Introduction to Econometrics (3 rd Updated Edition) Solutions to Odd- Numbered End- of- Chapter Exercises: Chapter 6

Transcription

1 Introduction to Econometrics (3 rd Updated Edition) by James H. Stock and Mark W. Watson Solutions to Odd- Numbered End- of- Chapter Exercises: Chapter 6 (This version August 17, 014) 015 Pearson Education, Inc.

2 Stock/Watson - Introduction to Econometrics - 3 rd Updated Edition - Answers to Exercises: Chapter By equation (6.15) in the text, we know Thus, that values of R n 1 = 1 (1 R ). n k 1 R are 0.16, 0.180, and for columns (1) (3). 015 Pearson Education, Inc.

3 Stock/Watson - Introduction to Econometrics - 3 rd Updated Edition - Answers to Exercises: Chapter (a) On average, a worker earns $0.51/hour more for each year he ages. (b) Sally s earnings prediction is = 1.17 dollars per hour. Betsy s earnings prediction is = 3.7 dollars per hour. The difference is $.55/hour. 015 Pearson Education, Inc.

4 Stock/Watson - Introduction to Econometrics - 3 rd Updated Edition - Answers to Exercises: Chapter (a) $3,400 (recall that Price is measured in $1000s). (b) In this case ΔBDR = 1 and ΔHsize = 100. The resulting expected change in price is = 39.0 thousand dollars or $39,000. (c) The loss is $48,800. (d) From the text R = 1 (1 R ), so n 1 n k 1 R = 1 (1 R ), thus, R = n k 1 n Pearson Education, Inc.

5 Stock/Watson - Introduction to Econometrics - 3 rd Updated Edition - Answers to Exercises: Chapter (a) The proposed research in assessing the presence of gender bias in setting wages is too limited. There might be some potentially important determinants of salaries: type of engineer, amount of work experience of the employee, and education level. The gender with the lower wages could reflect the type of engineer among the gender, the amount of work experience of the employee, or the education level of the employee. The research plan could be improved with the collection of additional data as indicated and an appropriate statistical technique for analyzing the data would be a multiple regression in which the dependent variable is wages and the independent variables would include a dummy variable for gender, dummy variables for type of engineer, work experience (time units), and education level (highest grade level completed). The potential importance of the suggested omitted variables makes a difference in means test inappropriate for assessing the presence of gender bias in setting wages. (b) The description suggests that the research goes a long way towards controlling for potential omitted variable bias. Yet, there still may be problems. Omitted from the analysis are characteristics associated with behavior that led to incarceration (excessive drug or alcohol use, gang activity, and so forth), that might be correlated with future earnings. Ideally, data on these variables should be included in the analysis as additional control variables. 015 Pearson Education, Inc.

6 Stock/Watson - Introduction to Econometrics - 3 rd Updated Edition - Answers to Exercises: Chapter For omitted variable bias to occur, two conditions must be true: X 1 (the included regressor) is correlated with the omitted variable, and the omitted variable is a determinant of the dependent variable. Since X 1 and X are uncorrelated, the estimator of β 1 does not suffer from omitted variable bias. 015 Pearson Education, Inc.

7 Stock/Watson - Introduction to Econometrics - 3 rd Updated Edition - Answers to Exercises: Chapter (a) ( Y bx b X ) i 1 1i i (b) ( Yi b1x1 i bxi) b ( Yi b1x1 i bxi) b 1 = X ( Y b X b X ) 1i i 1 1i i = X ( Y b X b X ) i i 1 1i i (c) From (b), ˆβ 1 satisfies ˆ ˆ XY 1 ˆ βx1 X X1 i( Yi β1x1 i β1xi) = 0, or β1 = X and the result follows immediately. ˆ i i i i 1i XY ˆ β1x1 X (d) Following analysis as in (c) β = X expression for ˆβ 1 in (c) yields Solving for ˆβ 1 yields: ˆ β ˆ i i i i i ˆ β. X ˆ iyi β1 X1iXi XY 1i X1 ix X i i 1 = X1 i X X Y X X X Y i 1i i 1i i i i 1 = X1 ixi ( X1 ixi) and substituting this into the (continued on the next page) 015 Pearson Education, Inc.

8 Stock/Watson - Introduction to Econometrics - 3 rd Updated Edition - Answers to Exercises: Chapter (continued) (e) The least squares objective function is derivative with respect to b 0 is ( Y i b 0 bx 1 1i b X i) and the partial ( Yi b0 bx 1 1i bxi) b 0 = ( Y b bx b X ). i 0 1 1i i Setting this to zero and solving for ˆ β yields: ˆ 0 β0 = Y ˆ β ˆ. 1X1 βx (f) Substituting ˆ β0 = Y ˆ β ˆ. 1X1 βx into the least squares objective function yields ˆ ( Y ( ) i β 0 b1x1 i bxi) = ( Yi Y) b1( X1 i X1) b( Xi X), which is identical to the least squares objective function in part (a), except that all variables have been replaced with deviations from sample means. The result then follows as in (c). Notice that the estimator for β 1 is identical to the OLS estimator from the regression of Y onto X 1, omitting X. Said differently, when ( X1 i X1)( Xi X) = 0, the estimated coefficient on X 1 in the OLS regression of Y onto both X 1 and X is the same as estimated coefficient in the OLS regression of Y onto X Pearson Education, Inc.