Universität Hamburg
D. Örsal, M. Paetz,
E. Offner, V. van Rienen
Angewandte Ökonometrie II
Wooldridge (2009)
Vorlesung: Prof. Dr. D. Örsal
Übung: Dr. M. Paetz, E. Offner, V. van Rienen
Problems
8.1 Which of the following are consequences of heteroskedasticity?
(i) The OLS estimators, β̂j , are inconsistent.
(ii) The usual F statistic no longer has an F distribution.
(iii) The OLS estimators are no longer BLUE.
8.2 Consider a linear model to explain monthly beer consumption:
beer = β0 + β1 inc + β2 price + β3 educ + β4 female + u
E(u|inc, price, educ, female) = 0
Var(u|inc, price, educ, female) = σ 2 inc2 ).
Write the transformed equation that has a homoskedastic error term.
8.3 True or False: WLS is preferred to OLS, when an important variable has been
omitted from the model.
8.4 Using the data in GPA3.DTA, the following equation was estimated for the fall
and second semester students:
\ = −2.12 + .900 crsgpa + .193 cumgpa + .0014 tothrs
trmgpa
(.55) (.175)
(.064)
(.0012)
[.55] [.166]
[.074]
[.0012]
+.0018 sat − .0039 hsperc + .351 female − .157 season
(.0002)
(.0018)
(.085)
(.098)
[.0002]
[.0019]
[.079]
[.080]
n = 269, R2 = .465.
Here, trmgpa is term GPA, crsgpa is a weighted average of overall GPA in
courses taken, cumgpa is GPA prior to the current semester, tothrs is total credit
hours prior to the semester, sat is SAT score, hsperc is graduating percentile in
high school class, female is a gender dummy, and season is a dummy variable
equal to unity if the student’s sport is in season during the fall. The usual and
heteroskedasticity-robust standard errors are reported in parentheses and brackets,
respectively.
(i) Do the variables crsgpa, cumgpa, and tothrs have the expected estimated
effects? Which of these variables are statistically significant at the 5% level?
Does it matter which standard errors are used?
(ii) Why does the hypothesis H0 : βcrsgpa = 1 make sense? Test this hypothesis
against the two-sided alternative at the 5% level, using both standard errors.
Describe your conclusions.
Problem Set: Chapter 8
1 / 18
Universität Hamburg
D. Örsal, M. Paetz,
E. Offner, V. van Rienen
Angewandte Ökonometrie II
Wooldridge (2009)
(iii) Test whether there is an in-season effect on term GPA, using both standard
errors. Does the significance level at which the null can be rejected depend
on the standard error used?
8.7 Consider a model at the employee level,
yi,e = β0 + β1 xi,e,1 + β2 xi,e,2 + . . . + βk xi,e,k + fi + υi,e ,
where the unobserved variable fi is a “firm effect” to each employee at a given firm
i. The error term υi,e is a specific to employee e at firm i. The composite error is
ui,e = fi + υi,e , such as in equation (8.28).
(i) Assume that Var(fi ) = σf2 , Var(υi,e ) = συ2 , and fi and υi,e are uncorrelated.
Show that Var(u, i) = σf2 + συ2 ; call this σ 2 .
(ii) Now suppose that for e ̸= g, υi,e and υi,g are uncorrelated.
Show that Cov(ui,e , ui,g ) = σf2 .
mi
(iii) Let ūi = m−1
i
ê=1 ui,e be the average of the composite errors within a firm.
Show that Var(ūi ) = σf2 + συ2 /mi .
P
(iv) Discuss the relevance of part (iii) for WLS estimation using data averaged at
the firm level, where the weight used for observation i is the usual firm size.
Computer Exercises
C8.2
(i) Use the data in HPRICE1.DTA to obtain the heteroskedasticity-robust standard errors for equation (8.17). Discuss any important differences with the
usual standard errors.
(ii) Repeat part (i) for equation (8.18).
(iii) What does this example suggest about heteroskedasticity and the transformation used for the dependent variable?
C8.3 Apply the full White test for heteroskedasticity [see equation (8.19)] to equation
(8.18). Using the chi-square form of the statistic, obtain the p-value. What do
you conclude?
C8.4 Use VOTE1.DTA for this exercise.
(i) Estimate a model with voteA as the dependent variable and prtystrA,
democA, log(expendA), and log(expendB) as independent variables. Obtain
the OLS residuals, ûi , and regress these on all of the independent variables.
Explain why you obtain R2 = 0.
(ii) Now compute the Breusch-Pagan test for heteroskedasticity. Use the F statistic version and report the p-value.
(iii) Compute the special case of the White test for heteroskedasticity, again using
the F statistic form. How strong is the evidence for heteroskedasticity now?
Problem Set: Chapter 8
2 / 18
Universität Hamburg
D. Örsal, M. Paetz,
E. Offner, V. van Rienen
Angewandte Ökonometrie II
Wooldridge (2009)
Problems
9.1 In Problem 4.11, the R-squared from estimating the model
log(salary) = β0 + β1 log(sales) + β2 log(mktval) + β3 profmarg
+β4 ceoten + β5 comten + u,
using the data in CEOSAL2.DTA, is R2 = .353 (n = 177). When ceoten2 and
comten2 are added, R2 = .375. Is there evidence of functional form misspecification
in this model?
9.2 Let us modify Exercise C8.4 by using voting outcomes in 1990 for incumbents
who were elected in 1988. Candidate A was elected in 1988 and was seeking
reelection in 1990; voteA90 is Candidate A’s share of the two-party vote in 1990.
The 1988 voting share of Candidate A is used as a proxy variable for quality of the
candidate. All other variables are for the 1990 election. The following equations
were estimated, using the data in VOTE2.DTA:
\ = 75.71 + .312 prtystrA + 4.93 democA
voteA90
(9.25) (.046)
(1.01)
−.929 log(expendA) − 1.950 log(expendB)
(.684)
(0.281)
n = 186, R2 = .495, R̄2 = .483,
and
\ = 70.81 + .282 prtystrA + 4.52 democA
voteA90
(10.01) (.052)
(1.06)
−.839 log(expendA) − 1.846 log(expendB) + .067 voteA88
(.687)
(0.292)
(.053)
n = 186, R2 = .499, R̄2 = .485
(i) Interpret the coefficient on voteA88 and discuss its statistical significance.
(ii) Does adding voteA88 have much effect on the other coefficients?
9.3 Let math10 denote the percentage of students at a Michigan high school receiving a
passing score on a standardized math test (see also Example 4.2). We are interested
in estimating the effect of per student spending on math performance. A simple
model is
math10 = β0 + β1 log(expend) + β2 log(enroll) + β3 poverty + u,
where poverty is the percentage of students living in poverty.
(i) The variable lnchprg is the percentage of students eligible for the federally funded school lunch program. Why is this a sensible proxy variable
for poverty?
Problem Set: Chapter 9
3 / 18
Universität Hamburg
D. Örsal, M. Paetz,
E. Offner, V. van Rienen
Angewandte Ökonometrie II
Wooldridge (2009)
(ii) The table that follows contains OLS estimates, with and without lnchprg as
an explanatory variable.
Explain why the effect of expenditures on math10 is lower in column (2) than
in column (1). Is the effect in column (2) still statistically greater than zero?
(iii) Does it appear that pass rates are lower at larger schools, other factors being
equal? Explain.
(iv) Interpret the coefficient on lnchprg in column (2).
(v) What do you make of the substantial increase in R2 from column (1) to
column (2)?
9.4 The following equation explains weekly hours of television viewing by a child in
terms of the child’s age, mother’s education, father’s education, and number of
siblings:
tvhours∗ = β0 + β1 age + β2 age2 + β3 motheduc + β4 f atheduc + β5 sibs + u.
We are worried that tvhours∗ is measured with error in our survey. Let tvhours
denote the reported hours of television viewing per week:
tvhours = tvhours∗ + e.
The error u and the explanatory variables are independent of each other. The
error u and the measurement error e are uncorrelated.
(i) Assume that the measurement error e and the explanatory variables are independent. Are the OLS estimators consistent and unbiased? What is the
effect of the measurement error on the variances of the OLS estimators in this
case?
(ii) Discuss whether the assumption cov(e, X) = 0 is likely to hold.
Problem Set: Chapter 9
4 / 18
Universität Hamburg
D. Örsal, M. Paetz,
E. Offner, V. van Rienen
Angewandte Ökonometrie II
Wooldridge (2009)
(iii) Now assume that the measurement error e is correlated with motheduc. Are
the OLS estimators consistent and unbiased?
Now, consider the following model:
tvhours = β0 + β1 age + β2 age2 + β3 motheduc + β4 f atheduc∗ + β5 sibs + u.
The variable f atheduc∗ is the true education of the father. We are now worried
that—instead of the dependent variable—the reported education of the father
f atheduc is measured with error:
f atheduc = f atheduc∗ + v.
(iv) What do the classical errors-in-variables (CEV) assumptions require in this
application?
(v) Under the CEV assumptions, are the OLS estimators consistent and unbiased?
(vi) Do you think the CEV assumptions are likely to hold? Explain.
9.5 In Example 4.4, we estimated a model relating number of campus crimes to student
enrollment for a sample of colleges. The sample we used was not a random sample
of colleges in the United States, because many schools in 1992 did not report
campus crimes. Do you think that college failure to report crimes can be viewed
as exogenous sample selection? Explain.
Computer Exercises
C9.1
(i) Apply RESET from equation (9.3) to the model estimated in Computer Exercise C7.5. Is there evidence of functional from misspecification?
(ii) Compute a heteroskedasticity-robust form of RESET. Does your conclusion
from part (i) change?
C9.2 Use the data set WAGE2.DTA for this exercise.
(i) Use the variable KW W (the “knowledge of the world of work” test score)
as a proxy for ability in place of IQ in Example 9.3. What is the estimated
return to education in this case?
(ii) Now use IQ and KW W together as proxy variables. What happens to the
estimated return to education?
(iii) In part (ii), are IQ and KW W individually significant? Are they jointly
significant?
C9.4 Use the data for the year 1990 in INFMRT.DTA for this exercise. You have data
at the state level of all 51 states in the US. The table describes the variables.
Problem Set: Chapter 9
5 / 18
Universität Hamburg
D. Örsal, M. Paetz,
E. Offner, V. van Rienen
Variable
infmort
pcinc
physic
popul
Angewandte Ökonometrie II
Wooldridge (2009)
Definition
number of deaths within the first year per 1,000 live births
yearly income per capita (USD)
physicians per 100,000 members of the civilian population
population in thousands
You have the following estimation result:
\
inf
mort = 33.86 − 4.68 log(pcinc) + 4.15 log(physic)
(20.43) (2.60)
(1.51)
−.088 log(popul)
(.287)
n = 51, R2 = .139, R̄2 = .084,
(i) Interpret the coefficent on log(physic) and comment on its significance.
(ii) Plot infmort and log(physic) for the year 1990 (Hint: scatter y x if...). What
do you notice? How does this help to explain the results of (i)?
(iii) Reestimate the model without the District of Columbia (called DC). How
does the estimated coefficient on log(physic) change?
(iv) Reestimate the model with the District of Columbia (called DC), but now
include a dummy variable for the observation on the District of Columbia.
Compare the result with (iii).
The objective function of the least absolute deviation (LAD) estimation method
is given by:
min
b0 ,b1 ,...,bk
n
X
|yi − b0 − b1 xi1 − ... − bk xik |
(1)
i=1
(v) How does the LAD objective function differ from the OLS objective function?
What does this imply for outliers?
(vi) Estimate the original model again i.e. with all 51 states and without a dummy
for DC, but apply the LAD estimation method (Stata: qreg y x1 x2...).
Report the results using the usual form.
(vii) Compare the coefficient on log(physic) with the OLS estimates.
(viii) Are there other reasons than outliers why OLS and LAD estimates might
substantially differ?
C9.5 Use the data in RDCHEM.RAW to further examine the effects of outliers on OLS
estimates and to see how LAD is less sensitive to outliers. The model is
rdintens = β0 + β1 sales + β2 sales2 + β3 prof marg + u,
where you should first change sales to be in billions of dollars to make the estimates
easier to interpret.
Problem Set: Chapter 9
6 / 18
Universität Hamburg
D. Örsal, M. Paetz,
E. Offner, V. van Rienen
Angewandte Ökonometrie II
Wooldridge (2009)
(i) Estimate the above equation by OLS, both with and without the firm having
annual sales of almost $40 billion. Discuss any notable differences in the
estimated coefficients.
(ii) Estimate the same equation by LAD, again with and without the largest firm.
Discuss any important differences in estimated coefficients.
(iii) Based on our findings in (i) and (ii), would you say OLS or LAD is more
resilient to outliers?
C9.7 Use the data in LOANAPP.DTA for this exercise.
(i) How many observations have obrat > 40, that is, other debt obligations more
than 40% of total income?
(ii) Reestimate the model in part (iii) of Exercise 7.16, excluding observations
with obrat > 40. What happens to the estimate and t statistic on white?
(iii) Does it appear that the estimate of βwhite is overly sensitive to the sample
used?
C9.11 Use the data in WAGE1.DTA for this exercise. You are interested in the effect of
education on wages. The model is
log(wage) = β0 + β1 educ + β2 exper + β3 tenure + β4 f emale + β5 married + u
(i) Estimate the model and present the results in the usual form. What is the
estimated impact of education on wages? Is the effect significant?
(ii) Assume that you only have data about workers with low wages. Estimate the
model again, but only with workers who earn less than 6$ per hour. What is
the estimated impact of education on wages? Is the effect significant?
(iii) Plot wage against education. Use the graph to explain the different results
from (i) and (ii).
Problem Set: Chapter 9
7 / 18
Universität Hamburg
D. Örsal, M. Paetz,
E. Offner, V. van Rienen
Angewandte Ökonometrie II
Wooldridge (2009)
Problems
10.1 Decide if you agree or disagree with each of the following statements and give a
brief explanation of your decision:
(i) Like cross-sectional observations, we can assume that most time series observations are independently distributed.
(ii) The OLS estimator in a time series regression is unbiased under the first three
Gauss-Markov assumptions.
(iii) A trending variable cannot be used as the dependent variable in multiple
regression analysis.
(iv) Seasonality is not an issue when using annual time series observations.
10.2 Let gGDPt denote the annual percentage change in gross domestic product and let
intt denote a short-term interest rate. Suppose that gGDPt is related to interest
rates by
gGDPt = α0 + δ0 intt + δ1 intt−1 + ut ,
where ut is uncorrelated with intt , intt−1 , and all other past values of interest rates.
Suppose that the Federal Reserve follows the policy rule:
intt = γ0 + γ1 (gGDPt−1 − 3) + υt ,
where γ1 > 0. (When last year’s GDP growth is above 3%, the Fed increases
interest rates to prevent an “overheated” economy.) If υt is uncorrelated with all
past values of intt and ut , argue that intt must be correlated with ut−1 . (Hint: Lag
the first equation for one time period and substitute for gGDPt−1 in the second
equation.) Which Gauss-Markov assumption does this violate?
10.3 Suppose yt follows a second order FDL model:
yt = α0 + δ0 zt + δ1 zt−1 + δ2 zt−2 + ut .
Let z ∗ denote the equilibrium value of zt and let y ∗ be the equilibrium value of yt ,
such that
y ∗ = α 0 + δ0 z ∗ + δ1 z ∗ + δ2 z ∗ .
Show that the change in y ∗ , due to a change in z ∗ , equals the long-run propensity
times the change in z ∗ :
∆y ∗ = LRP · ∆z ∗ .
This gives an alternative way of interpreting the LRP.
10.4 When the three event indicators befile6 , affile6 , and afdec6 are dropped from
equation (10.22), we obtain R2 = .281 and R̄2 = .264. Are the event indicators
jointly significant at the 10% level?
Problem Set: Chapter 10
8 / 18
Universität Hamburg
D. Örsal, M. Paetz,
E. Offner, V. van Rienen
Angewandte Ökonometrie II
Wooldridge (2009)
10.5 Suppose you have quarterly data on new housing starts, interest rates, and real
per capita income. Specify a model for housing starts that accounts for possible
trends and seasonality in the variables.
Computer Exercises
C10.2 Use the data in BARIUM.DTA for this exercise.
(i) Add a linear time trend to equation (10.22). Are any variables, other than
the trend, statistically significant?
(ii) In the equation estimated in part (i), test for joint significance of all variables
except the time trend. What do you conclude?
(iii) Add monthly dummy variables to this equation and test for seasonality. Does
including the monthly dummies change any other estimates or their standard
errors in important ways?
C10.6 Use the data in FERTIL3.DTA for this exercise.
(i) Regress gfr t on t and t2 and save the residuals. This gives a detrended gfr t ,
say g f̈t .
(ii) Regress g f̈t on all of the variables in equation (10.35), including t and t2 .
Compare the R-squared with that from (10.35). What do you conclude?
(iii) Reestimate equation (10.35) but add t3 to the equation. Is this additional
term statistically significant?
Problem Set: Chapter 10
9 / 18
Universität Hamburg
D. Örsal, M. Paetz,
E. Offner, V. van Rienen
Angewandte Ökonometrie II
Wooldridge (2009)
Problems
13.1 In example 13.1 assume that the averages of all factors other than educ have
remained constant over time and that the average level of education is 12.2 for
the 1972 sample and 13.3 in the 1984 sample. Using the estimates reported in the
table below, find the estimated change in average fertility between 1972 and 1984.
(Be sure to account for the intercept change and the change in average education.)
Problem Set: Chapter 13
10 / 18
Universität Hamburg
D. Örsal, M. Paetz,
E. Offner, V. van Rienen
Angewandte Ökonometrie II
Wooldridge (2009)
13.2 Using the data in KIELMC.DTA, the following equations were estimated using
the years 1978 and 1981:
log\
(price) = 11.49 − .547 nearinc + .394 y81 · nearinc
(.26) (.058)
(.080)
n = 321, R2 = .220
and
log\
(price) = 11.18 + .563 y81 − .403 y81 · nearinc
(.27) (.044)
(.067)
n = 321, R2 = .337
Compare the estimates on the interaction term y81 · nearinc with those from
equation (13.9). Why are the estimates so different?
Equation 13.9:
log\
(price) = 11.29 + .457 y81 − .340 nearinc − .063 y81 · nearinc
(.31) (.045)
(.055)
(.083)
n = 321, R2 = .409
13.3 Why can we not use first differences when we have independent cross sections in
two years (as opposed to panel data)?
13.6 Suppose that in 2003 a school district with two high schools was given a sharp
increase in funding. Call the high schools Eastern and Western. The district
decided that the funding would go to Western, mostly to hire more teachers to
reduce class sizes. To evaluate the effectiveness of the increased spending, the
district collected random samples of seniors in 2002 and 2004 at both high schools.
The measure of student performance is a compulsory standardized test score given
to all seniors in the district.
(i) Let score be the test score. Without controlling for other factors, write down
a model that allows you to determine whether the additional funding led
to improved test scores. What is the treatment and control group in this
application? Be sure to state which coefficient measures the effect of the new
funding.
The following table gives the sample test score averages:
Western
Eastern
2002
60.0
55.0
2004
65.0
57.0
(ii) Use the table to write down the estimates of the model from (i). What is the
estimated impact of the additional funding on test scores?
(iii) Give a graphical representation of the estimated model.
Problem Set: Chapter 13
11 / 18
Universität Hamburg
D. Örsal, M. Paetz,
E. Offner, V. van Rienen
Angewandte Ökonometrie II
Wooldridge (2009)
(iv) How might estimating the model from (i) be biased if Western gets an influx
of better students because of the increased funding?
C13.3 Use the data in KIELMC.DTA for this exercise.
(i) The variable dist is the distance from each home to the incinerator site, in
feet. Consider the model
log(price) = β0 + δ0 y81 + β1 log(dist) + δ1 y81 · log(dist) + u
If building the indicator reduces the value of homes closer to the site, what
is the sign of δ1 ? What does it mean if β1 > 0?
(ii) Estimate the model from part (i) and report the results in the usual form.
Interpret the coefficient on y81 · log(dist). What do you conclude?
(iii) Add age, age2 , rooms, baths, log(intst), log(land), and log(area) to the
equation. Now, what do you conclude about the effect of the incinerator on
housing values?
(iv) How come the coefficient on log(dist) is positive and statistically significant
in part (ii) but not in part (iii)? What does it say about the controls used in
part (iii)?
C13.5 Use the data in RENTAL.DTA for this exercise. The data for the years 1980 and
1990 include rental prices and other variables for college towns. The idea is to
see whether a stronger presence of students affects rental rates. The unobserved
effects model is
log(rentit ) = β0 + δ0 y90t + β1 log(popit ) + β2 log(avgincit ) + β3 pctstuit + ai + uit ,
where pop is city population, avginc is average income and pctstu is student population as a percentage of city population (during the school year).
(i) Estimate the equation by pooled OLS and report the results in standard form.
What do you make of the estimate on the 1990 dummy variable? What do
you get for βbpctstu ?
(ii) Are the standard errors you report in part (i) valid? Explain.
(iii) Now, difference the equation and estimate by OLS. Compare your estimate of
βpctstu with that from part (ii). Does the relative size of the student population
appear to affect rental prices?
(iv) Obtain the heteroskedasticity-robust standard errors for the first-differenced
equation in part (iii). Does this change your conclusions?
C13.7 Use GPA3.DTA for this exercise. The data set is for 336 student-athletes from a
large university for fall and spring semesters. [A similar analysis is in Maloney and
McCormick (1993), but here we use a true panel data set.] Because you have two
terms of data for each student, an unobserved effects model is appropriate. The
Problem Set: Chapter 13
12 / 18
Universität Hamburg
D. Örsal, M. Paetz,
E. Offner, V. van Rienen
Angewandte Ökonometrie II
Wooldridge (2009)
primary question of interest is this: Do athletes perform more poorly in school
during the semester their sport is in season?
(i) Use pooled OLS to estimate a model with term GPA (trmgpa) as the dependent variable. the explanatoy variables are spring, sat, hsperc, f emale,
black, white, f rstsem, tothrs, crsgpa, and season. Interpret the coefficient
on season. Is it statistically significant?
(ii) Most of the athletes who play their their sport only in the fall are football
players. Suppose the ability levels of football players differ systematically
from those of other athletes. If ability is not adequately captured by SAT
score and high school percentile, explain why the pooled OLS estimator will
be biased.
(iii) Now, use the data differenced across the two terms. Which variables drop
out? Now, test for an in-season effect.
(iv) Can you think of one or more potentially important, time varying variables
that have been omitted from the analysis?
Problem Set: Chapter 13
13 / 18
Universität Hamburg
D. Örsal, M. Paetz,
E. Offner, V. van Rienen
Angewandte Ökonometrie II
Wooldridge (2009)
Problems
15.1 Consider a simple model to estimate the effect of personal computer (PC) ownership on college grade point average for graduating seniors at a large public university:
GP A = β0 + β1 P C + u
where PC is a binary variable indicating PC ownership.
(i) Why might PC ownership be correlated with u?
(ii) Explain why PC is likely to be related to parents’ annual income. Does this
mean parental income is a good IV for PC? Why or why not?
(iii) Suppose that, four years ago, the university gave grants to buy computers
to roughly one-half of the incoming students, and the students who received
grants were randomly chosen. Carefully explain how you would use this
information to construct an instrumental variable for PC.
15.2 Suppose that you wish to estimate the effect of class attendance on student performance, as in Example 6.3. A basic model is
stndf nl = β0 + β1 atndrte + β2 priGP A + β3 ACT + u
where the variables are defined as in Chapter 6.
(i) Let dist be the distance from the students’ living quarters to the lecture hall.
Do you think dist is uncorrelated with u?
(ii) Assuming that dist and u are uncorrelated, what other assumption must dist
satisfy to be a valid IV for atndrte?
(iii) Suppose, as in equation (6.18), we add the interaction term priGP A·atndrte:
stndf nl = β0 + β1 atndrte + β2 priGP A + β3 ACT + β4 + priGP A · atndrte + u
If atndrte is correlated with u, then, in general, so is priGP A · atndrte. What
might be a good IV for priGP A·atndrte? [Hint: If E(u|priGP A, ACT, dist) =
0, as happens when priGP A, ACT , and dist are all exogenous, then any
function of priGP A and dist is uncorrelated with u.]
15.3 Consider the simple regression model
y = β0 + β1 x + u
and let z be a binary instrumental variable for x. Use (15.10) to show that the IV
estimator βˆ1 can be written as
βˆ1 = (ȳ1 − ȳ0 )/(x̄1 − x̄0 ),
where ȳ0 and x̄0 are the sample averages of yi and xi over the part of the sample
Problem Set: Chapter 15
14 / 18
Universität Hamburg
D. Örsal, M. Paetz,
E. Offner, V. van Rienen
Angewandte Ökonometrie II
Wooldridge (2009)
with zi = 0, and where ȳ1 and x̄1 are the sample averages of yi and xi over the
part of the sample with zi = 1. This estimator, known as a grouping estimator,
was first suggested by Wald (1940).
15.4 Suppose that, for a given state in the United States, you wish to use annual
time series data to estimate the effect of the state-level minimum wage on the
employment of those 18 to 25 years old (EM P ). A simple model is
gEM Pt = β0 + β1 gM INt + β2 gP OPt + β3 gGSPt + β4 gGDPt + ut ,
where M INt is the minimum wage, in real dollars, P OPt is the population from
18 to 25 years old, GSPt is gross state product, and GDPt is U.S. gross domestic
product. The g prefix indicates the growth rate from year t − 1 to year t, which
would typically be approximated by the difference in the logs.
(i) If we are worried that the state chooses its minimum wage partly based on
unobserved (to us) factors that affect youth employment, what is the problem
with OLS estimation?
(ii) Let U SM INt be the U.S. minimum wage, which is also measured in real
terms. Do you think gU SM INt is uncorrelated with ut ?
(iii) By law, any state’s minimum wage must be at least as large as the U.S.
minimum. Explain why this makes gU SM INt a potential IV candidate for
gM INt .
15.7 The following is a simple model to measure the effect of a school choice program
on standardized test performance [see Rouse (1998) for motivation]:
score = β0 + β1 choice + β2 f aminc + u1 ,
where score is the score on a statewide test, choice is a binary variable indicating
whether a student attended a choice school in the last year, and f aminc is family
income. The IV for choice is grant, the dollar amount granted to students to use
for tuition at choice schools. The grant amount differed by family income level,
which is why we control for f aminc in the equation.
(i) Even with f aminc in the equation, why might choice be correlated with u1 ?
(ii) If within each income class, the grant amounts were assigned randomly, is
grant uncorrelated with u1 ?
(iii) Write the reduced form equation for choice. What is needed for grant to be
partially correlated with choice?
(iv) Write the reduced form equation for score. Explain why this is useful. (Hint:
How do you interpret the coefficient on grant?)
15.10 Evans and Schwab (1995) studied the effects of attending a Catholic high school on
the probability of attending college. For concreteness, let college be a binary variable equal to unity if a student attends college, and zero otherwise. Let CathHS
Problem Set: Chapter 15
15 / 18
Universität Hamburg
D. Örsal, M. Paetz,
E. Offner, V. van Rienen
Angewandte Ökonometrie II
Wooldridge (2009)
be a binary variable equal to one if the student attends a Catholic high school. A
linear probability model is
college = β0 + β1 CathHS + otherf actors + u,
where the other factors include gender, race, family income, and parental education.
(i) Why might CathHS be correlated with u?
(ii) Evans and Schwab have data on a standardized test score taken when each
student was a sophomore. What can be done with this variable to improve
the ceteris paribus estimate of attending a Catholic high school?
(iii) Let CathRel be a binary variable equal to one if the student is Catholic.
Discuss the two requirements needed for this to be a valid IV for CathHS in
the preceding equation. Which of these can be tested?
(iv) Not surprisingly, being Catholic has a significant effect on attending a
Catholic high school. Do you think CathRel is a convincing instrument
for CathHS?
Computer Exercises
C15.1 Use the data in WAGE2.RAW for this exercise.
(i) In Example 15.2, using sibs as an instrument for educ, the IV estimate of the
return to education is .122. To convince yourself that using sibs as an IV for
educ is not the same as just plugging sibs in for educ and running an OLS
regression, run the regression of log(wage) on sibs and explain your findings.
(ii) The variable brthord is birth order (brthord is one for a first-born child, two
for a second-born child, and so on). Explain why educ and brthord might be
negatively correlated. Regress educ on brthord to determine whether there is
a statistically significant negative correlation.
(iii) Use brthord as an IV for educ in equation (15.1). Report and interpret the
results.
(iv) Now, suppose that we include number of siblings as an explanatory variable
in the wage equation; this controls for family background, to some extent:
log(wage) = β0 + β1 educ + β2 sibs + u.
Suppose that we want to use brthord as an IV for educ, assuming that sibs
is exogenous. The reduced form for educ is
educ = π0 + π1 sibs + π2 brthord + v.
Problem Set: Chapter 15
16 / 18
Universität Hamburg
D. Örsal, M. Paetz,
E. Offner, V. van Rienen
Angewandte Ökonometrie II
Wooldridge (2009)
State and test the identification assumption.
(v) Estimate the equation from part (iv) using brthord as an IV for educ (and
sibs as its own IV). Comment on the standard errors for β̂educ and β̂sibs .
d compute the correlation between
(vi) Using the fitted values from part (iv), educ,
d and sibs. Use this result to explain your findings from part (v).
educ
C15.5 Use the data in CARD.RAW for this exercise.
(i) In Table 15.1, the difference between the IV and OLS estimates of the return
to education is economically important. Obtain the reduced form residuals,v̂2 ,
from the reduced form regression educ on nearc4, exper, exper2 , black, smsa,
south, smsa66, reg662, ..., reg669 — see Table 15.1. Use these to test whether
educ is exogenous; that is, determine if the difference between OLS and IV is
statistically significant.
(ii) Estimate the equation by 2SLS, adding nearc2 as an instrument. Does the
coefficient on educ change much?
(iii) Test the single overidentifying restriction from part (ii).
C15.6 Use the data in MURDER.RAW for this exercise. The variable mrdrte is the
murder rate, that is, the number of murders per 100,000 people. The variable exec
is the total number of prisoners executed for the current and prior two years; unem
is the state unemployment rate.
(i) How many states executed at least one prisoner in 1991, 1992, or 1993? Which
state had the most executions?
(ii) Using the two years 1990 and 1993, do a pooled regression of mrdrte on d93,
exec, and unem. What do you make of the coefficient on exec?
(iii) Using the changes from 1990 to 1993 only (for a total of 51 observations),
estimate the equation
∆mrdrte = δ0 + β1 ∆exec + β2 ∆unem + ∆u
by OLS and report the results in the usual form. Now, does capital punishment appear to have a deterrent effect?
(iv) The change in executions may be at least partly related to changes in the
expected murder rate, so that ∆exec is correlated with ∆u in part (iii). It
might be reasonable to assume that ∆exec−1 is uncorrelated with ∆u. (After
all, ∆exec−1 depends on executions that occurred three or more years ago.)
Regress ∆exec on ∆exec−1 to see if they are sufficiently correlated; interpret
the coefficient on ∆exec−1 .
(v) Reestimate the equation from part (iii), using ∆exec−1 as an IV for ∆exec.
Assume that ∆unem is exogenous. How do your conclusions change from
part (iii)?
Problem Set: Chapter 15
17 / 18
Universität Hamburg
D. Örsal, M. Paetz,
E. Offner, V. van Rienen
Angewandte Ökonometrie II
Wooldridge 2009
References
Wooldridge, Jeffrey (2009). Introductory Econometrics: A Modern Approach. 4th ed.
South-Western College Pub.