Wooldridge J. Introductory Econometrics: A Modern Approach (Basic Text - 3d ed.)
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Chapter 8 Heteroskedasticity 297
3. Construct the estimated variances in equation (8.40).
4. Estimate the equation
y
0
1
x
1
…
k
x
k
u
by WLS, using weights 1/h
ˆ
.
EXAMPLE 8.9
(Determinants of Personal Computer Ownership)
We use the data in GPA1.RAW to estimate the probability of owning a computer. Let PC denote
a binary indicator equal to unity if the student owns a computer, and zero otherwise. The vari-
able hsGPA is high school GPA, ACT is achievement test score, and parcoll is a binary indicator
equal to unity if at least one parent attended college. (Separate college indicators for the mother
and the father do not yield individually significant results, as these are pretty highly correlated.)
The equation estimated by OLS is
(PC .0004)(.065)hsGPA (.0006)ACT (.221)parcoll
P
ˆ
C (.4905)(.137)hsGPA (.0155)ACT (.093)parcoll
P
ˆ
C [.4888][.139]hsGPA [.0158]ACT [.087]parcoll
n 141, R
2
.0415.
(8.41)
Just as with Example 8.8, there are no striking differences between the usual and robust stan-
dard errors. Nevertheless, we also estimate the model by WLS. Because all of the OLS fitted
values are inside the unit interval, no adjustments are needed:
PC (.026)(.033)hsGPA (.0043)ACT (.215)parcoll
P
ˆ
C (.477)(.130)hsGPA (.0155)ACT (.086)parcoll
n 141, R
2
.0464.
(8.42)
There are no important differences in the OLS and WLS estimates. The only significant explana-
tory variable is parcoll, and in both cases we estimate that the probability of PC ownership is
about .22 higher if at least one parent attended college.
SUMMARY
We began by reviewing the properties of ordinary least squares in the presence of het-
eroskedasticity. Heteroskedasticity does not cause bias or inconsistency in the OLS esti-
mators, but the usual standard errors and test statistics are no longer valid. We showed
how to compute heteroskedasticity-robust standard errors and t statistics, something that
is routinely done by many regression packages. Most regression packages also compute a
heteroskedasticity-robust, F-type statistic.
We discussed two common ways to test for heteroskedasticity: the Breusch-Pagan test
and a special case of the White test. Both of these statistics involve regressing the squared
298 Part 1 Regression Analysis with Cross-Sectional Data
OLS residuals on either the independent variables (BP) or the fitted and squared fitted val-
ues (White). A simple F test is asymptotically valid; there are also Lagrange multiplier
versions of the tests.
OLS is no longer the best linear unbiased estimator in the presence of heteroskedastic-
ity. When the form of heteroskedasticity is known, generalized least squares (GLS) esti-
mation can be used. This leads to weighted least squares as a means of obtaining the BLUE
estimator. The test statistics from the WLS estimation are either exactly valid when the error
term is normally distributed or asymptotically valid under nonnormality. This assumes, of
course, that we have the proper model of heteroskedasticity.
More commonly, we must estimate a model for the heteroskedasticity before applying
WLS. The resulting feasible GLS estimator is no longer unbiased, but it is consistent and
asymptotically efficient. The usual statistics from the WLS regression are asymptotically
valid. We discussed a method to ensure that the estimated variances are strictly positive
for all observations, something needed to apply WLS.
As we discussed in Chapter 7, the linear probability model for a binary dependent vari-
able necessarily has a heteroskedastic error term. A simple way to deal with this problem
is to compute heteroskedasticity-robust statistics. Alternatively, if all the fitted values (that
is, the estimated probabilities) are strictly between zero and one, weighted least squares
can be used to obtain asymptotically efficient estimators.
KEY TERMS
Breusch-Pagan Test for
Heteroskedasticity (BP
Test)
Feasible GLS (FGLS)
Estimator
Generalized Least Squares
(GLS) Estimators
Heteroskedasticity of
Unknown Form
Heteroskedasticity-Robust
F Statistic
Heteroskedasticity-Robust
LM Statistic
Heteroskedasticity-Robust
Standard Error
Heteroskedasticity-Robust t
Statistic
Weighted Least Squares
(WLS) Estimators
White Test for
Heteroskedasticity
PROBLEMS
8.1 Which of the following are consequences of heteroskedasticity?
(i) The OLS estimators,
ˆ
j
,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
0
1
inc
2
price
3
educ
4
female u
E(uinc,price,educ,female) 0
Var(uinc,price,educ,female)
2
inc
2
.
Write the transformed equation that has a homoskedastic error term.
Chapter 8 Heteroskedasticity 299
8.3 True or False: WLS is preferred to OLS, when an important variable has been omit-
ted from the model.
8.4 Using the data in GPA3.RAW, the following equation was estimated for the fall and
second semester students:
(trmgpa 2.12)(.900)crsgpa (.193)cumgpa (.0014)tothrs
2trmgpa (.55)(.175)crsgpa (.064)cumgpa (.0012)tothrs
2trmgpa [.55][.166]crsgpa [.074]cumgpa [.0012]tothrs
(.0018)sat (.0039)hsperc (.351)female (.157)season
(.0002)sat (.0018)hsperc (.085)female (.098)season
[.0002]sat [.0019]hsperc [.079]female [.080]season
n 269, R
2
.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 semes-
ter, 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 H
0
:
crsgpa
1 make sense? Test this hypothesis
against the two-sided alternative at the 5% level, using both standard
errors. Describe your conclusions.
(iii) Test whether there is an in-season effect on term GPA, using both stan-
dard errors. Does the significance level at which the null can be rejected
depend on the standard error used?
8.5 The variable smokes is a binary variable equal to one if a person smokes, and zero oth-
erwise. Using the data in SMOKE.RAW, we estimate a linear probability model for smokes:
smokes (.656)(.069)log(cigpric) (.012)log(income) (.029)educ
smo
ˆ
kes (.855)(.204)log(cigpric) (.026)log(income) (.006)educ
smo
ˆ
kes [.856][.207]log(cigpric) [.026]log(income) [.006]educ
(.020)age (.00026)age
2
(.101)restaurn (.026)white
(.006)age (.00006)age
2
(.039)restaurn (.052)white
[.005]age [.00006]age
2
[.038]restaurn [.050]white
n 807, R
2
.062.
The variable white equals one if the respondent is white, and zero otherwise; the other inde-
pendent variables are defined in Example 8.7. Both the usual and heteroskedasticity-robust
standard errors are reported.
(i) Are there any important differences between the two sets of standard errors?
(ii) Holding other factors fixed, if education increases by four years, what
happens to the estimated probability of smoking?
(iii) At what point does another year of age reduce the probability of smoking?
300 Part 1 Regression Analysis with Cross-Sectional Data
(iv) Interpret the coefficient on the binary variable restaurn (a dummy variable
equal to one if the person lives in a state with restaurant smoking restrictions).
(v) Person number 206 in the data set has the following characteristics:
cigpric 67.44, income 6,500, educ 16, age 77, restaurn 0,
white 0, and smokes 0. Compute the predicted probability of smok-
ing for this person and comment on the result.
8.6 There are different ways to combine features of the Breusch-Pagan and White tests
for heteroskedasticity. One possibility not covered in the text is to run the regression
u
i
ˆ
2
on x
i1
,x
i2
,…,x
ik
,y
ˆ
i
2
, i 1,…,n,
where the u
ˆ
i
are the OLS residuals and the y
ˆ
i
are the OLS fitted values. Then, we would
test joint significance of x
i1
,x
i2
,…,x
ik
and y
ˆ
i
2
. (Of course, we always include an intercept in
this regression.)
(i) What are the df associated with the proposed F test for heteroskedasticity?
(ii) Explain why the R-squared from the regression above will always be at
least as large as the R-squareds for the BP regression and the special case
of the White test.
(iii) Does part (ii) imply that the new test always delivers a smaller p-value
than either the BP or special case of the White statistic? Explain.
(iv) Suppose someone suggests also adding y
ˆ
i
to the newly proposed test. What
do you think of this idea?
8.7 Consider a model at the employee level,
y
i,e
0
1
x
i,e,1
2
x
i,e,2
…
k
x
i,e,k
f
i
v
i,e
,
where the unobserved variable f
i
is a “firm effect” to each employee at a given firm i. The
error term v
i,e
is specific to employee e at firm i. The composite error is u
i,e
f
i
v
i,e
,
such as in equation (8.28).
(i) Assume that Var( f
i
)
f
2
,Var(v
i,e
)
v
2
, and f
i
and v
i,e
are uncorrelated.
Show that Var(u
i,e
)
f
2
v
2
; call this
2
.
(ii) Now suppose that for e g, v
i,e
and v
i,g
are uncorrelated. Show that
Cov(u
i,e
,u
i,g
)
f
2
.
(iii) Let u
¯
i
m
i
1
m
i
i1
u
i,e
be the average of the composite errors within a firm.
Show that Var(u
¯
i
)
f
2
v
2
/m
i
.
(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
C8.1 Consider the following model to explain sleeping behavior:
sleep
0
1
totwrk
2
educ
3
age
4
age
2
5
yngkid
6
male u.
(i) Write down a model that allows the variance of u to differ between men
and women. The variance should not depend on other factors.
Chapter 8 Heteroskedasticity 301
(ii) Use the data in SLEEP75.RAW to estimate the parameters of the model
for heteroskedasticity. (You have to estimate the sleep equation by OLS,
first, to obtain the OLS residuals.) Is the estimated variance of u higher
for men or for women?
(iii) Is the variance of u statistically different for men and for women?
C8.2 (i) Use the data in HPRICE1.RAW 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 trans-
formation 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.RAW 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, uˆ
i
, and regress these on all of the independent
variables. Explain why you obtain R
2
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 heteroskedas-
ticity now?
C8.5 Use the data in PNTSPRD.RAW for this exercise.
(i) The variable sprdcvr is a binary variable equal to one if the Las Vegas
point spread for a college basketball game was covered. The expected
value of sprdcvr,say
, is the probability that the spread is covered in a
randomly selected game. Test H
0
:
.5 against H
1
:
.5 at the 10%
significance level and discuss your findings. (Hint: This is easily done
using a t test by regressing sprdcvr on an intercept only.)
(ii) How many games in the sample of 553 were played on a neutral court?
(iii) Estimate the linear probability model
sprdcvr
0
1
favhome
2
neutral
3
fav25
4
und25 u
and report the results in the usual form. (Report the usual OLS standard
errors and the heteroskedasticity-robust standard errors.) Which variable
is most significant, both practically and statistically?
(iv) Explain why, under the null hypothesis H
0
:
1
2
3
4
0, there
is no heteroskedasticity in the model.
(v) Use the usual F statistic to test the hypothesis in part (iv). What do you
conclude?
(vi) Given the previous analysis, would you say that it is possible to system-
atically predict whether the Las Vegas spread will be covered using
information available prior to the game?
302 Part 1 Regression Analysis with Cross-Sectional Data
C8.6 In Example 7.12, we estimated a linear probability model for whether a young man
was arrested during 1986:
arr86
0
1
pcnv
2
avgsen
3
tottime
4
ptime86
5
qemp86 u.
(i) Estimate this model by OLS and verify that all fitted values are strictly
between zero and one. What are the smallest and largest fitted values?
(ii) Estimate the equation by weighted least squares, as discussed in
Section 8.5.
(iii) Use the WLS estimates to determine whether avgsen and tottime are
jointly significant at the 5% level.
C8.7 Use the data in LOANAPP.RAW for this exercise.
(i) Estimate the equation in part (iii) of Computer Exercise C7.8, comput-
ing the heteroskedasticity-robust standard errors. Compare the 95% con-
fidence interval on
white
with the nonrobust confidence interval.
(ii) Obtain the fitted values from the regression in part (i). Are any of them
less than zero? Are any of them greater than one? What does this mean
about applying weighted least squares?
C8.8 Use the data set GPA1.RAW for this exercise.
(i) Use OLS to estimate a model relating colGPA to hsGPA, ACT, skipped,
and PC. Obtain the OLS residuals.
(ii) Compute the special case of the White test for heteroskedasticity. In the
regression of û
i
2
on colGPA
i
, colGPA
i
2
, obtain the fitted values, say h
ˆ
i
.
(iii) Verify that the fitted values from part (ii) are all strictly positive. Then,
obtain the weighted least squares estimates using weights 1/h
ˆ
i
. Compare
the weighted least squares estimates for the effect of skipping lectures and
the effect of PC ownership with the corresponding OLS estimates. What
about their statistical significance?
(iv) In the WLS estimation from part (iii), obtain heteroskedasticity-robust
standard errors. In other words, allow for the fact that the variance func-
tion estimated in part (ii) might be misspecified. (See Question 8.4.) Do
the standard errors change much from part (iii)?
C8.9 In Example 8.7, we computed the OLS and a set of WLS estimates in a cigarette
demand equation.
(i) Obtain the OLS estimates in equation (8.35).
(ii) Obtain the h
ˆ
i
used in the WLS estimation of equation (8.36) and repro-
duce equation (8.36). From this equation, obtain the unweighted residu-
als and fitted values; call these û
i
and yˆ
i
,respectively. (For example, in
Stata, the unweighted residuals and fitted values are given by default.)
(iii) Let ˘u
i
û
i
/
h
ˆ
i
and y˘y
i
yˆ
i
/
h
ˆ
i
be the weighted quantities. Carry out
the special case of the White test for heteroskedasticity by regressing ˘u
i
2
on ˘y
i
, ˘y
i
2
, being sure to include an intercept, as always. Do you find het-
eroskedasticity in the weighted residuals?
(iv) What does the finding from part (iii) imply about the proposed form of
heteroskedasticity used in obtaining (8.36)?
Chapter 8 Heteroskedasticity 303
(v) Obtain valid standard errors for the WLS estimates that allow the vari-
ance function to be misspecified.
C8.10 Use the data set 401KSUBS.RAW for this exercise.
(i) Using OLS, estimate a linear probability model for e401k, using as
explanatory variables inc, inc
2
, age, age
2
, and male. Obtain both the usual
OLS standard errors and the heteroskedasticity-robust versions. Are
there any important differences?
(ii) In the special case of the White test for heteroskedasticity, where we
regress the squared OLS residuals on a quadratic in the OLS fitted val-
ues, û
i
2
on yˆ
i
,yˆ
i
2
, i 1,...,n,argue that the probability limit of the coeffi-
cient on yˆ
i
should be one, the probability limit of the coefficient on yˆ
i
2
should be 1, and the probability limit of the intercept should be
zero. {Hint: Remember that Var(yx
1
,...,x
k
) p(x)[1 p(x)], where p(x)
0
1
x
1
...
k
x
k
.}
(iii) For the model estimated from part (i), obtain the White test and see if
the coefficient estimates roughly correspond to the theoretical values
described in part (ii).
(iv) After verifying that the fitted values from part (i) are all between zero and
one, obtain the weighted least squares estimates of the linear probability
model. Do they differ in important ways from the OLS estimates?
C8.11 Use the data in 401KSUBS.RAW to answer this question, restricting attention to
single-person households (fsize 1).
(i) Estimate the equation
nettfa
0
1
inc
2
age
3
age
2
4
male
5
e401k
by OLS. Report the usual standard errors as well as the heteroskedasticity-
robust standard errors. Comment on any notable differences.
(ii) Using the residuals from part (i), estimate a heteroskedasticity model
using the regression in (8.32), obtaining the h
ˆ
i
from equation (8.33). What
are the smallest and largest values of the h
ˆ
i
?
(iii) Estimate the model from part (i) using WLS, where the h
ˆ
i
are obtained
in part (ii). Obtain the usual WLS standard errors and the standard errors
that are robust to misspecification of the heteroskedasticity function.
Comment on how the two sets compare.
(iv) Consider the variable e401k. Is the robust OLS standard error smaller or
larger than the robust WLS standard error? What might you conclude
about asymptotic efficiency of OLS versus WLS?
More on Specification
and Data Problems
I
n Chapter 8, we dealt with one failure of the Gauss-Markov assumptions. While het-
eroskedasticity in the errors can be viewed as a problem with a model, it is a relatively
minor one. The presence of heteroskedasticity does not cause bias or inconsistency in the
OLS estimators. Also, it is fairly easy to adjust confidence intervals and t and F statistics
to obtain valid inference after OLS estimation, or even to get more efficient estimators by
using weighted least squares.
In this chapter, we return to the much more serious problem of correlation between
the error, u, and one or more of the explanatory variables. Remember from Chapter 3 that
if u is, for whatever reason, correlated with the explanatory variable x
j
, then we say that
x
j
is an endogenous explanatory variable. We also provide a more detailed discussion on
three reasons why an explanatory variable can be endogenous; in some cases, we discuss
possible remedies.
We have already seen in Chapters 3 and 5 that omitting a key variable can cause cor-
relation between the error and some of the explanatory variables, which generally leads
to bias and inconsistency in all of the OLS estimators. In the special case that the omit-
ted variable is a function of an explanatory variable in the model, the model suffers from
functional form misspecification.
We begin in the first section by discussing the consequences of functional form
misspecification and how to test for it. In Section 9.2, we show how the use of proxy
variables can solve, or at least mitigate, omitted variables bias. In Section 9.3, we derive
and explain the bias in OLS that can arise under certain forms of measurement error.
Additional data problems are discussed in Section 9.4.
All of the procedures in this chapter are based on OLS estimation. As we will see,
certain problems that cause correlation between the error and some explanatory variables
cannot be solved by using OLS on a single cross section. We postpone a treatment of
alternative estimation methods until Part 3.
9.1 Functional Form Misspecification
A multiple regression model suffers from functional form misspecification when it does not
properly account for the relationship between the dependent and the observed explanatory
Chapter 9 More on Specification and Data Problems 305
variables. For example, if hourly wage is determined by log(wage)
0
1
educ
2
exper
3
exper
2
u,but we omit the squared experience term, exper
2
, then we are
committing a functional form misspecification. We already know from Chapter 3 that
this generally leads to biased estimators of
0
,
1
, and
2
. (We do not estimate
3
because exper
2
is excluded from the model.) Thus, misspecifying how exper affects
log(wage) generally results in a biased estimator of the return to education,
1
. The
amount of this bias depends on the size of
3
and the correlation among educ, exper,
and exper
2
.
Things are worse for estimating the return to experience: even if we could get an unbi-
ased estimator of
2
, we would not be able to estimate the return to experience because it
equals
2
2
3
exper (in decimal form). Just using the biased estimator of
2
can be mis-
leading, especially at extreme values of exper.
As another example, suppose the log(wage) equation is
log(wage)
0
1
educ
2
exper
3
exper
2
4
female
5
female·educ u,
(9.1)
where female is a binary variable. If we omit the interaction term, femaleeduc,
then we are misspecifying the functional form. In general, we will not get unbiased
estimators of any of the other parameters, and since the return to education depends
on gender, it is not clear what return we would be estimating by omitting the interac-
tion term.
Omitting functions of independent variables is not the only way that a model can suffer
from misspecified functional form. For example, if (9.1) is the true model satisfying the
first four Gauss-Markov assumptions, but we use wage rather than log(wage) as the depen-
dent variable, then we will not obtain unbiased or consistent estimators of the partial
effects. The tests that follow have some ability to detect this kind of functional form prob-
lem, but there are better tests that we will mention in the subsection on testing against
nonnested alternatives.
Misspecifying the functional form of a model can certainly have serious consequences.
Nevertheless, in one important respect, the problem is minor: by definition, we have data
on all the necessary variables for obtaining a functional relationship that fits the data well.
This can be contrasted with the problem addressed in the next section, where a key vari-
able is omitted on which we cannot collect data.
We already have a very powerful tool for detecting misspecified functional form: the
F test for joint exclusion restrictions. It often makes sense to add quadratic terms of any
significant variables to a model and to perform a joint test of significance. If the additional
quadratics are significant, they can be added to the model (at the cost of complicating the
interpretation of the model). However, significant quadratic terms can be symptomatic of
other functional form problems, such as using the level of a variable when the logarithm
is more appropriate, or vice versa. It can be difficult to pinpoint the precise reason that a
functional form is misspecified. Fortunately, in many cases, using logarithms of certain
variables and adding quadratics are sufficient for detecting many important nonlinear rela-
tionships in economics.