38) (0.18) 4.27 (0.059) ,876 Note: Standard d errors in parentheses. Corr. Between living i in frat/sor house and being a member of a Greek organization is.42 32

33 The Picture 21 Living in frat house X =0.19 X 2 ˆ M 2 =2.26 Drinks ik per 30 days Y Member of fraternity y X 1 M 1 ˆ =

34 Accounting for the total t effect ˆ B ˆ M ˆ M Total effect = Direct effect + indirect effect 21 Living in frat house X 2 =0.19 M ˆ = Drinks per 30 days Y Member of fraternity X 1 ˆ M 1 =

35 Accounting for the effects of frat house living and Greek membership on drinking From bivariate i regressions From multiple regressions From accounting identity: T=D+I Effect Total Direct Indirect Member of Greek org. (85%) (15%) Live in frat/ sor. house (51%) (49%) 35

36 MIT OpenCourseWare Political Science Laboratory Spring 2012 For information about citing these materials or our Terms of Use, visit: