Solution
Robert answered on
Dec 25 2021
CHAPTER 7
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CHAPTER 7
TEACHING NOTES
This is a fairly standard chapter on using qualitative information in regression analysis, although
I try to emphasize examples with policy relevance (and only cross-sectional applications are
included.).
In allowing for different slopes, it is important, as in Chapter 6, to appropriately interpret the
parameters and to decide whether they are of direct interest. For example, in the wage equation
where the return to education is allowed to depend on gender, the coefficient on the female
dummy variable is the wage differential between women and men at zero years of education. It
is not surprising that we cannot estimate this very well, nor should we want to. In this particular
example we would drop the interaction term because it is insignificant, but the issue of
interpreting the parameters can arise in models where the interaction term is significant.
In discussing the Chow test, I think it is important to discuss testing for differences in slope
coefficients after allowing for an intercept difference. In many applications, a significant Chow
statistic simply indicates intercept differences. (See the example in Section 7.4 on student-
athlete GPAs in the text.) From a practical perspective, it is important to know whether the
partial effects differ across groups or whether a constant differential is sufficient.
I admit that an unconventional feature of this chapter is its introduction of the linear probability
model. I cover the LPM here for several reasons. First, the LPM is being used more and more
ecause it is easier to interpret than probit or logit models. Plus, once the proper parameter
scalings are done for probit and logit, the estimated effects are often similar to the LPM partial
effects near the mean or median values of the explanatory variables. The theoretical drawbacks
of the LPM are often of secondary importance in practice. Computer Exercise C7.9 is a good one
to illustrate that, even with over 9,000 observations, the LPM can deliver fitted values strictly
etween zero and one for all observations.
If the LPM is not covered, many students will never know about using econometrics to explain
qualitative outcomes. This would be especially unfortunate for students who might need to read
an article where an LPM is used, or who might want to estimate an LPM for a term paper or
senior thesis. Once they are introduced to purpose and interpretation of the LPM, along with its
shortcomings, they can tackle nonlinear models on their own or in a subsequent course.
A useful modification of the LPM estimated in equation (7.29) is to drop kidsge6 (because it is
not significant) and then define two dummy variables, one for kidslt6 equal to one and the other
for kidslt6 at least two. These can be included in place of kidslt6 (with no young children being
the base group). This allows a diminishing marginal effect in an LPM. I was a bit surprised
when a diminishing effect did not materialize.
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SOLUTIONS TO PROBLEMS
7.1 (i) The coefficient on male is 87.75, so a man is estimated to sleep almost one and one-half
hours more per week than a comparable woman. Further, tmale = 87.75/34.33 2.56, which is
close to the 1% critical value against a two-sided alternative (about 2.58). Thus, the evidence for
a gender differential is fairly strong.
(ii) The t statistic on totwrk is .163/.018 9.06, which is very statistically significant. The
coefficient implies that one more hour of work (60 minutes) is associated with .163(60) 9.8
minutes less sleep.
(iii) To obtain 2rR , the R-squared from the restricted regression, we need to estimate the
model without age and age
2
. When age and age
2
are both in the model, age has no effect only if
the parameters on both terms are zero.
7.2 (i) If cigs = 10 then log( )bwght = .0044(10) = .044, which means about a 4.4% lower
irth weight.
(ii) A white child is estimated to weigh about 5.5% more, other factors in the first equation
fixed. Further, twhite 4.23, which is well above any commonly used critical value. Thus, the
difference between white and nonwhite babies is also statistically significant.
(iii) If the mother has one more year of education, the child’s birth weight is estimated to be
.3% higher. This is not a huge effect, and the t statistic is only one, so it is not statistically
significant.
(iv) The two regressions use different sets of observations. The second regression uses fewer
observations because motheduc or fatheduc are missing for some observations. We would have
to reestimate the first equation (and obtain the R-squared) using the same observations used to
estimate the second equation.
7.3 (i) The t statistic on hsize
2
is over four in absolute value, so there is very strong evidence that
it belongs in the equation. We obtain this by finding the turnaround point; this is the value of
hsize that maximizes ˆsat (other things fixed): 19.3/(2 2.19) 4.41. Because hsize is measured
in hundreds, the optimal size of graduating class is about 441.
(ii) This is given by the coefficient on female (since black = 0): nonblack females have SAT
scores about 45 points lower than nonblack males. The t statistic is about –10.51, so the
difference is very statistically significant. (The very large sample size certainly contributes to
the statistical significance.)
(iii) Because female = 0, the coefficient on black implies that a black male has an estimated
SAT score almost 170 points less than a comparable nonblack male. The t statistic is over 13 in
absolute value, so we easily reject the hypothesis that there is no ceteris paribus difference.
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(iv) We plug in black = 1, female = 1 for black females and black = 0 and female = 1 for
nonblack females. The difference is therefore –169.81 + 62.31 = 107.50. Because the estimate
depends on two coefficients, we cannot construct a t statistic from the information given. The
easiest approach is to define dummy variables for three of the four race/gender categories and
choose nonblack females as the base group. We can then obtain the t statistic we want as the
coefficient on the black female dummy variable.
7.4 (i) The approximate difference is just the coefficient on utility times 100, or –28.3%. The t
statistic is .283/.099 2.86, which is very statistically significant.
(ii) 100 [exp(.283) – 1) 24.7%, and so the estimate is somewhat smaller in magnitude.
(iii) The proportionate difference is .181 .158 = .023, or about 2.3%. One equation that can
e estimated to obtain the standard e
or of this difference is
log(salary) = 0 + 1 log(sales) + 2 roe + 1 consprod + 2 utility + 3 trans + u,
where trans is a dummy variable for the transportation industry. Now, the base group is finance,
and so the coefficient 1 directly measures the difference between the consumer products and
finance industries, and we can use the t statistic on consprod.
7.5 (i) Following the hint, colGPA =
0̂ + 0̂ (1 – noPC) + 1̂ hsGPA + 2̂ ACT = ( 0̂ + 0̂ )
0̂ noPC + 1̂ hsGPA + 2̂ ACT. For the specific estimates in equation (7.6), 0̂ = 1.26 and 0̂
= .157, so the new intercept is 1.26 + .157 = 1.417. The coefficient on noPC is –.157.
(ii) Nothing happens to the R-squared. Using noPC in place of PC is simply a different way
of including the same information on PC ownership.
(iii) It makes no sense to include both dummy variables in the regression: we cannot hold
noPC fixed while changing PC. We have only two groups based on PC ownership so, in
addition to the overall intercept, we need only to include one dummy variable. If we try to
include both along with an intercept we have perfect multicollinearity (the dummy variable trap).
7.6 In Section 3.3 – in particular, in the discussion su
ounding Table 3.2 – we discussed how to
determine the direction of bias in the OLS estimators when an important variable (ability, in this
case) has been omitted from the regression. As we discussed there, Table 3.2 only strictly holds
with a single explanatory variable included in the regression, but we often ignore the presence of
other independent variables and use this table as a rough guide. (Or, we can use the results of
Problem 3.10 for a more precise analysis.) If less able workers are more likely to receive
training, then train and u are negatively co
elated. If we ignore the presence of educ and exper,
or at least assume that train and u are negatively co
elated after netting out educ and exper, then
we can use Table 3.2: the OLS estimator of 1 (with ability in the e
or term) has a downward
ias. Because we think 1 0, we are less likely to conclude that the training program was
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effective. Intuitively, this makes sense: if those chosen for training had not received training,
they would have lowers wages, on average, than the control group.
7.7 (i) Write the population model underlying (7.29) as
inlf = 0 + 1 nwifeinc + 2 educ + 3 exper + 4 expe
2
+ 5 age
+ 6 kidslt6 + 7 kidsage6 + u,
plug in inlf = 1 – outlf, and rea
ange:
1 – outlf = 0 + 1 nwifeinc + 2 educ + 3 exper + 4 expe
2
+ 5 age
+ 6 kidslt6 + 7 kidsage6 + u,
or
outlf = (1 0 ) 1 nwifeinc 2 educ 3 exper 4 expe
2
5 age
6 kidslt6 7 kidsage6 u,
The new e
or term, u, has the same properties as u. From this we see that if we regress outlf on
all of the independent variables in (7.29), the new intercept is 1 .586 = .414 and each slope
coefficient takes on the opposite sign from when inlf is the dependent variable. For example, the
new coefficient on educ is .038 while the new coefficient on kidslt6 is .262.
(ii) The standard e
ors will not change. In the case of the slopes, changing the signs of the
estimators does not change their variances, and therefore the standard e
ors are unchanged (but
the t statistics change sign). Also, Var(1
0̂ ) = Var( 0̂ ), so the standard e
or of the intercept
is the same as before.
(iii) We know that changing the units of measurement of independent variables, or entering
qualitative information using different sets of dummy variables, does not change the R-squared.
But here we are changing the dependent variable. Nevertheless, the R-squareds from the
egressions are still the same. To see this, part (i) suggests that the squared residuals will be
identical in the two regressions. For each i the e
or in the equation for outlfi is just the negative
of the e
or in the other equation for inlfi, and the same is true of the residuals. Therefore, the
SSRs are the same. Further, in this case, the total sum of squares are the same. For outlf we
have
SST = 2 2
1 1
( ) [(1 ) (1 )]
n n
i i
i i
outlf outlf inlf inlf
= 2 2
1 1
( ) ( )
n n
i i
i i
inlf inlf inlf inlf
,
which is the SST for inlf. Because R
2
= 1 – SSR/SST, the R-squared is the same in the two
egressions.
7.8 (i) We want to have a constant semi-elasticity model, so a standard wage equation with
marijuana usage included would be
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log(wage) = 0 + 1 usage + 2 educ + 3 exper + 4 expe
2
+ 5 female + u.
Then 100 1 is the approximate percentage change in wage when marijuana usage increases by
one time per month.
(ii) We would add an interaction term in female and usage:
log(wage) = 0 + 1 usage + 2 educ + 3 exper + 4 expe
2
+ 5 female
+ 6 female usage + u.
The null hypothesis that the effect of marijuana usage does not differ by gender is H0: 6 = 0.
(iii) We take the base group to be nonuser. Then we need dummy variables for the other
three groups: lghtuser, moduser, and hvyuser. Assuming no interactive effect with gender, the
model would be
log(wage) = 0 + 1 lghtuser + 2 moduser + 3 hvyuser + 2 educ + 3 exper
+ 4 expe
2
+ 5 female + u.
(iv) The null hypothesis is H0: 1 = 0, 2 = 0, 3 = 0, for a total of q = 3 restrictions. If n is
the sample size, the df in the unrestricted model – the denominator df in the F distribution – is
n – 8. So we would obtain the critical value from the Fq,n-8 distribution.
(v) The e
or term could contain factors, such as family background (including parental
history of drug abuse) that could directly affect wages and also be co
elated with marijuana
usage. We are interested in the effects of a person’s drug usage on his or her wage, so we would
like to hold other confounding factors fixed. We could try to collect data on relevant background
information.
7.9 (i) Plugging in u = 0 and d = 1 gives 1 0 0 1 1( ) ( ) ( )f z z .
(ii) Setting * *0 1( ) ( )f z f z gives
* *
0 1 0 0 1 1( ) ( )z z or
*
0 10 z .
Therefore, provided 1 0 , we have
*
0 1/z . Clearly,
*z is positive if and only if 0 1/ is
negative, which means 0 1 and must have opposite signs.
(iii) Using part (ii) we have * .357 / .030 11.9totcoll years.
(iv) The estimated years of college where women catch up to men is much too high to be
practically relevant. While the estimated coefficient on female totcoll shows...