Chapter 12

Introduction to Econometrics (3rd Updated Edition)
by
James H. Stock and Mark W. Watson
Solutions to End-of-Chapter Exercises: Chapter 12*
(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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Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 12
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cigarettes
cigarettes
12.1. (a) The change in the regressor, ln( Pi ,1995
) − ln( Pi ,1985
), from a $0.50 per pack
increase in the retail price is ln(8.00) − ln(7.50) = 0.0645. The expected
percentage change in cigarette demand is −0.94 × 0.0645× 100% = −6.07%. The
95% confidence interval is (−0.94 ± 1.96 × 0.21) × 0.0645 × 100% = [−8.72%,
−3.41%].
(b) With a 2% reduction in income, the expected percentage change in cigarette
demand is 0.53 × (−0.02) × 100% = −1.06%.
(c) The regression in column (1) will not provide a reliable answer to the question
in (b) when recessions last less than 1 year. The regression in column (1) studies
the long-run price and income elasticity. Cigarettes are addictive. The response
of demand to an income decrease will be smaller in the short run than in the
long run.
(d) The instrumental variable would be too weak (irrelevant) if the F-statistic in
column (1) was 3.7 instead of 33.7, and we cannot rely on the standard methods
for statistical inference. Thus the regression would not provide a reliable answer
to the question posed in (a).
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12.2. (a) When there is only one X, we only need to check that the instrument enters the first
stage population regression. Since the instrument is Z = X, the regression of X onto Z
will have a coefficient of 1.0 on Z, so that the instrument enters the first stage
population regression. Key Concept 4.3 implies corr(Xi, ui) = 0, and this implies
corr(Zi, ui) = 0. Thus, the instrument is exogenous.
(b) Condition 1 is satisfied because there are no W’s. Key Concept 4.3 implies that
condition 2 is satisfied because (Xi, Zi, Yi) are i.i.d. draws from their joint
distribution. Condition 3 is also satisfied by applying assumption 3 in Key
Concept 4.3. Condition 4 is satisfied because of conclusion in part (a).
(c) The TSLS estimator is βˆ1TSLS =
sZY
sZX
using Equation (10.4) in the text. Since Zi =
Xi, we have
s
s
βˆ1TSLS = ZY = XY2 = βˆ1OLS .
sZX s X
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Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 12
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12.3. (a) The estimator σˆ a2 = n−1 2 ∑in=1 (Yi − βˆ0TSLS − βˆ1TSLS Xˆ i )2 is not consistent. Write this as
σˆ a2 = n−1 2 ∑in=1 (uˆi − βˆ1TSLS ( Xˆ i − X i ))2 , where uˆi = Yi − βˆ0TSLS − βˆ1TSLS X i . Replacing
βˆ1TSLS with β1, as suggested in the question, write this as
σˆ a2 ≈ 1n ∑in=1 (ui − β1 ( Xˆ i − X i ))2 = 1n ∑in=1 ui2 + 1n ∑in=1[β12 ( Xˆ i − X i )2 + 2ui β1 ( Xˆ i − X i )].
The first term on the right hand side of the equation converges to σˆ u2 , but the
second term converges to something that is non-zero. Thus σˆ a2 is not consistent.
(b) The estimator σˆb2 = n−1 2 Σin=1 (Yi − βˆ0TSLS − βˆ1TSLS X i )2 is consistent. Using the same
notation as in (a), we can write σˆ b2 ≈ 1n Σin=1ui2 , and this estimator converges in
probability to σ u2 .
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Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 12
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12.4. Using Xˆ i = πˆ0 + πˆ1Zi , we have Xˆ = πˆ0 + πˆ1Z and
n
n
ˆ
ˆ
s XY
ˆ = ∑ ( X i − X )(Yi − Y ) = πˆ1 ∑ ( Z i − Z )(Yi − Y ) = πˆ1s ZY ,
i =1
i =1
n
n
i =1
i =1
s X2ˆ = ∑ ( Xˆ i − Xˆ ) 2 = πˆ12 ∑ ( Z i − Z ) 2 = πˆ12 sZ2 .
Using the formula for the OLS estimator in Key Concept 4.2, we have
πˆ1 =
sZX
.
sZ2
Thus the TSLS estimator
βˆ1TSLS =
sXY
ˆ
s
2
Xˆ
=
πˆ1sZY sZY
s
s
=
= s ZY 2 = ZY .
2 2
2
πˆ1 sZ πˆ1sZ s × sZ sZX
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ZX
2
Z
Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 12
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12.5. (a) Instrument relevance. Z i does not enter the population regression for X i
(b) Z is not a valid instrument. Xˆ * will be perfectly collinear with W. (Alternatively,
the first stage regression suffers from perfect multicollinearity.)
(c) W is perfectly collinear with the constant term.
(d) Z is not a valid instrument because it is correlated with the error term.
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12.6. Use R 2 to compute the homoskedasitic-only F statistic as
0.05
FHomoskedasitcOnly = 1−RR2 /T/ k−k −1 = 0.95/98
= 5.16 with 100 observations in which case we
2
conclude that the instrument may be week.
With 500 observations the FHomoskedasitcOnly = 26.2 so the instrument is not weak.
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12.7. (a) Under the null hypothesis of instrument exogeneity, the J statistic is distributed
as a χ12 random variable, with a 1% critical value of 6.63. Thus the statistic is
significant, and instrument exogeneity E(ui |Z1i, Z2i) = 0 is rejected.
(b) The J test suggests that E(ui |Z1i, Z2i) ≠ 0, but doesn’t provide evidence about
whether the problem is with Z1 or Z2 or both.
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Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 12
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12.8. (a) Solving for P yields P =
γ 0 − β0
β1
+
uid −uis
β1
; thus Cov( P, u s ) =
−σ 2s
u
β1
(b) Because Cov(P,u) ≠ 0, the OLS estimator is inconsistent (see (6.1)).
(c) We need a instrumental variable, something that is correlated with P but
uncorrelated with us. In this case Q can serve as the instrument, because demand
is completely inelastic (so that Q is not affected by shifts in supply). γ0 can be
estimated by OLS (equivalently as the sample mean of Qi).
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12.9. (a) There are other factors that could affect both the choice to serve in the military
and annual earnings. One example could be education, although this could be
included in the regression as a control variable. Another variable is “ability”
which is difficult to measure, and thus difficult to control for in the regression.
(b) The draft was determined by a national lottery so the choice of serving in the
military was random. Because it was randomly selected, the lottery number is
uncorrelated with individual characteristics that may affect earning and hence the
instrument is exogenous. Because it affected the probability of serving in the
military, the lottery number is relevant.
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12.10.
Cov( Zi , Yi ) Cov( Z i , β 0 + β1 X i + β 2Wi + ui ) β1Cov( Z i , X i ) + β 2Cov( Z i ,Wi )
βˆTSLS =
=
=
Cov( Zi , X i )
Cov( Z i , X i )
Cov( Z i , X i )
(a) If Cov( Zi ,Wi ) = 0 the IV estimator is consistent.
(b) If Cov( Zi ,Wi ) ≠ 0the IV estimator is not consistent.
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