Introduction to Econometrics (3 rd Updated Edition, Global Edition) Solutions to Odd-Numbered End-of-Chapter Exercises: Chapter 13

Transcription

1 Introducton to Econometrcs (3 rd Updated Edton, Global Edton by James H. Stock and Mark W. Watson Solutons to Odd-Numbered End-of-Chapter Exercses: Chapter 13 (Ths verson August 17, 014

2 Stock/Watson - Introducton to Econometrcs - 3 rd Updated Edton - Answers to Exercses: Chapter The small class treatment effect and ade treatment effect cannot be ndependently calculated of one another. The coeffcent on small class treatment effect of measures the hgher test score n small classes relatve to large classes wth an ade. To correctly estmate both effects, one of the control groups must be a small class wth an ade.

3 Stock/Watson - Introducton to Econometrcs - 3 rd Updated Edton - Answers to Exercses: Chapter (a The estmated average treatment effect s X X = = TreatmentGroup Control 47 ponts. (b There would be nonrandom assgnment f men (or women had dfferent probabltes of beng assgned to the treatment and control groups. Let p Men denote the probablty that a male s assgned to the treatment group. Random assgnment means p Men = 0.5. Testng ths null hypothess results n a t-statstc of t ˆp Me n = Men = 1 pˆ 1 Men (1 pˆ Men 0.60( nmen 100 =.04 so that the null of random assgnment s rejected at the 5% level. A smlar result s found for women.

4 Stock/Watson - Introducton to Econometrcs - 3 rd Updated Edton - Answers to Exercses: Chapter (a Ths s an example of attrton, whch poses a threat to nternal valdty. After the male athletes leave the experment, the remanng subjects are representatve of a populaton that excludes male athletes. If the average causal effect for ths populaton s the same as the average causal effect for the populaton that ncludes the male athletes, then the attrton does not affect the nternal valdty of the experment. On the other hand, f the average causal effect for male athletes dffers from the rest of populaton, nternal valdty has been compromsed. (b Ths s an example of partal complance whch s a threat to nternal valdty. The local area network s a falure to follow treatment protocol, and ths leads to bas n the OLS estmator of the average causal effect. (c Ths poses no threat to nternal valdty. As stated, the study s focused on the effect of dorm room Internet connectons. The treatment s makng the connectons avalable n the room; the treatment s not the use of the Internet. Thus, the art majors receved the treatment (although they chose not to use the Internet. (d As n part (b ths s an example of partal complance. Falure to follow treatment protocol leads to bas n the OLS estmator.

5 Stock/Watson - Introducton to Econometrcs - 3 rd Updated Edton - Answers to Exercses: Chapter From the populaton regresson Y = α + β X + β ( D W + β D + v, t 1 t t 0 t t we have Y Y = β ( X X + β [( D D W] + β ( D D + ( v v By defnng DY = Y Y 1, DX = X X 1 (a bnary treatment varable and u = v v 1, and usng D 1 = 0 and D = 1, we can rewrte ths equaton as Y = + X + W + u β0 β1 β, whch s Equaton (13.5 n the case of a sngle W regressor.

6 Stock/Watson - Introducton to Econometrcs - 3 rd Updated Edton - Answers to Exercses: Chapter The covarance between β X and X s cov( β X, X = E{[ β X E( β X ][ X E( X ]} = E{ β X E( β X X β XEX ( + E( β X EX ( } = E( β X E( β X E( X Because X s randomly assgned, X s dstrbuted ndependently of b. The ndependence means E( β X = E( β EX ( and E( β X = E( β EX (. Thus cov( β 1 X, X can be further smplfed: cov( β1 X, X = E( β1 [ EX ( E( X] = E σ ( β1 X. So cov( β X, X E( β σ = = E( β1. σ X X σ X

7 Stock/Watson - Introducton to Econometrcs - 3 rd Updated Edton - Answers to Exercses: Chapter Followng the notaton used n Chapter 13, let π denote the coeffcent on dstance to the nearest hosptal offerng cardac catheterzaton n the frst stage IV regresson, and let β denote the effect of cardac catheterzaton on patent survval tmes. From (13.11 p E(β ˆβ TSLS π = E(β E(π + Cov(β, π E(π = Average Treatment Effect + Cov(β,π, E(π where the frst equalty uses the uses propertes of covarances (equaton (.34, and the second equalty uses the defnton of the average treatment effect. Evdently, the local average treatment effect wll devate from the average treatment effect when Cov ( β, π 0. As dscussed n Secton 13.6, ths covarance s zero when β or π are constant. Ths seems lkely. But, for the sake of argument, suppose that they are not constant; that s, suppose the mpact of catheterzaton on survval tmes vares across patents (β s not constant as does the effect of dstances on the use of catheterzaton (π s not constant. Are β and π related? They mght be f we beleve that the estmated effect s dfferent for people who are senstve to dstance from a hosptal. For nstance, very old ndvduals may be more senstve to dstance from a hosptal for gettng treatment. They mght also be more or less senstve to the treatment compared to other types of patents. For example, let us assume that older people beneft more from the use of catheterzaton. Ths suggests that β and π are postvely related, so that Cov(β, π > 0. Because E(π 1 < 0, the local treatment effect s less than the average treatment effect when β vares between older and younger patents.