Stock Watson 3U Exercise Solutions Chapter 9 Instructors PDF

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     Introduction to Econometrics rd (3 Updated Edition, Global Edition) by

James H. Stock and Mark W. Watson

Solutions to End-of-Chapter Exercises: Chapter 9* (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, please pass them along to us at [email protected].

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

 9.1. As explained in the text, potential threats to external validity arise from differences between the population and setting studied and the population and setting of interest. The statistical results based on West Africa in 2000 might apply to some parts of India in 2000 but not to the United Kingdom in 2000. In 2000, many parts of India, particularly rural India, were plagued by high malnutrition and poor child health. In contrast, the United Kingdom had very high standards of public health as well as higher per capita food consumption, so the distribution of vitamins would most likely not have a high impact on test scores. The results from West Africa in 2000 may continue to apply to the same region in 2015 but this depends on improvements in public health and per capita consumption that may have taken place over this time period.

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

9.2. (a) When Yi is measured with error, we have

or

Substituting

the 2nd equation into the regression model Yi  0  1 X i  ui gives

Thus vi  ui  wi .

or

(b) (1) The error term vi has conditional mean zero given Xi:

E(vi |X i )  E (ui  wi |Xi )  E (ui |Xi )  E (wi |Xi )  0 0  0. (2)

is i.i.d since both Yi and wi are i.i.d. and mutually independent; Xi and

are independent since Xi is independent of both Yj and wj.

Thus,

are i.i.d. draws from their joint distribution.

(3) vi  ui  wi has a finite fourth moment because both ui and wi have finite fourth moments and are mutually independent. So (Xi, vi) have nonzero finite fourth moments. (c) The OLS estimators are consistent because the least squares assumptions hold. (d) Because of the validity of the least squares assumptions, we can construct the confidence intervals in the usual way. (e) The answer here is the economists’ “On the one hand, and on the other hand.” On the one hand, the statement is true: i.i.d. measurement error in X means that the OLS estimators are inconsistent and inferences based on OLS are invalid. OLS estimators are consistent and OLS inference is valid when Y has i.i.d. measurement error. On the other hand, even if the measurement error in Y is i.i.d. and independent of Yi and Xi, it increases the variance of the regression error ( v2   u2   w2 ), and this will increase the variance of the OLS estimators. Also, measurement error that is not i.i.d. may change these results, although this would need to be studied on a case-by-case basis.

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

9.3. The key is that the selected sample contains only employed women. Consider two women, Beth and Julie. Beth has no children; Julie has one child. Beth and Julie are otherwise identical. Both can earn $25,000 per year in the labor market. Each must compare the $25,000 benefit to the costs of working. For Beth, the cost of working is forgone leisure. For Julie, it is forgone leisure and the costs (pecuniary and other) of child care. If Beth is just on the margin between working in the labor market or not, then Julie, who has a higher opportunity cost, will decide not to work in the labor market. Instead, Julie will work in “home production,” caring for children, and so forth. Thus, on average, women with children who decide to work are women who earn higher wages in the labor market.

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

9.4. Estimated effect of a 10% increase in the percentage of students eligible for free lunch In points In Std. Dev. State % Standard deviation of scores Calif. –0.398 19.1 –3.98 –0.21 Mass. –0.521 15.1 –5.21 –0.35 A 10% point increase in the number of students eligible for a free lunch has a 66% larger negative impact on test scores in Massachusetts compared to California.

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

9.5. (a) Q 

 1 0  0  1  1u  1v  . 1  1 1  1

and P 

0   0 u  v  .  1  1 1  1

(b) E (Q ) 

 1  0   0 1   , E ( P)  0 0  1  1  1  1

(c) 2

2

 1   1  2 2 2 2 2 2 Var (Q )    ( 1  u  1  v ), Var ( P)    ( u   v ), and        1 1  1 1 2

 1  2 2 Cov( P, Q)    (  1 u  1 V )   1  1 

p (d) (i) ˆ1 

Cov (Q , P )  1 u2  1 V2 ,   u2   V2 Var (P )

ˆ0  E (Q )  E (P ) p

Cov( P, Q ) Var (P )

p 2 (    ) (ii) ˆ1   1  u 2 1  21  0, using the fact that 1  0 (supply curves slope up) and u

V

1  0 (demand curves slope down).

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

9.6. (a) The parameter estimates do not change. Nor does the R2. The sum of squared residuals from the 50 observation regression is SSR50  (50  2)  23.52  26, 508, and the sum of squared residuals from the

100 observation regression is twice this value: SSR100  2 26,508  53,016. Thus, the SER from the 100 observation regression is SER100 

1 100  2

SSR100  23.26. The standard errors for the regression

2 2 ˆ i and coefficients are now computed using equation (5.4) where 100 i1 ( X i  X ) u 2 100 i 1 ( X i  X ) are twice their value from the 50 observation regression. Thus,

the standard errors for the 100 observation regression are the standard errors in the 50 observation regression multiplied by

50 2 100  2

 0.700. In summary, the

results for the 100 observation regression are

Yˆ  49.2  73.9 X, SER  23. 26, R 2  0. 78 (16.45) (11.48) (b) The observations are not i.i.d.: half of the observations are identical to the other half, so that the observations are not independent.

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

9.7. (a) False. Correlation between the dependent variable and the regressors reduce the precision of the OLS estimator, but do not induce bias. (b) False. If the error term exhibits heteroskedasticity, the standard errors need to be corrected for this to produce unbiased estimates of X. If heteroskedasticity-robust standard errors are not used, then estimates of X will be biased.

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

9.8. Not directly. Test scores in California and Massachusetts are for different tests and have different means and variances. However, converting (9.5) into units for Massachusetts yields the implied regression to TestScore(MA units)  740.9  1.80 STR, which is similar to the regression using Massachusetts data shown in Column 1 of Table 9.2. After this adjustment the regression could be somewhat useful, hower the regression in Column 1 of Table 9.2 has a low R2 suggesting that it will not provide an accurate forecast of test scores.

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

9.9. Both regressions suffer from omitted variable bias so that they will not provide reliable estimates of the causal effect of income on test scores. However, the nonlinear regression in (8.18) fits the data well, so that it could be used for forecasting.

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

9.10. There are several reasons for concern. Here are a few. Internal consistency: omitted variable bias as explained in the last paragraph of the box. Internal consistency: sample selection may be a problem as the regressions were estimated using a sample of full-time workers. (See exercise 9.3 for a related problem.) External consistency: Returns to education may change over time because of the relative demands and supplies of skilled and unskilled workers in the economy. To the extent that this is important, the results shown in the box (based on 2008 data) may not accurately estimate today’s returns to education.

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

9.11. There are several reasons for concern. Here are a few. Internal consistency: To the extent that price is affected by demand, there may be simultaneous equation bias. External consistency: The internet and introduction of “E-journals” may induce important changes in the market for academic journals so that the results for 2000 may not be relevant for today’s market.

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

9.12. (a) See the answer to part (c) of exercise 2.27. (b) Because E(w| (c) Xi =

= E(X | Z), then E(X |

) = E[(X −

|

)] = E(X |

+ wi, so that Yi = 0 + 1(

 1 wi. Because E(u | Z) = 0 and E(u|

)= ) – E(

. Thus |

)=

−

+ wi) + ui = 0 + 1

= 0. + vi, where vi = ui +

depends only on Z (that is,

= E(X | Z)), then

) = 0. Together with the result in (b), this implies that E(vi |

) = 0. You

can then verify the other assumptions in Key Concept (4.3), and the result follows from the consistency of the OLS estimator under these assumptions.

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

9.13. (a)

. Because all of the Xi’s are used (although some are

used for the wrong values of Yj), X = X , and



n i1

( X i  X ) 2 . Also,

Y i Y  1 (X i  X )  u i  u . Using these expressions:

 

0.8 n

ˆ1  1

i1 n i 1

(X i  X )2

( X i  X )2

 1



n i0.8 n1



(X i  X )(X i  X ) n i 1

( X i  X )2

 

n i1

(X i  X )(u i  u )



n i 1

( X i  X )2

1 0.8n 1 n 1 n  (X i  X ) 2 ( X i  X )( X i  X )    i1( X i  X )(u i  u ) i1 i0.8 n1 n n n  1  1  1 n 1 n 1 n 2 2 ( ) ( )   ( X i  X )2 X X X X    i i   i 1 1 1 i i n n n where n = 300, and the last equality uses an ordering of the observations so that the first 240 observations (= 0.8×n) correspond to the correctly measured observations ( X i = Xi). As is done elsewhere in the book, we interpret n = 300 as a large sample, so we use the approximation of n tending to infinity. The solution provided here thus shows that these expressions are approximately true for n large and hold in the limit that n tends to infinity. Each of the averages in the expression for ˆ1 have the following probability limits: p 1 n 2 X X   X2 , ( )  i i 1 n p 1 0.8 n 2 2   ( ) 0.8 X , X X  i i 1 n

, and

,

(continued on next page)

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

9.13 (continued)

where the last result follows because Xi ≠ Xi for the scrambled observations and p

Xj is independent of Xi for i ≠ j. Taken together, these results imply 1ˆ  0.81 . p

p

(b) Because ˆ1  0.81 , ˆ 1 / 0.8  1 , so a consistent estimator of  1 is the OLS estimator divided by 0.8. (c) Yes, the estimator based on the first 240 observations is better than the adjusted estimator from part (b). Equation (4.21) in Key Concept 4.4 (page 129) implies that the estimator based on the first 240 observations has a variance that is 1 var  ( X i   X )ui  . var(ˆ1 (240obs ))  240  var(X i )2

From part (a), the OLS estimator based on all of the observations has two sources of sampling error. The first is

which is the usual

source that comes from the omitted factors (u). The second is , which is the source that comes from scrambling the data. These two terms are uncorrelated in large samples, and their respective large-sample variances are: .

and .

(continued on next page)

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

9.13 (continued)

Thus  ˆ (300obs)  1 var  1    0.8   0.64

 1 var ( X i   X )u i   2 0.2    1 2 300  var( X i )  300

which is larger than the variance of the estimator that only uses the first 240 observations.

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