Stock Watson 3U Exercise Solutions Chapter 6 Instructors PDF

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! ! ! ! Introduction!to!Econometrics!(3rd!Updated!Edition)! ! ! by! ! ! James!H.!Stock!and!Mark!W.!Watson! ! ! ! ! !

Solutions!to!End7of7Chapter!Exercises:!Chapter!6*" ! ! (This version August 17, 2014)

! ! ! ! ! ! ! ! ! ! ! *Limited!distribution:!For!Instructors!Only.!!Answers!to!all!odd=numbered! questions!are!provided!to!students!on!the!textbook!website.!!!If!you!find!errors!in! the!solutions,[email protected].!!

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Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 6 1 _____________________________________________________________________________________________________

! 6.1. By equation (6.15) in the text, we know

R 2 = 1−

n −1 (1 − R 2 ). n − k −1

Thus, that values of R2 are 0.162, 0.180, and 0.181 for columns (1)–(3).

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Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 6 2 _____________________________________________________________________________________________________

6.2. (a) Workers with college degrees earn $8.31/hour more, on average, than workers with only high school degrees.

(b) Men earn $3.85/hour more, on average, than women.

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Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 6 3 _____________________________________________________________________________________________________

6.3. (a) On average, a worker earns $0.51/hour more for each year he ages. (b) Sally’s earnings prediction is 1.87 + 8.32 × 1− 3.81 × 1+ 0.51 × 29 = 21.17 dollars per hour. Betsy’s earnings prediction is 1.87 + 8.32 × 1− 3.81 × 1+ 0.51 × 34 = 23.72 dollars per hour. The difference is $2.55/hour.

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Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 6 4 _____________________________________________________________________________________________________

6.4. (a) Workers in the Northeast earn $0.18 more per hour than workers in the West, on average, controlling for other variables in the regression. Workers in the Midwest earn $1.23 less per hour than workers in the West, on average, controlling for other variables in the regression. Workers in the South earn $0.43 less than workers in the West, controlling for other variables in the regression. (b) The regressor West is omitted to avoid perfect multicollinearity. If West is included, then the intercept can be written as a perfect linear function of the four regional regressors. (c) The expected difference in earnings between Juanita and Jennifer is −0.43 − (−1.23) = $0.80/hour.

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Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 6 5 _____________________________________________________________________________________________________

6.5. (a) $23,400 (recall that Price is measured in $1000s). (b) In this case ΔBDR = 1 and ΔHsize = 100. The resulting expected change in price is 23.4 + 0.156 × 100 = 39.0 thousand dollars or $39,000. (c) The loss is $48,800. (d) From the text R 2 = 1 − n −nk−1−1 (1 − R2 ), so R 2 = 1 − n −n−k 1−1 (1 − R 2 ), thus, R2 = 0.727.

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Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 6 6 _____________________________________________________________________________________________________

6.6. (a) There are other important determinants of a country’s crime rate, including demographic characteristics of the population.

(b) Suppose that the crime rate is positively affected by the fraction of young males in the population, and that counties with high crime rates tend to hire more police. In this case, the size of the police force is likely to be positively correlated with the fraction of young males in the population leading to a positive value for the omitted variable bias so that βˆ1 > β1.

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Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 6 7 _____________________________________________________________________________________________________

6.7. (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.

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Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 6 8 _____________________________________________________________________________________________________

6.8. Omitted from the analysis are reasons why the survey respondents slept more or less than average. People with certain chronic illnesses might sleep more than 8 hours per night. People with other illnesses might sleep less than 5 hours. This study says nothing about the causal effect of sleep on mortality.

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Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 6 9 _____________________________________________________________________________________________________

6.9. For omitted variable bias to occur, two conditions must be true: X1 (the included regressor) is correlated with the omitted variable, and the omitted variable is a determinant of the dependent variable. Since X1 and X2 are uncorrelated, the estimator of β1 does not suffer from omitted variable bias.

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Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 6 10 _____________________________________________________________________________________________________

6.10. (a)

1⎡ 1 σ 2ˆβ = ⎢ n ⎣⎢ 1 − ρ X2 , X 1

1

⎤ σ u2 ⎥ 2 ⎥σ X1 2 ⎦

Assume X1 and X2 are uncorrelated: ρ 2X1 X2 = 0

1 ⎡ 1 ⎤4 400 ⎣⎢ 1− 0⎦⎥ 6 1 4 1 = ⋅ = = 0.00167 400 6 600

σ 2ˆβ = 1

(b) With ρ X 1 , X 2 = 0.5

σ β2ˆ = 1

=

1 ⎡ 1 ⎤4 400 ⎣⎢ 1− 0.52 ⎦⎥ 6 1 ⎡ 1 ⎤4 = .0022 400 ⎢⎣ 0.75⎥⎦ 6

(c) The statement correctly says that the larger is the correlation between X1 and X2 the larger is the variance of βˆ1 , however the recommendation “it is best to leave X2 out of the regression” is incorrect. If X2 is a determinant of Y, then leaving X2 out of the regression will lead to omitted variable bias in βˆ1 .

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Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 6 11 _____________________________________________________________________________________________________

6.11. (a)

∑(Y − b X i

1

1i

− b2 X 2i )2

(b)

∂ ∑ (Yi − b1 X1 i − b2 X 2 i )2 = − 2∑ X 1 i (Y i − b1X 1 i − b 2X 2 i ) ∂b1 ∂ ∑ (Yi − b1 X1i − b2 X 2i )2 = − 2∑ X 2 i (Yi − b1 X 1i − b2 X 2 i ) ∂ b2

∑ X 1 iYi − βˆ2 ∑ X 1 i X 2 i (c) From (b), βˆ1 satisfies ∑ X 1i ( Yi − βˆ1 X 1i − βˆ1 X 2i ) = 0 , or βˆ1 = ∑ X 12i

and the result follows immediately.

∑ X 2iYi − βˆ1 ∑ X 1i X 2i (d) Following analysis as in (c) βˆ2 = and substituting this into the ∑ X 22i X Y − βˆ X 1 iX 2 i ∑ X1 iY ∑ 2 i i 1X∑ ∑ X1 iX2 i 2 i 2 ∑ expression for βˆ1 in (c) yields βˆ1 = . ∑ X 12i

∑ X 22i ∑ X 1iYi − ∑ X 1i X 2 i ∑ X 2 iYi Solving for βˆ1 yields: βˆ1 = 2 2 2 ∑ X 1i ∑ X 2i − ( ∑ X 1i X 2i )

(continued on the next page)

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Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 6 12 _____________________________________________________________________________________________________

6.11 (continued)

(e) The least squares objective function is

∑( Y − b i

0

− b1 X1i − b2 X 2 i) 2 and the partial

derivative with respect to b0 is

∂ ∑ (Yi − b0 − b1 X1 i − b2 X2 i )2 = −2∑ (Yi − b0 − b1 X1i − b2 X2 i ). ∂b0 Setting this to zero and solving for βˆ0 yields: βˆ0 = Y − βˆ1 X1 − βˆ2 X 2.

(f) Substituting βˆ0 = Y − βˆ1 X1 − βˆ2 X 2. into the least squares objective function yields

∑( Y − βˆ i

0

− b1 X 1i − b2 X 2 i) 2 =

∑(( Y − Y) − b ( X i

1

1i

2

− X1) − b2( X 2 i − X 2) ) , 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 X1, omitting X2. Said differently, when

∑( X

1i

− X 1)( X 2i − X 2) = 0 , the

estimated coefficient on X1 in the OLS regression of Y onto both X1 and X2 is the same as estimated coefficient in the OLS regression of Y onto X1.

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