Solutions to Odd-Numbered End-of-Chapter Exercises: Chapter 9

Transcription

1 Itroductio to Ecoometrics (3 rd Updated Editio) by James H. Stock ad Mark W. Watso Solutios to Odd-Numbered Ed-of-Chapter Exercises: Chapter 9 (This versio July 0, 04)

2 Stock/Watso - Itroductio to Ecoometrics - 3 rd Updated Editio - Aswers to Exercises: Chapter As explaied i the text, potetial threats to exteral validity arise from differeces betwee the populatio ad settig studied ad the populatio ad settig of iterest. The statistical results based o New York i the 970 s are likely to apply to Bosto i the 970 s but ot to Los Ageles i the 970 s. I 970, New York ad Bosto had large ad widely used public trasportatio systems. Attitudes about smokig were roughly the same i New York ad Bosto i the 970s. I cotrast, Los Ageles had a cosiderably smaller public trasportatio system i 970. Most residets of Los Ageles relied o their cars to commute to work, school, ad so forth. The results from New York i the 970 s are ulikely to apply to New York i 04. Attitudes towards smokig chaged sigificatly from 970 to 04.

3 Stock/Watso - Itroductio to Ecoometrics - 3 rd Updated Editio - Aswers to Exercises: Chapter The key is that the selected sample cotais oly employed wome. Cosider two wome, Beth ad Julie. Beth has o childre; Julie has oe child. Beth ad Julie are otherwise idetical. Both ca ear $5,000 per year i the labor market. Each must compare the $5,000 beefit to the costs of workig. For Beth, the cost of workig is forgoe leisure. For Julie, it is forgoe leisure ad the costs (pecuiary ad other) of child care. If Beth is just o the margi betwee workig i the labor market or ot, the Julie, who has a higher opportuity cost, will decide ot to work i the labor market. Istead, Julie will work i home productio, carig for childre, ad so forth. Thus, o average, wome with childre who decide to work are wome who ear higher wages i the labor market.

4 Stock/Watso - Itroductio to Ecoometrics - 3 rd Updated Editio - Aswers to Exercises: Chapter u v 9.5. (a) Q. 0 0 u v ad P. EQ ( ), (b) 0 0 EP ( ) 0 0 (c) Var( Q) ( u v), Var( P) ( u v),ad Cov( P, Q) ( u V) (d) (i) ˆ Cov( Q, P), p u V Var( P) u V ˆ p Cov( P, Q) 0 EQ ( ) EP ( ) Var ( P ) (ii) ˆ p 0, usig the fact that 0 (supply curves slope up) ad u ( ) u V 0 (demad curves slope dow).

5 Stock/Watso - Itroductio to Ecoometrics - 3 rd Updated Editio - Aswers to Exercises: Chapter (a) True. Correlatio betwee regressors ad error terms meas that the OLS estimator is icosistet. (b) True.

6 Stock/Watso - Itroductio to Ecoometrics - 3 rd Updated Editio - Aswers to Exercises: Chapter Both regressios suffer from omitted variable bias so that they will ot provide reliable estimates of the causal effect of icome o test scores. However, the oliear regressio i (8.8) fits the data well, so that it could be used for forecastig.

7 Stock/Watso - Itroductio to Ecoometrics - 3 rd Updated Editio - Aswers to Exercises: Chapter There are several reasos for cocer. Here are a few. Iteral cosistecy: To the extet that price is affected by demad, there may be simultaeous equatio bias. Exteral cosistecy: The iteret ad itroductio of E-jourals may iduce importat chages i the market for academic jourals so that the results for 000 may ot be relevat for today s market.

8 Stock/Watso - Itroductio to Ecoometrics - 3 rd Updated Editio - Aswers to Exercises: Chapter (a). Because all of the X i s are used (although some are used for the wrog values of Y j ), X = X, ad Y ( ) i Y Xi X ui u. Usig these expressios: i ( X ) i X. Also, ( X X) ( X X)( X X) ( X X)( u u) ˆ 0.8 i i i0.8 i i i i i ( X ) ( ) ( ) i i X X i i X X i i X 0.8 ( X i X) ( X )( ) ( )( ) 0.8 i X Xi X X i X ui u i i i ( X ) ( ) i i X X i i X ( X ) i i X where = 300, ad the last equality uses a orderig of the observatios so that the first 40 observatios (= 0.8 ) correspod to the correctly measured observatios ( X i = X i ). As is doe elsewhere i the book, we iterpret = 300 as a large sample, so we use the approximatio of tedig to ifiity. The solutio provided here thus shows that these expressios are approximately true for large ad hold i the limit that teds to ifiity. Each of the averages i the expressio for ˆ have the followig probability limits: p ( X ) i i X X, p 0.8 ( X ) 0.8 i i X X,, ad, (cotiued o ext page)

9 Stock/Watso - Itroductio to Ecoometrics - 3 rd Updated Editio - Aswers to Exercises: Chapter (cotiued) where the last result follows because X i X i for the scrambled observatios ad X j is idepedet of X i for i j. Take together, these results imply ˆ p 0.8. (b) Because ˆ p 0.8 p, ˆ /0.8, so a cosistet estimator of is the OLS estimator divided by 0.8. (c) Yes, the estimator based o the first 40 observatios is better tha the adjusted estimator from part (b). Equatio (4.) i Key Cocept 4.4 (page 9) implies that the estimator based o the first 40 observatios has a variace that is ˆ var ( X ) u var( (40 obs)). i X i 40 var( X ) i From part (a), the OLS estimator based o all of the observatios has two sources of samplig error. The first is which is the usual source that comes from the omitted factors (u). The secod is, which is the source that comes from scramblig the data. These two terms are ucorrelated i large samples, ad their respective large-sample variaces are:. ad. (cotiued o ext page)

10 Stock/Watso - Itroductio to Ecoometrics - 3 rd Updated Editio - Aswers to Exercises: Chapter (cotiued) Thus ˆ (300 obs) var ( Xi X) u 0. i var var( X ) 300 i which is larger tha the variace of the estimator that oly uses the first 40 observatios.