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dicted when y 0 is 80, and the percent correctly predicted when y 1
is 40. Find the overall percent correctly predicted.
17.2 Let grad be a dummy variable for whether a student-athlete at a large university
graduates in five years. Let hsGPA and SAT be high school grade point average and SAT
score. Let study be the number of hours spent per week in organized study hall. Suppose
that, using data on 420 student-athletes, the following logit model is obtained:
P(grad 1
ˆ
兩hsGPA,SAT,study) (1.17 .24 hsGPA .00058 SAT .073
study),
where (z) exp(z)/[1 exp(z)] is the logit function. Holding hsGPA fixed at 3.0 and
SAT fixed at 1,200, compute the estimated difference in the graduation probability for
someone who spent 10 hours per week in study hall and someone who spent five hours
per week.
17.3 (Requires calculus) (i) Suppose in the Tobit model that x
1
log(z
1
), and this is the
only place z
1
appears in x. Show that
(
1
/z
1
){1
(x

/
)[x

/
(x

/
)]}, (17.49)
where
1
is the coefficient on log(z
1
).
(ii) If x
1
z
1
, and x
2
z
1
2
, show that
(
1
2
2
z
1
){1
(x

/
)[x

/
(x

/
)]},
where
1
is the coefficient on z
1
, and
2
is the coefficient on z
1
2
.
17.4 Let mvp
i
be the marginal value product for worker i, which is the price of a firm’s
good multiplied by the marginal product of the worker. Assume that
log(mvp
i
)
0
1
x
i1
…
k
x
ik
u
i
wage
i
max(mvp
i
,minwage
i
),
where the explanatory variables include education, experience, and so on, and minwage
i
is the minimum wage relevant for person i. Write log(wage
i
) in terms of log(mvp
i
) and
log(minwage
i
).
17.5 (Requires calculus) Let patents be the number of patents applied for by a firm
during a given year. Assume that the conditional expectation of patents given sales and
RD is
E(patents兩sales,RD) exp[
0
1
log(sales)
2
RD
3
RD
2
],
where sales is annual firm sales, and RD is total spending on research and development
over the past 10 years.
(i) How would you estimate the
j
? Justify your answer by discussing the
nature of patents.
E(y兩y 0,x)
z
1
E(y兩y 0,x)
z
1
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(ii) How do you interpret
1
?
(iii) Find the partial effect of RD on E(patents兩sales,RD).
17.6 Consider a family saving function for the population of all families in the United
States:
sav
0
1
inc
2
hhsize
3
educ
4
age u,
where hhsize is household size, educ is years of education of the household head, and
age is age of the household head. Assume that E(u兩inc,hhsize,educ,age) 0.
(i) Suppose that the sample includes only families whose head is over 25
years old. If we use OLS on such a sample, do we get unbiased estima-
tors of the
j
? Explain.
(ii) Now suppose our sample includes only married couples without chil-
dren. Can we estimate all of the parameters in the saving equation?
Which ones can we estimate?
(iii) Suppose we exclude from our sample families that save more than
$25,000 per year. Does OLS produce consistent estimators of the
j
?
17.7 Suppose you are hired by a university to study the factors that determine whether
students admitted to the university actually come to the university. You are given a large
random sample of students who were admitted the previous year. You have information
on whether each student chose to attend, high school performance, family income, finan-
cial aid offered, race, and geographic variables. Someone says to you, “Any analysis of
that data will lead to biased results because it is not a random sample of all college appli-
cants, but only those who apply to this university.” What do you think of this criticism?
COMPUTER EXERCISES
17.8 Use the data in PNTSPRD.RAW for this exercise.
(i) The variable favwin is a binary variable if the team favored by the Las
Vegas point spread wins. A linear probability model to estimate the
probability that the favored team wins is
P( favwin 1兩spread)
0
1
spread.
Explain why, if the spread incorporates all relevant information, we
expect
0
.5.
(ii) Estimate the model from part (i) by OLS. Test H
0
:
0
.5 against a
two-sided alternative. Use both the usual and heteroskedasticity-robust
standard errors.
(iii) Is spread statistically significant? What is the estimated probability that
the favored team wins when spread 10?
(iv) Now, estimate a probit model for P( favwin 1兩spread). Interpret and
test the null hypothesis that the intercept is zero. [Hint: Remember that
(0) .5.]
(v) Use the probit model to estimate the probability that the favored team
wins when spread 10. Compare this with the LPM estimate from part
(iii).
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(vi) Add the variables favhome, fav25, and und25 to the probit model and
test joint significance of these variables using the likelihood ratio test.
(How many df are in the chi-square distribution?) Interpret this result,
focusing on the question of whether the spread incorporates all observ-
able information prior to a game.
17.9 Use the data in LOANAPP.RAW for this exercise; see also Problem 7.16.
(i) Estimate a probit model of approve on white. Find the estimated prob-
ability of loan approval for both whites and nonwhites. How do these
compare with the linear probability estimates?
(ii) Now, add the variables hrat, obrat, loanprc, unem, male, married, dep,
sch, cosign, chist, pubrec, mortlat1, mortlat2, and vr to the probit
model. Is there statistically significant evidence of discrimination
against nonwhites?
(iii) Estimate the model from part (ii) by logit. Compare the coefficient on
white to the probit estimate.
(iv) How would you compare the size of the discrimination effect between
probit and logit?
17.10 Use the data in FRINGE.RAW for this exercise.
(i) For what percentage of the workers in the sample is pension equal to
zero? What is the range of pension for workers with nonzero pension
benefits? Why is a Tobit model appropriate for modeling pension?
(ii) Estimate a Tobit model explaining pension in terms of exper, age,
tenure, educ, depends, married, white, and male. Do whites and males
have statistically significant higher expected pension benefits?
(iii) Use the results from part (ii) to estimate the difference in expected pen-
sion benefits for a white male and a nonwhite female, both of whom are
35 years old, single with no dependents, have 16 years of education, and
10 years of experience.
(iv) Add union to the Tobit model and comment on its significance.
(v) Apply the Tobit model from part (iv) but with peratio, the pension-
earnings ratio, as the dependent variable. (Notice that this is a fraction
between zero and one, but, while it often takes on the value zero, it never
gets close to being unity. Thus, a Tobit model is fine as an approxima-
tion.) Does gender or race have an effect on the pension-earnings ratio?
17.11 In Example 9.1, we added the quadratic terms pcnv
2
, ptime86
2
, and inc86
2
to a
linear model for narr86.
(i) Use the data in CRIME1.RAW to add these same terms to the Poisson
regression in Example 17.3.
(ii) Compute the estimate of
2
given by
ˆ
2
(n k 1)
1
兺
n
i1
u
ˆ
i
2
/y
ˆ
i
. Is
there evidence of overdispersion? How should the Poisson MLE stan-
dard errors be adjusted?
(iii) Use the results from parts (i) and (ii) and Table 17.3 to compute the
quasi-likelihood ratio statistic for joint significance of the three qua-
dratic terms. What do you conclude?
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17.12 Refer to Table 13.1 in Chapter 13. There, we used the data in FERTIL1.RAW to
estimate a linear model for kids, the number of children ever born to a woman.
(i) Estimate a Poisson regression model for kids, using the same variables
in Table 13.1. Interpret the coefficient on y82.
(ii) What is the estimated percentage difference in fertility between a black
woman and a nonblack woman, holding other factors fixed?
(iii) Obtain
ˆ
. Is there evidence of over- or underdispersion?
(iv) Compute the fitted values from the Poisson regression and the
R-squared as the squared correlation between kids
i
and ki
ˆ
ds
i
. Compare
this with the R-squared for the linear regression model.
17.13 Use the data in RECID.RAW to estimate the model from Example 17.4 by OLS,
using only the 552 uncensored durations. Comment generally on how these estimates
compare with those in Table 17.4.
17.14 Use the MROZ.RAW data for this exercise.
(i) Using the 428 women who were in the work force, estimate the return
to education by OLS including exper, exper
2
, nwifeinc, age, kidslt6, and
kidsge6 as explanatory variables. Report your estimate on educ and its
standard error.
(ii) Now estimate the return to education by Heckit, where all exogenous
variables show up in the second-stage regression. In other words, the
regression is log(wage) on educ, exper, exper
2
, nwifeinc, age, kidslt6,
kidsge6, and
ˆ
. Compare the estimated return to education and its stan-
dard error to that from part (i).
(iii) Using only the 428 observations for working women, regress
ˆ
on educ,
exper, exper
2
, nwifeinc, age, kidslt6, and kidsge6. How big is the
R-squared? How does this help explain your findings from part (ii)?
(Hint: Think multicollinearity.)
APPENDIX 17A
Asymptotic Standard Errors in Limited Dependent
Variable Models
Derivations of the asymptotic standard errors for the models and methods introduced in
this chapter are well beyond the scope of this text. Not only do the derivations require
matrix algebra, but they also require advanced asymptotic theory of nonlinear estima-
tion. The background needed for a careful analysis of these methods and several deriva-
tions are given in Wooldridge (1999).
It is instructive to see the formulas for obtaining the asymptotic standard errors for
at least some of the methods. Given the binary response model P(y 1兩x) G(x

),
where G() is the logit or probit function, and

is the k 1 vector of parameters, the
asymptotic variance matrix of

ˆ
is estimated as
Chapter 17 Limited Dependent Variable Models and Sample Selection Corrections
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Ava
ˆ
r(

ˆ
) ⬅
冸
兺
n
i1
冹
1
, (17.50)
which is a k k matrix. (See Appendix D for a summary of matrix algebra.) Without
the terms involving g() and G(), this formula looks a lot like the estimated variance
matrix for the OLS estimator, minus the term
ˆ
2
. The expression in (17.50) accounts
for the nonlinear nature of the response probability—that is, the nonlinear nature of
G()—as well as the particular form of heteroskedasticity in a binary response mode:
Var(y兩x) G(x

)[1 G(x

)].
The square roots of the diagonal elements of (17.50) are the asymptotic standard
errors of the
ˆ
j
, and they are routinely reported by econometrics software that supports
logit and probit analysis. Once we have these, (asymptotic) t statistics and confidence
intervals are obtained in the usual ways.
The matrix in (17.50) is also the basis for Wald tests of multiple restrictions on

[see Wooldridge (1999, Chapter 15)].
The asymptotic variance matrix for Tobit is more complicated but has a similar
structure. Note that we can obtain a standard error for
ˆ
as well. The asymptotic vari-
ance for Poisson regression, allowing for
2
1 in (17.32), has a form much like
(17.50):
Ava
ˆ
r(

ˆ
)
ˆ
2
冸
兺
n
i1
exp(x
i

ˆ
)x
i
ⴕx
i
冹
1
.
The square roots of the diagonal elements of this matrix are the asymptotic standard
errors. If the Poisson assumption holds, we can drop
ˆ
2
from the formula (because
2
1).
Asymptotic standard errors for censored regression, truncated regression, and the
Heckit sample selection correction are more complicated, although they share features
with the previous formulas. See Wooldridge (1999) for details.
[g(x
i

ˆ
)]
2
x
i
ⴕx
i
G(x
i

ˆ
)[1 G(x
i

ˆ
)]
Part 3 Advanced Topics
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I
n this chapter, we cover some more advanced topics in time series econometrics. In
Chapters 10, 11, and 12, we emphasized in several places that using time series data
in regression analysis requires some care due to the trending, persistent nature of
many economic time series. In addition to studying topics such as infinite distributed
lag models and forecasting, we also discuss some recent advances in analyzing time
series processes with unit roots.
In Section 18.1, we describe infinite distributed lag models, which allow a change
in an explanatory variable to affect all future values of the dependent variable.
Conceptually, these models are straightforward extensions of the finite distributed lag
models in Chapter 10; but estimating these models poses some interesting challenges.
In Section 18.2, we show how to formally test for unit roots in a time series process.
Recall from Chapter 11 that we excluded unit root processes to apply the usual asymp-
totic theory. Because the presence of a unit root implies that a shock today has a long-
lasting impact, determining whether a process has a unit root is of interest in its own
right.
We cover the notion of spurious regression between two time series processes, each
of which has a unit root, in Section 18.3. The main result is that even if two unit root
series are independent, it is quite likely that the regression of one on the other will yield
a statistically significant t statistic. This emphasizes the potentially serious conse-
quences of using standard inference when the dependent and independent variables are
integrated processes.
The issue of cointegration applies when two series are I(1), but a linear combina-
tion of them is I(0); in this case, the regression of one on the other is not spurious, but
instead tells us something about the long-run relationship between them. Cointegration
between two series also implies a particular kind of model, called an error correction
model, for the short-term dynamics. We cover these models in Section 18.4.
In Section 18.5, we provide an overview of forecasting and bring together all of the
tools in this and previous chapters to show how regression methods can be used to fore-
cast future outcomes of a time series. The forecasting literature is vast, so we focus only
on the most common regression-based methods. We also touch on the related topic of
Granger causality.
571
Chapter Eighteen
Advanced Time Series Topics
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18.1 INFINITE DISTRIBUTED LAG MODELS
Let {(y
t
,z
t
): t …,2,1,0,1,2,…} be a bivariate time series process (which is only
partially observed). An infinite distributed lag (IDL) model relating y
t
to current and
all past values of z is
y
t
0
z
t
1
z
t1
2
z
t2
… u
t
, (18.1)
where the sum on lagged z extends back to the indefinite past. This model is only an
approximation to reality, as no economic process started infinitely far into the past.
Compared with a finite distributed lag model, an IDL model does not require that we
truncate the lag at a particular value.
In order for model (18.1) to make sense, the lag coefficients,
j
, must tend to zero
as j
*
. This is not to say that
2
is smaller in magnitude than
1
; it only means that
the impact of z
tj
on y
t
must eventually become small as j gets large. In most applica-
tions, this makes economic sense as well: the distant past of z should be less important
for explaining y than the recent past of z.
Even if we decide that (18.1) is a useful model, we clearly cannot estimate it with-
out some restrictions. For one, we only observe a finite history of data. Equation (18.1)
involves an infinite number of parameters,
0
,
1
,
2
, …, which cannot be estimated with-
out restrictions. Later, we place restrictions on the
j
that allow us to estimate (18.1).
As with finite distributed lag models, the impact propensity in (18.1) is simply
0
(see Chapter 10). Generally, the
h
have the same interpretation as in an FDL. Suppose
that z
s
0 for all s 0 and that z
0
1 and z
s
0 for all s 1; in other words, at time
t 0, z increases temporarily by one unit and then reverts to its initial level of zero. For
any h 0, we have y
h
h
u
h
for all h 0, and so
E(y
h
)
h
, (18.2)
where we use the standard assumption that u
h
has zero mean. It follows that
h
is the
change in E(y
h
), given a one-unit, temporary change in z at time zero. We just said that
h
must be tending to zero as h gets large for the IDL to make sense. This means that a
temporary change in z has no long-run effect on expected y:E(y
h
)
h
*
as
h * .
We assumed that the process z starts at z
s
0 and that the one-unit increase
occurred at t 0. These were only for the purpose of illustration. More generally, if z
temporarily increases by one unit (from any initial level) at time t, then
h
measures the
change in the expected value of y after h periods. The lag distribution, which is
h
plot-
ted as a function of h, shows the expected path that future y follow given the one-unit,
temporary increase in z.
The long run propensity in model (18.1) is the sum of all of the lag coefficients:
LRP
0
1
2
3
…, (18.3)
where we assume that the infinite sum is well-defined. Because the
j
must converge
to zero, the LRP can often be well-approximated by a finite sum of the form
0
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1
…
p
for sufficiently large p. To interpret the LRP, suppose that the process z
t
is steady at z
s
0 for s 0. At t 0, the process permanently increases by one unit.
For example, if z
t
is the percentage change in the money supply and y
t
is the inflation
rate, then we are interested in the effects of a permanent increase of one percentage
point in money supply growth. Then, by substituting z
s
0 for s 0 and z
t
1 for
t 0, we have
y
h
0
1
…
h
u
h
,
where h 0 is any horizon. Because u
t
has a zero mean for all t, we have
E(y
h
)
0
1
…
h
. (18.4)
[It is useful to compare (18.4) and (18.2).] As the horizon increases, that is, as h * ,
the right-hand side of (18.4) is, by definition, the long run propensity. Thus, the LRP
measures the long-run change in the ex-
pected value of y given a one-unit, perma-
nent increase in z.
The previous derivation of the LRP, and
the interpretation of
j
, used the fact that
the errors have a zero mean; as usual, this is
not much of an assumption, provided an
intercept is included in the model. A closer examination of our reasoning shows that we
assumed that the change in z during any time period had no effect on the expected value
of u
t
. This is the infinite distributed lag version of the strict exogeneity assumption that
we introduced in Chapter 10 (in particular, Assumption TS.2). Formally,
E(u
t
兩…,z
t2
,z
t1
,z
t
,z
t1
,…) 0, (18.5)
so that the expected value of u
t
does not depend on the z in any time period. While
(18.5) is natural for some applications, it rules out other important possibilities. In
effect, (18.5) does not allow feedback from y
t
to future z because z
th
must be uncor-
related with u
t
for h 0. In the inflation/money supply growth example, where y
t
is
inflation and z
t
is money supply growth, (18.5) rules out future changes in money sup-
ply growth that are tied to changes in today’s inflation rate. Given that money supply
policy often attempts to keep interest rates and inflation at certain levels, this might be
unrealistic.
One approach to estimating the
j
, which we cover in the next subsection, requires
a strict exogeneity assumption in order to produce consistent estimators of the
j
. A
weaker assumption is
E(u
t
兩z
t
,z
t1
,…) 0. (18.6)
Under (18.6), the error is uncorrelated with current and past z, but it may be correlated
with future z; this allows z
t
to be a variable that follows policy rules that depend on
past y. Sometimes, (18.6) is sufficient to estimate the
j
; we explain this in the next
subsection.
Chapter 18 Advanced Time Series Topics
573
QUESTION 18.1
Suppose that z
s
0 for s 0 and that z
0
1, z
1
1, and z
s
0
for s 1. Find E(y
1
), E(y
0
), and E(y
h
) for h 1. What happens as
h * ?
d 7/14/99 8:36 PM Page 573
One thing to remember is that neither (18.5) nor (18.6) says anything about the ser-
ial correlation properties of {u
t
}. (This is just as in finite distributed lag models.) If any-
thing, we might expect the {u
t
} to be serially correlated because (18.1) is not generally
dynamically complete in the sense discussed in Section 11.4. We will study the serial
correlation problem later.
How do we interpret the lag coefficients and the LRP if (18.6) holds but (18.5) does
not? The answer is: the same way as before. We can still do the previous thought (or
counterfactual) experiment, even though the data we observe are generated by some
feedback between y
t
and future z. For example, we can certainly ask about the long-run
effect of a permanent increase in money supply growth on inflation, even though the
data on money supply growth cannot be characterized as strictly exogenous.
The Geometric (or Koyck) Distributed Lag
Because there are generally an infinite number of
j
, we cannot consistently estimate
them without some restrictions. The simplest version of (18.1), which still makes the
model depend on an infinite number of lags, is the geometric (or Koyck) distributed
lag. In this model, the
j
depend on only two parameters:
j
j
, 兩
兩 1, j 0,1,2, …. (18.7)
The parameters
and
may be positive or negative, but
must be less than one in
absolute value. This ensures that
j
* 0 as j * 0. In fact, this convergence happens at
a very fast rate. (For example, with
.5 and j 10,
j
1/1024 .001.)
The impact propensity in the GDL is simply
0
, and so the sign of the IP is
determined by the sign of
. If
0, say, and
0, then all lag coefficients are pos-
itive. If
0, the lag coefficients alternate in sign (
j
is negative for odd j). The long
run propensity is more difficult to obtain, but we can use a standard result on the sum
of a geometric series: for 兩
兩 1, 1
2
…
j
… 1/(1
), and so
LRP
/(1
).
The LRP has the same sign as
.
If we plug (18.7) into (18.1), we still have a model that depends on the z back to the
indefinite past. Nevertheless, a simple subtraction yields an estimable model. Write the
IDL at times t and t 1 as:
y
t
z
t
z
t1
2
z
t2
… u
t
(18.8)
and
y
t1
z
t1
z
t2
2
z
t3
… u
t1
. (18.9)
If we multiply the second equation by
and subtract it from the first, all but a few of
the terms cancel:
y
t
y
t1
(1
)
z
t
u
t
u
t1
,
Part 3 Advanced Topics
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which we can write as
y
t
0
z
t
y
t1
u
t
u
t1
, (18.10)
where
0
(1
)
. This equation looks like a standard model with a lagged depen-
dent variable, where z
t
appears contemporaneously. Because
is the coefficient on z
t
and
is the coefficient on y
t1
, it appears that we can estimate these parameters. [If, for
some reason, we are interested in
, we can always obtain
ˆ
ˆ
0
/(1
ˆ
) after esti-
mating
and
0
.]
The simplicity of (18.10) is somewhat misleading. The error term in this equation,
u
t
u
t1
, is generally correlated with y
t1
. From (18.9), it is pretty clear that u
t1
and
y
t1
are correlated. Therefore, if we write (18.10) as
y
t
0
z
t
y
t1
v
t
, (18.11)
where v
t
⬅ u
t
u
t1
, then we generally have correlation between v
t
and y
t1
. Without
further assumptions, OLS estimation of (18.11) produces inconsistent estimates of
and
.
One case where v
t
must be correlated with y
t1
occurs when u
t
is independent of z
t
and all past values of z and y. Then, (18.8) is dynamically complete, and u
t
is uncorre-
lated with y
t1
. From (18.9), the covariance between v
t
and y
t1
is
Var(u
t1
)
u
2
, which is zero only if
0. We can easily see that v
t
is serially correlated:
because {u
t
} is serially uncorrelated, E(v
t
v
t1
) E(u
t
u
t1
)
E(u
t
2
1
)
E(u
t
u
t2
)
2
E(u
t1
u
t2
)
u
2
. For j 1, E(v
t
v
tj
) 0. Thus, {v
t
} is a moving average process
of order one (see Section 11.1). This gives an example of a model—which is derived
from the original model of interest—that has a lagged dependent variable and a partic-
ular kind of serial correlation.
If we make the strict exogeneity assumption (18.5), then z
t
is uncorrelated with u
t
and u
t1
, and therefore with v
t
. Thus, if we can find a suitable instrumental variable for
y
t1
, then we can estimate (18.11) by IV. What is a good IV candidate for y
t1
? By
assumption, u
t
and u
t1
are both uncorrelated with z
t1
, and so v
t
is uncorrelated with
z
t1
. If
0, z
t1
and y
t1
are correlated, even after partialling out z
t
. Therefore, we
can use instruments (z
t
,z
t1
) to estimate (18.11). Generally, the standard errors need to
be adjusted for serial correlation in the {v
t
}, as we discussed in Section 15.7.
An alternative to IV estimation exploits the fact that {u
t
} may contain a specific
kind of serial correlation. In particular, in addition to (18.6), suppose that {u
t
} follows
the AR(1) model
u
t
u
t1
e
t
(18.12)
E(e
t
兩z
t
,y
t1
,z
t1
,…) 0. (18.13)
It is important to notice that the
appearing in (18.12) is the same parameter multiply-
ing y
t1
in (18.11). If (18.12) and (18.13) hold, we can write
y
t
0
z
t
y
t1
e
t
, (18.14)
Chapter 18 Advanced Time Series Topics
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