12
Instrumental Chapter Variables Regression
Solutions to Exercises
1.
(a) The change in the regressor, ln(Pi ,cigarettes ) ln( Pi ,cigarettes ), from a $0.10 per pack increase in the
1995
1985
retail price is ln 2.10 ln 2.00 = 0.0488. The expected percentage change in cigarette demand
is 9.94 0.0488 100% = 4.5872%. The 95% confidence interval is (0.94 1.96 0.21)
0.0488 100% = [6.60%, 2.58%].
(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 3.6
instead of 33.6, 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).
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 Ws. 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).
s
(c) The TSLS estimator is TSLS = ZY using Equation (10.4) in the text. Since Z = X , we have
1
sZX
i
i
s
s
1TSLS = ZY = XY = 1OLS .
2
sZX sX
3.
TSLS
2
(a) The estimator a = n 1 2 in=1 (Yi 0 1TSLS Xi )2 is not consistent. Write this as
2
a = n 1 2 in=1 (ui 1TSLS ( Xi Xi ))2 , where ui = Yi 0TSLS 1TSLS Xi . Replacing 1TSLS with 1,
as suggested in the question, write this as
1
1
2 1
a n in=1 (ui 1 ( Xi Xi ))2 = n in=1 ui2 + n in=1 [ 12 ( Xi Xi )2 + 2ui 1 ( Xi Xi )]. The first term on
2
the right hand side of the equation converges to u , but the second term converges to something
2
that is non-zero. Thus a is not consistent.
2
(b) The estimator b =
1
n 2
TSLS
in=1 (Yi 0 1TSLS Xi )2 is consistent. Using the same notation as in (a),
2
2 n
we can write b 1 in=1ui2 , and this estimator converges in probability to u .
58
4.
Stock/Watson - Introduction to Econometrics - Second Edition
Using Xi = 0 + 1Zi , we have X = 0 + 1Z and
n
n
sXY ( = Xi X )(Yi Y ) = 1 (Zi Z )(Yi Y ) = 1sZY ,
i =1
i =1
n
n
i =1
i =1
2
2
sX = ( Xi X )2 = 12 (Zi Z )2 = 12 sZ .
Using the formula for the OLS estimator in Key Concept 4.2, we have
1 =
sZX
.
2
sZ
Thus the TSLS estimator
s
s
s
s
s
1TSLS = XY = 12 ZY = ZY2 = sZX ZY 2 = ZY .
2
2
sX 1 sZ 1sZ s2 sZ sZX
Z
5.
(a) Instrument relevance. Zi does not enter the population regression for Xi
(b) Z is not a valid instrument. X * 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.
6.
0.05
Use R 2 to compute the homoskedasitic-only F statistic as FHomoskedasitcOnly = 1 RR/ T/ k k 1 = 0.95/ 98 = 5.16 with
2
2
100 observations in which case we conclude that the instrument may be week. With 500 observations
the FHomoskedasitcOnly = 26.2 so the instrument is not weak.
7.
(a) Under the null hypothesis of instrument exogeneity, the J statistic is distributed as a 12 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 doesnt provide evidence about whether the problem
is with Z1 or Z2 or both.
8.
(a) Solving for P yields P =
0 0
1
+
uid uis
1
; thus Cov( P, u s ) =
2s
u
1
(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). 0 can be estimated by OLS (equivalently as the sample mean of Qi).
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.
Solutions to Exercises in Chapter 12
10.
Cov( Z i , Yi ) Cov( Z i , 0 + 1 X i + 2Wi + ui ) 1Cov( Z i , X i ) + 2Cov( Z i ,Wi )
TSLS =
=
=
Cov ( Z i , X i )
Cov (Z i , X i )
Cov ( Z i , X i )
(a) If Cov ( Z i ,Wi ) = 0 the IV estimator is consistent.
(b) If Cov ( Z i ,Wi ) 0 the IV estimator is not consistent.
59