Lecture_11_Prof._Arkonac's_Slides_(Ch_8_-_9.3)

# Lecture_11_Prof._Arkonac's_Slides_(Ch_8_-_9.3) - Nonlinear...

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Nonlinear Regression III (Fall 2010) Lecture 11 Prof: Seyhan Erden Arkonac, PhD Problem Set 4 is due NOW! Problem Set 5 will be posted today. It is due at the beginning of class on Tuesday October 19 th . 1

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2 Interactions Between Independent Variables (SW Section 8.3) Perhaps a class size reduction is more effective in some circumstances than in others… Perhaps smaller classes help more if there are many English learners, who need individual attention That is, TestScore STR might depend on PctEL More generally, 1 Y X might depend on X 2 How to model such “interactions” between X 1 and X 2 ? We first consider binary X ’s, then continuous X ’s
3 (a) Interactions between two binary variables Y i = 0 + 1 D 1 i + 2 D 2 i + u i D 1 i , D 2 i are binary 1 is the effect of changing D 1 =0 to D 1 =1. In this specification, this effect doesn’t depend on the value of D 2 . To allow the effect of changing D 1 to depend on D 2 , include the “interaction term” D 1 i D 2 i as a regressor: Y i = 0 + 1 D 1 i + 2 D 2 i + 3 ( D 1 i D 2 i ) + u i

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4 Interpreting the coefficients Y i = 0 + 1 D 1 i + 2 D 2 i + 3 ( D 1 i D 2 i ) + u i General rule: compare the various cases E ( Y i | D 1 i =0, D 2 i = d 2 ) = 0 + 2 d 2 (b) E ( Y i | D 1 i =1, D 2 i = d 2 ) = 0 + 1 + 2 d 2 + 3 d 2 (a) subtract (a) – (b): E ( Y i | D 1 i =1, D 2 i = d 2 ) – E ( Y i | D 1 i =0, D 2 i = d 2 ) = 1 + 3 d 2 The effect of D 1 depends on d 2 (what we wanted) 3 = increment to the effect of D 1 , when D 2 = 1
5 Example : TestScore, STR, English learners Let HiSTR = 1 if 20 0 if 20 STR STR and HiEL = 1 if l0 0 if 10 PctEL PctEL TestScore = 664.1 – 18.2 HiEL – 1.9 HiSTR – 3.5( HiSTR HiEL ) (1.4) (2.3) (1.9) (3.1) “Effect” of HiSTR when HiEL = 0 is –1.9 “Effect” of HiSTR when HiEL = 1 is –1.9 – 3.5 = –5.4 Class size reduction is estimated to have a bigger effect when the percent of English learners is large This interaction isn’t statistically significant: t = 3.5/3.1

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Wage example: (1)Wage = β 0 + β 1 Female + β 2 Married + ……. single male: β 0 married male: β 0 + β 2 single female : β 0 + β 1 married female: β 0 + β 1 + β 2 difference: β 1 difference: β 1 (2)Wage = β 0 1 Female+β 2 Married+β 3 FemxMar+. .. single male: β 0 married male: β 0 + β 2 single female : β 0 + β 1 married female:β 0 1 2 + β 3 difference: β 1 difference: β 1 + β 3 6
Wage example (cont’): . reg lwage exper expersq tenure tenursq female married femxmar,r Linear regression Number of obs = 526 F( 7, 518) = 35.31 Prob > F = 0.0000 R-squared = 0.3160 Root MSE = .44257 ------------------------------------------------------------------------------ | Robust lwage | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- exper | .0260293 .0057505 4.53 0.000 .0147321 .0373265 expersq | -.0006615 .0001186 -5.58 0.000 -.0008946 -.0004285 tenure | .0345734 .0080995 4.27 0.000 .0186615 .0504853 tenursq | -.000643

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Lecture_11_Prof._Arkonac's_Slides_(Ch_8_-_9.3) - Nonlinear...

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