Wooldridge J. Introductory Econometrics: A Modern Approach (Basic Text - 3d ed.)
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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 1spread). 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).
(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
observable information prior to a game.
C17.2 Use the data in LOANAPP.RAW for this exercise; see also Computer Exercise C7.8.
(i) Estimate a probit model of approve on white. Find the estimated
probability 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 pro-
bit 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) Use equation (17.17) to estimate the sizes of the discrimination
effects for probit and logit.
C17.3 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
pension benefits for a white male and a nonwhite female, both of
whom are 35 years old, are single with no dependents, have 16 years
of education, and have 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, though it often takes on the value zero, it never
626 Part 3 Advanced Topics
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?
C17.4 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 Pois-
son regression in Example 17.3.
(ii) Compute the estimate of
2
given by
ˆ
2
(n k 1)
1
n
i1
u
ˆ
i
2
/y
ˆ
i
.
Is there evidence of overdispersion? How should the Poisson MLE
standard 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 quad-
ratic terms. What do you conclude?
C17.5 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 vari-
ables 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 obtain the
R-squared as the squared correlation between kids
i
and kids
i
. Com-
pare this with the R-squared for the linear regression model.
C17.6 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 com-
pare with those in Table 17.4.
C17.7 Use the MROZ.RAW data for this exercise.
(i) Using the 428 women who were in the workforce, 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
standard 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.)
C17.8 The file JTRAIN2.RAW contains data on a job training experiment for a group
of men. Men could enter the program starting in January 1976 through about mid-1977.
The program ended in December 1977. The idea is to test whether participation in the
job training program had an effect on unemployment probabilities and earnings in 1978.
Chapter 17 Limited Dependent Variable Models and Sample Selection Corrections 627
(i) The variable train is the job training indicator. How many men in the
sample participated in the job training program? What was the high-
est number of months a man actually participated in the program?
(ii) Run a linear regression of train on several demographic and pretrain-
ing variables: unem74, unem75, age, educ, black, hisp, and married.
Are these variables jointly significant at the 5% level?
(iii) Estimate a probit version of the linear model in part (ii). Compute the
likelihood ratio test for joint significance of all variables. What do
you conclude?
(iv) Based on your answers to parts (ii) and (iii), does it appear that par-
ticipation in job training can be treated as exogenous for explaining
1978 unemployment status? Explain.
(v) Run a simple regression of unem78 on train and report the results in
equation form. What is the estimated effect of participating in the job
training program on the probability of being unemployed in 1978? Is
it statistically significant?
(vi) Run a probit of unem78 on train. Does it make sense to compare the
probit coefficient on train with the coefficient obtained from the lin-
ear model in part (v)?
(vii) Find the fitted probabilities from parts (v) and (vi). Explain why they
are identical. Which approach would you use to measure the effect
and statistical significance of the job training program?
(viii) Add all of the variables from part (ii) as additional controls to the
models from parts (v) and (vi). Are the fitted probabilities now iden-
tical? What is the correlation between them?
C17.9 Use the data in APPLE.RAW for this exercise. These are telephone survey data
attempting to elicit the demand for a (fictional) “ecologically friendly” apple. Each family
was (randomly) presented with a set of prices for regular apples and the eco-labeled apples.
They were asked how many pounds of each kind of apple they would buy.
(i) Of the 660 families in the sample, how many report wanting none of
the eco-labeled apples at the set price?
(ii) Does the variable ecolbs seem to have a continuous distribution over
strictly positive values? What implications does your answer have for
the suitability of a Tobit model for ecolbs?
(iii) Estimate a Tobit model for ecolbs with ecoprc, regprc, faminc, and
hhsize as explanatory variables. Which variables are significant at the
1% level?
(iv) Are the signs of the coefficients on the price variables from part (iii)
what you expect? Explain.
(v) Let
1
be the coefficient on ecoprc and let
2
be the coefficient on
regprc. Test the hypothesis H
0
:
1
2
against the two-sided alter-
native. Report the p-value of the test. (You might want to refer to Sec-
tion 4.4 if your regression package does not easily compute such
tests.)
628 Part 3 Advanced Topics
(vi) Obtain the estimates of E(ecolbsx) for all observations in the sam-
ple. [See equation (17.25).] Call these ecolbs
i
. What are the smallest
and largest fitted values?
(vii) Compute the squared correlation between ecolbs
i
and ecolbs
i
.
(viii) Now, estimate a linear model for ecolbs using the same explanatory
variables from part (iii). Why are the OLS estimates so much smaller
than the Tobit estimates? In terms of goodness-of-fit, is the Tobit
model better than the linear model?
(ix) Evaluate the following statement: “Because the R-squared from the
Tobit model is so small, the estimated price effects are probably
inconsistent.”
C17.10 Use the data in SMOKE.RAW for this exercise.
(i) The variable cigs is the number of cigarettes smoked per day. How
many people in the sample do not smoke at all? What fraction of peo-
ple claim to smoke 20 cigarettes a day? Why do you think there is a
pileup of people at 20 cigarettes?
(ii) Given your answers to part (i), does cigs seem a good candidate for
having a conditional Poisson distribution?
(iii) Estimate a Poisson regression model for cigs, including log(cigpric),
log(income), white, educ, age, and age
2
as explanatory variables.
What are the estimated price and income elasticities?
(iv) Using the maximum likelihood standard errors, are the price and
income variables statistically significant at the 5% level?
(v) Obtain the estimate of
2
described after equation (17.35). What is
ˆ? How should you adjust the standard errors from part (iv)?
(vi) Using the adjusted standard errors from part (v), are the price and
income elasticities now statistically different from zero? Explain.
(vii) Are the education and age variables significant using the more robust
standard errors? How do you interpret the coefficient on educ?
(viii) Obtain the fitted values, yˆ
i
,from the Poisson regression model. Find
the minimum and maximum values and discuss how well the expo-
nential model predicts heavy cigarette smoking.
(ix) Using the fitted values from part (viii), obtain the squared correlation
coefficient between yˆ
i
and y
i
.
(x) Estimate a linear model for cigs by OLS, using the explanatory vari-
ables (and same functional forms) as in part (iii). Does the linear
model or exponential model provide a better fit? Is either R-squared
very large?
C17.11 Use the data in CPS91.RAW for this exercise. These data are for married women,
where we also have information on each husband’s income and demographics.
(i) What fraction of the women report being in the labor force?
(ii) Using only the data for working women—you have no choice—
estimate the wage equation
log(wage)
0
1
educ
2
exper
3
exper
2
4
black
5
hispanic u
Chapter 17 Limited Dependent Variable Models and Sample Selection Corrections 629
by ordinary least squares. Report the results in the usual form. Do
there appear to be significant wage differences by race and ethnicity?
(iii) Estimate a probit model for inlf that includes the explanatory vari-
ables in the wage equation from part (ii) as well as nwifeinc and
kidlt6. Do these last two variables have coefficients of the expected
sign? Are they statistically significant?
(iv) Explain why, for the purposes of testing and, possibly, correcting the
wage equation for selection into the workforce, it is important for
nwifeinc and kidlt6 to help explain inlf. What must you assume about
nwifeinc and kidlt6 in the wage equation?
(v) Compute the inverse Mills ratio (for each observation) and add it as
an additional regressor to the wage equation from part (ii). What is
its two-sided p-value? Do you think this is particularly small with
3,286 observations?
(vi) Does adding the inverse Mills ratio change the coefficients in the
wage regression in important ways? Explain.
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 estimation.
The background needed for a careful analysis of these methods and several derivations are
given in Wooldridge (2002).
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 1x) G(x
), where
G() is the logit or probit function, and
is the k 1 vector of parameters, the asymp-
totic variance matrix of
ˆ
is estimated as
Avar (
ˆ
)
n
i1
1
, (17.53)
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 (17.53) accounts for the nonlinear
nature of the response probability—that is, the nonlinear nature of G()—as well as the par-
ticular form of heteroskedasticity in a binary response model: Var(yx) G(x
)[1 G(x
)].
The square roots of the diagonal elements of (17.53) 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.
[g(x
i
ˆ
)]
2
x
i
x
i
G(x
i
ˆ
)[1 G(x
i
ˆ
)]
630 Part 3 Advanced Topics
The matrix in (17.53) is also the basis for Wald tests of multiple restrictions on
(see
Wooldridge [2002, Chapter 15]).
The asymptotic variance matrix for Tobit is more complicated but has a similar struc-
ture. Note that we can obtain a standard error for
ˆ
as well. The asymptotic variance for
Poisson regression, allowing for
2
1 in (17.35), has a form much like (17.53):
Avar(
ˆ
)
ˆ
2
n
i1
exp(x
i
ˆ
)x
i
x
i
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 (2002) for details.
Chapter 17 Limited Dependent Variable Models and Sample Selection Corrections 631
Advanced Time Series Topics
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 eco-
nomic 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 sta-
tistically significant t statistic. This emphasizes the potentially serious consequences of
using standard inference when the dependent and independent variables are integrated
processes.
The notion of cointegration applies when two series are I(1), but a linear combination
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.
Chapter 18 Advanced Time Series Topics 633
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 par-
tially 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. Com-
pared 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 applications,
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 without
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 without restric-
tions. Later, we place restrictions on the
j
that allow us to estimate (18.1).
As with finite distributed lag (FDL) 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
plotted 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
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, plus a. Thus, the
LRP measures the long-run change in the
expected value of y given a one-unit, per-
manent 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 inter-
cept 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.3). Formally,
E(u
t
…,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. Although
(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
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.
634 Part 3 Advanced Topics
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 → ?
QUESTION 18.1
Chapter 18 Advanced Time Series Topics 635
One thing to remember is that neither (18.5) nor (18.6) says anything about the serial
correlation properties of {u
t
}. (This is just as in finite distributed lag models.) If anything,
we might expect the {u
t
} to be serially correlated because (18.1) is not generally dynami-
cally 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 coun-
terfactual) 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
,
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 → . In fact, this convergence happens at a very fast
rate. (For example, with
.5 and j 10,
j
1/1024 .001.)
The impact propensity (IP) in the GDL is simply
0
, so the sign of the IP is
determined by the sign of
. If
0, say, and
0, then all lag coefficients are posi-
tive. 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
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
,