Solutions to Odd Numbered End of Chapter Exercises: Chapter 4

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 4 (This versio July 2, 24)

2 Stock/Watso - Itroductio to Ecoometrics - 3 rd Updated Editio - Aswers to Exercises: Chapter (a) The predicted average test score is (b) The predicted chage i the classroom average test score is (c) Usig the formula for ˆ i Equatio (4.8), we kow the sample average of the test scores across the classrooms is TestScore ˆ ˆ CS (d) Use the formula for the stadard error of the regressio (SER) i Equatio (4.9) to get the sum of squared residuals: Use the formula for SSR SER The sample variace is s Y s Y ( 2) ( 2) R i Equatio (4.6) to get the total sum of squares: TSS SSR R sy TSS Thus, stadard deviatio is 99

3 Stock/Watso - Itroductio to Ecoometrics - 3 rd Updated Editio - Aswers to Exercises: Chapter (a) The coefficiet 9.6 shows the margial effect of Age o AWE; that is, AWE is expected to icrease by $9.6 for each additioal year of age is the itercept of the regressio lie. It determies the overall level of the lie. (b) SER is i the same uits as the depedet variable (Y, or AWE i this example). Thus SER is measures i dollars per week. (c) R 2 is uit free. (d) (i) $936.7; (ii) $,28.7 (e) No. The oldest worker i the sample is 65 years old. 99 years is far outside the rage of the sample data. (f) No. The distributio of earig is positively skewed ad has kurtosis larger tha the ormal. (g) ˆ ˆ, Y X so that Y ˆ ˆ. X Thus the sample mea of AWE is $,96.6.

4 Stock/Watso - Itroductio to Ecoometrics - 3 rd Updated Editio - Aswers to Exercises: Chapter (a) u i represets factors other tha time that ifluece the studet s performace o the exam icludig amout of time studyig, aptitude for the material, ad so forth. Some studets will have studied more tha average, other less; some studets will have higher tha average aptitude for the subject, others lower, ad so forth. (b) Because of radom assigmet u i is idepedet of X i. Sice u i represets deviatios from average E(u i ). Because u ad X are idepedet E(u i X i ) E(u i ). (c) (2) is satisfied if this year s class is typical of other classes, that is, studets i this year s class ca be viewed as radom draws from the populatio of studets that eroll i the class. (3) is satisfied because Y i ad X i ca take o oly two values (9 ad 2). (d) (i) ; ; (ii)

5 Stock/Watso - Itroductio to Ecoometrics - 3 rd Updated Editio - Aswers to Exercises: Chapter The expectatio of ˆ is obtaied by takig expectatios of both sides of Equatio (4.8): ˆ ˆ ˆ E( ) E( Y X) E X ui X i ˆ E( ) X E( ui ) i where the third equality i the above equatio has used the facts that E(u i ) = ad E[( ˆ ) X ] = E[(E( ˆ ) X ) X ] = because E[( ˆ ) X] (see text equatio (4.3).)

6 Stock/Watso - Itroductio to Ecoometrics - 3 rd Updated Editio - Aswers to Exercises: Chapter (a) With ˆ ˆ, Y, ad Yˆ ˆ i Y. Thus ESS ad R 2. (b) If R 2, the ESS, so that Y ˆi Y for all i. But Yˆ ˆ ˆ i Xi, so that Y ˆi for all i, which implies that ˆ, or that X i is costat for all i. If X i is costat Y for all i, the ( ) 2 X i i X ad ˆ is udefied (see equatio (4.7)).

7 Stock/Watso - Itroductio to Ecoometrics - 3 rd Updated Editio - Aswers to Exercises: Chapter (a) The least squares objective fuctio is 2 ( Y ). i i bx i Differetiatig with respect to b yields 2 i( YibXi) b X i Y i i bx i Settig this zero, ad 2 ( ). solvig for the least squares estimator yields ˆ i XY i i. 2 i Xi (b) Followig the same steps i (a) yields ˆ i Xi( Yi4) 2 i Xi

8 Stock/Watso - Itroductio to Ecoometrics - 3 rd Updated Editio - Aswers to Exercises: Chapter The aswer follows the derivatios i Appedix 4.3 i Large-Sample Normal Distributio of the OLS Estimator. I particular, the expressio for i is ow i = (X i X )u i, so that var( i ) = 3 var[(x i X )u i ], ad the term 2 carry through the rest of the calculatios.