Manajemen | Fakultas Ekonomi Universitas Maritim Raja Ali Haji 073500102288618739
Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Testing the Normality Assumption in the Sample
Selection Model With an Application to Travel
Demand
Bas van der Klaauw & Ruud H Koning
To cite this article: Bas van der Klaauw & Ruud H Koning (2003) Testing the Normality
Assumption in the Sample Selection Model With an Application to Travel Demand, Journal of
Business & Economic Statistics, 21:1, 31-42, DOI: 10.1198/073500102288618739
To link to this article: http://dx.doi.org/10.1198/073500102288618739
Published online: 01 Jan 2012.
Submit your article to this journal
Article views: 58
View related articles
Citing articles: 1 View citing articles
Full Terms & Conditions of access and use can be found at
http://www.tandfonline.com/action/journalInformation?journalCode=ubes20
Download by: [Universitas Maritim Raja Ali Haji]
Date: 13 January 2016, At: 00:38
Testing the Normality Assumption in the
Sample Selection Model With an
Application to Travel Demand
Bas van der Klaauw
Department of Economics, Free University Amsterdam, De Boelelaan 1105, NL-1081 HV Amsterdam,
The Netherlands and Scholar (klaauw@tinbergen.nl)
Ruud H. Koning
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
Department of Econometrics, University of Groningen, P.O. Box 800, NL-9700 AV Groningen,
The Netherlands (r.h.koning@eco.rug.nl)
In this article we introduce a test for the normality assumption in the sample selection model. The test is
based on a exible parametric specication of the density function of the error terms in the model. This
specication follows a Hermite series with bivariate normality as a special case. All parameters of the
model are estimated both under normality and under the more general exible parametric specication,
which enables testing for normality using a standard likelihood ratio test. If normality is rejected, then
the exible parametric specication provides consistent parameter estimates. The test has reasonable
power, as is shown by a simulation study. The test also detects some types of ignored heteroscedasticity.
Finally, we apply the exible specication of the density to a travel demand model and test for normality
in this model.
KEY WORDS:
1.
Flexible parametric density estimation; Hermite series; Heteroscedasticity; Sample
selection.
INTRODUCTION
eduction (Heckman, Tobias, and Vytlacil 2000), and to estimate the effect of union membership on earnings (Lee 1978).
In the switching regression model, many of the interesting
policy parameters, such as the treatment effect on the treated,
depend not only on the functional form of the relationship, but
also on the joint distribution of the error terms. Therefore, for
policy evaluation and also for simulations, both the parameters
of the functional form of the relationship and the joint distribution of the error terms must be known. Of course, this does
not hold for either the switching regression model, or also for
other models, like the sample selection model.
Usually, the sample selection model is estimated either by
maximum likelihood or by Heckman’s two-step procedure
(Heckman 1979), under the assumption that the error term
in the regression equation and the error term in the selection
equation follow a bivariate normal distribution. However, it
is often argued that the parameter estimates are sensitive to
the distributional assumptions (Manski 1989). In fact, Newey
(1999a) proved that only under very restrictive assumptions,
the parameter estimates are consistent against misspecication
of the distribution of the error terms. In many cases, the normal distribution might be overly restrictive. As mentioned earlier, the sample selection model is often used to model earnings, and it is well known from the economic literature that
the distribution of (the logarithm of) earnings in the population has fatter tails than the normal distribution function (e.g.,
Heckman et al. 2000).
Maximum likelihood is the most popular estimation method
in microeconometrics. The method yields consistent (in fact,
asymptotically efcient) estimators if the model is specied
correctly. However, correct specication may not be known
beforehand. Two major sources of misspecication are incorrect specication of the functional form of the relationship
under study (e.g., omitting exogenous variables or inappropriately assuming linearity) and misspecication of the stochastic
structure of the model (e.g., neglecting heteroscedasticity or
misspecication of the distribution of the random variables).
The maximum likelihood estimator is generally inconsistent
in these cases. This is particularly true in the case of limited
dependent variable models (e.g., Hurd 1979; Smith 1989). In
this article we focus on one particular form of misspecication: misspecication of the distribution of the error terms. We
retain the assumption of correct specication of the functional
form of the relationship.
The model that we study is the sample selection model
introduced by Heckman (1979). This type of model accounts
for problems arising because the outcome of the endogenous
variable is observed only for a (selective) part of the sample. For example, sample selection models are used to study
wage equations, where only the wages of employed workers
are observed (e.g., Buchinsky 1998; Melenberg and Van Soest
1993). This model is not only very useful when the outcome
variable is only observed for a (nonrandom) part of the population, but it can easily be extended to the switching regression
model. The switching regression model is the point of departure for many interesting problems in economics. For example, it is used to evaluate active labor market programs (Heckman, LaLonde, and Smith 1999), to investigate the effect of
migration on earnings (Tunalõ 2000), to study the returns to
© 2003 American Statistical Association
Journal of Business & Economic Statistics
January 2003, Vol. 21, No. 1
DOI 10.1198/073500102288618739
31
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
32
Journal of Business & Economic Statistics, January 2003
We introduce a formal test for the normality assumption
within the sample selection model. The test is motivated by the
seminonparametric maximum likelihood estimation method of
Gallant and Nychka (1987) in which the true distribution of
the error terms is approximated by a Hermite series. The
density of the error terms follows a very exible parametric
specication, which is estimated using maximum likelihood
together with all other parameters of the model. The test thus
provides an alternative distribution of the error terms in the
event that normality is rejected. Because the parameters are
estimated
p using maximum likelihood, the estimators are efcient, N consistent, and asymptotically normal. To examine the power of the normality test, we perform a simulation
study, which will also provide practical experience about the
actual exibility of the parametric specication of the density. For the binary choice model and duration models, Smith
(1989) proposed a similar type of polynomial specication to
test for normality. Gabler, Laisney, and Lechner (1993) provided Monte Carlo evidence on this test and applied it to a
model of female labor force participation.
So far we did not discuss heteroscedasticity of the error
terms. Ignored heteroscedasticity causes the usual estimators in limited dependent-variable models to be inconsistent.
For example, Hurd (1979) showed that the bias caused by
ignored heteroscedasticity is severe in the truncated regression model. Not many estimation procedures are robust against
heteroscedasticity. An exception is the quantile regression
approach of Buchinsky (1998). The disadvantage of quantile
regression methods is that the complete distribution of the
error terms is not estimated. This complicates performing simulations, and because the mean is not estimated, it is difcult
to dene relevant policy parameters (see Abadie, Angrist, and
Imbens 1998, who applied the quantile regression estimator to
the switching regression model). In our simulation study we
also focus on the robustness of the exible parametric density
against heteroscedastic error terms.
An advantage of the exible parametric density used in this
article is its general applicability; it is not specic to one particular econometric model. The approach taken herein can be
used to examine the sensitivity of estimation results to the
assumption of normality in other microeconometric models
where no formal test of normality is available. We illustrate
both the exible parametric density and the normality test in
a model of travel demand and car ownership.
The article is organized as follows. Section 2 discusses the
sample selection model and the exible parametric density
that naturally leads to a normality test for the sample selection
model. Section 3 reports some simulations to investigate the
performance of the exible parametric density and the power
of the normality test. Section 4 discusses an application of
the estimation method and the normality test to the model of
travel demand. Finally, Section 5 concludes.
2.
IDENTIFICATION AND ESTIMATION OF THE
SAMPLE SELECTION MODEL
In this section we discuss some identication and estimation issues associated with the sample selection model. In particular, we focus on the exible parametric density that leads
to a test for bivariate normality.
Within a population, we are interested in the distribution
of an outcome variable y conditional on a vector of exogenous variables x1 . Therefore, we specify for individual i 4i D
11 : : : 1 N 5 the regression equation
yi D ‚01 x1i C ˜1i 0
(1)
The dependent variable yi is observed for a subsample only. A
binary variable zi is introduced that equals 1 if the realization
of yi is observed and otherwise equals 0. The selection rule
depends on a set of exogenous variables x2 and is given by
züi D ‚02 x2i C ˜2i
and
zi D
(
11
01
züi > 0
züi µ 00
(2)
As mentioned in Section 1, we focus only on the joint distribution of the error terms 4˜1i 1 ˜2i 5, and thus we simply assume
linear specications in both the regression and the selection
equation. If the conditional expectation of ˜1i given zi D 1
does not equal 0, ordinary least squares (OLS) estimation of
(1) will not yield consistent estimates for the parameters of
interest, ‚1 .
If one is willing to assume that the error terms 4˜1i 1 ˜2i 5 follow a bivariate normal distribution, then the model can be estimated using Heckman’s two-stage procedure (see Heckman
1979). In the rst step, a probit model is estimated for the
selection equation to obtain ‚O 2 , in the second step, the regression equation
yi D
‚01 x1i
0
”4‚O 2 x2i 5
C ‡i
C ‘ 1
0
ê4‚O x 5
2
(3)
2i
is estimated (by OLS) for the subsample for which a realization of yi is actually observed 4zi D 15. The parameter
denotes the correlation between ˜1 and ˜2 , and ‘ 12 is the
variance of ˜1 (the variance of ˜2 is normalized to 1). If
the vector x1i equals the vector x2i , then identication of the
model hinges entirely on the functional form and distributional
assumptions, which makes the parameter estimates very sensitive to the distributional assumptions (see Manski 1989 for
an extensive overview). Therefore, in most economic applications, an exclusion restriction is imposed—at least one variable enters only the selection equation and not the regression
equation.
The bivariate normality assumption may be overly restrictive. Therefore, starting with Lee (1982), who modied Heckman’s two-step procedure to allow in a general way for
departures from normality, a number of estimation methods
have been introduced that relax the distributional assumptions. These (semiparametric) estimation methods generalize the probit estimation in the rst step to, for example,
kernel-based estimation and use more exible error correction terms in the regression equation (e.g., Ahn and Powell
1993; Cosslett 1991; Ichimura and Lee 1991; Newey 1999b).
This provides consistent estimators for the parameters of interest ‚1 , but computation of the corrected standard errors of
these parameters is quite cumbersome (e.g., Ahn and Powell
van der Klaauw and Koning: Testing the Normality Assumption
33
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
1993; Cosslet 1991). A main disadvantage of most semiparametric estimation methods is that the intercept in the outcome
equation remains unidentied (and thus cannot be estimated).
As stressed by Heckman (1990), in many economic applications the intercept is an important parameter of interest.
The alternative to two-step estimation is maximum likelihood estimation. This requires full specication of the joint
density function of the error terms, which is denoted by
f 4˜1 1 ˜2 5. Under the assumption of correct specication, maximum likelihood estimation yields consistent estimates of all
parameters of interest and straightforward computation of the
asymptotic distribution. The log-likelihood function of the
sample selection model is
log ¬ D
N
X
iD1
³Z
zi log
ˆ
ƒ‚02 x2i
f 4yi ƒ ‚01 x1i 1 ˜2 5d˜2
³Z
C 41 ƒ zi 5 log
ƒ‚02 x2i
ƒˆ
Z
ˆ
ƒˆ
´
´
f 4˜1 1 ˜2 5d˜ 1 d˜2 0
(4)
In maximum likelihood estimation, consistency depends
crucially on the correct specication of the joint density of the
error terms. Therefore, it is convenient to specify this density
so that it is exible. We use exible parametric specications
that are inspired by seminonparametric estimation as introduced by Gallant and Nychka (1987). The unknown density
function f 4˜1 1 ˜2 5 is approximated by a Hermite series.
We consider densities h4˜5 in a subclass ¨K that are of the
Hermite form
hü 4˜5 D PK2 4˜5”2 4˜—è51
(5)
where PK 4˜5 is a polynomial of degree K and ”4˜—è5 is the
normal density function with mean 0 and covariance matrix è.
In fact, Gallant and Nychka (1987) restricted è to being a
diagonal matrix. Because of the squaring, no restrictions are
necessary to ensure that hü 4˜5 is nonnegative. Some restrictions are required to ensure that the density integrates to 1 and
for identication. We return to these restrictions later.
Gallant and Nychka (1987) showed that a large class ¨
of density functions can be approximated arbitrarily well by
increasing the number of terms K of the polynomial PK 4˜5.
They gave the precise conditions dening ¨ . For our purposes, it sufces to note that the fattest tails allowed are t-like
tails, and the thinnest tails allowed are thinner than normal-like
tails. Any sort of skewness and kurtosis (especially in that part
of the distribution where most probability mass is observed)
is allowed, and only very violently oscillatory densities are
excluded from ¨ . Gallant and Nychka (1987) proved that the
densities in ¨ can be estimated consistently by increasing K
with the number of observations. In (5), the normal density
is used as the base class for ¨K , but this is not necessary;
any density with a moment-generating function could be used
(see, e.g., Cameron and Johansson 1997).
It is also possible to assume that the true density is a member of ¨K and hence to interpret ¨K as a exible class
of density functions. This is the interpretation that we follow in this article. This interpretation is especially appealing
if one wants to examine the sensitivity of estimation results
obtained by assuming normality to this distributional assumption, because it allows one to use the standard framework
of inference. Because we want to test for bivariate normality
(with unrestricted correlation) in the sample selection model,
we do not restrict the covariance matrix, unlike Gallant and
Nychka (1987). This provides a formal test for normality in
the sample selection model; that is, PK 4˜5 is constant for all
values of ˜. Such a test for normality has not yet been derived
in the literature. An alternative could be to perform Hausman
tests on the slope coefcients of the regressions equation. One
could either use Hausman tests to test normality against a
semiparametric alternative or also against the exible parametric alternative discussed in this article. The rst alternative
may be somewhat cumbersome, because it requires computing standard errors in the semiparametric estimation, which is
not always straightforward. The second alternative is easy to
implement, but these tests are not expected to be as powerful
as the tests derived in this article. The simulations in Section 3
indicate that the slope coefcients are not very sensitive to
departures from normality. Hausman tests leave out any information on the distribution of the error terms.
In line with Gallant and Nychka (1987), we parameterize
hü 4˜5 as
´2
³ K
X
ij ˜i1 ˜j2 exp4ƒ˜0 èƒ1 ˜5
hü 4˜5 D
i1 jD0
D
K
X
jCl
i1 j1 k1 lD0
ƒ 0 ƒ1
ij kl ˜iCk
1 ˜2 exp4 ˜ è ˜50
This bivariate density can be easily generalized to a density function for more variables, which is important when,
for example, studying switching regression models containing
three error terms. To ensure that this density integrates to 1,
a restriction on the parameters is necessary. This restriction
can take the form of an explicit restriction on the parameters
of the density. However, for computational convenience, we
follow Gabler et al. (1993), who ensured integration to 1 by
scaling the density. Dene S by
SD
D
Z
K
X
R2 i1 j1 k1 lD0
K
X
i1 j1 k1 lD0
jCl
ij kl ˜1iCk ˜2 exp4ƒ˜0 èƒ1 ˜5d˜
ij kl
Z
ˆ
ƒˆ
jCl
˜2
Z
ˆ
ƒˆ
˜iCk
exp4ƒ˜0 èƒ1 ˜5d˜ 1 d˜2 0
1
Because of the denition of S, the following density integrates
to 1:
h4˜5 D hü 4˜5=S0
It is clear that the parameters are identied up to a scale
only, so normalization is necessary. Therefore, we impose
00 D 1. For K D 0, h4˜5 now reduces to the bivariate normal
density. The exibility of this family of densities is illustrated
in Figures 1–3 (K D 2). It is clear that the contour lines differ
from the usual ellipsoids of the bivariate normal density.
In the sample
p selection model, identication is achieved by
setting è22 D 2, to ensure identication of the scale of (2).
To x the location of the distribution function, we optimize
the log-likelihood function (4) under the restrictions that the
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
34
Journal of Business & Economic Statistics, January 2003
Figure 1. Bivariate Flexible Parametric Density, 01 D .1, 10 D - .1,
and 11 D - .2.
Figure 2. Bivariate Flexible Parametric Density, 01 D .1, 10 D .1,
and 11 D 0.
means of ˜1 and ˜2 equal 0. Alternatively, one could impose
explicit restrictions on the parameters to ensure that the
means of ˜1 and ˜2 are 0. However, for K ¶ 2, these are rather
complex nonlinear restrictions. Another alternative was proposed by Melenberg and Van Soest (1993), who suggested not
imposing restrictions on the parameters of the density function
of ˜ to ensure a zero mean, but rather restricting the intercepts
of (1) and (2). For the purpose of this article, such a strategy is
not very useful. Our approach allows testing the null hypothesis that 4˜1 1 ˜2 5 is distributed according to a normal distribution function against the alternative hypothesis that 4˜1 1 ˜2 5
has some other mean zero bivariate distribution function in the
class of distribution functions ¨K for any xed K. In other
words, for some xed K, we test for joint signicance of all
ij with i C j ¶ 1. The additional number of parameters under
the alternative hypotheses compared to the null hypotheses
is 4K C 152 ƒ 1. Note that there are two restrictions on these
parameters to x the location of the distribution. Therefore,
the likelihood ratio test statistic is distributed according to a
chi-squared distribution with 4K C 152 ƒ 3 degrees of freedom.
Optimizing the log-likelihood function subject to the means
of ˜1 and ˜2 being equal to 0 requires evaluating the relevant
integrals in the log-likelihood function and the expressions for
the means. Because of the general covariance matrix in the
base function ”4˜—è5, these integrals do not have the same
computationally attractive properties as they do when using
the base function proposed by Gallant and Nychka (1987).
However, because of the special structure of the sample selection model, we are able to avoid evaluations of bivariate normal integrals, so the computational cost of this generalization
is limited. Appendix A gives the recursion formulas that can
be used to determine the relevant integrals explicitly.
Finally, the exible parametric density discussed earlier
assumes homoscedasticity of the error terms. As is clear from
(3), the estimators are not robust against ignored heteroscedasticity (i.e., ‘ 1 is not the same for all observations). In the
van der Klaauw and Koning: Testing the Normality Assumption
35
and
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
züi D ‚20 C ‚21 vi C ‚22 wi C ˜2i 1
i D 11 : : : 1 N 1
with true parameters ‚10 D 1, ‚11 D 05, ‚12 D ƒ05, ‚20 D 1,
‚21 D ƒ1, and ‚22 D 1. The exogenous variables xi and vi
are independently ® 401 35 distributed, and wi is distributed
uniformly on 6ƒ31 37. We perform ve experiments in which
we vary the distribution of ˜. The sample size N is 1000.
(Similar simulation experiments with a sample size N equal
to 500 gave the same results and are available on request.)
Within each experiment, we draw 100 samples. In the rst
three experiments, we draw ˜ from either a bivariate normal distribution with mean 0, a bivariate t distribution, or
a centered chi-squared distribution. The t distribution has
fatter tails than the normal distribution and the chi-squared
distribution is asymmetric, so both these cases are a deviation from normality. For all experiments, we set var4˜1 5 D 4,
var4˜2 5 D 1, and cov4˜1 1 ˜2 5 D 1. In the last two experiments, the error terms are bivariate normal distributed but
heteroscedastic. First, the distribution of ˜ depends on the
explanatory variable x. Second, we distinguish three groups
of individuals who have different covariance matrices. (This
simulation experiment has some similarities with experiments
in Hurd 1979.) The details of the simulation experiments are
given in Appendix B. The simulations were performed on Pentium workstations using the constrained maximum likelihood
(CML) library of GAUSS.
Because the computer time required for optimization of the
log-likelihood function increases quickly with K, we estimate
only the model for K D 1 and K D 2 and for the bivariate
normal distributed disturbances. Hence we consider the normality test against the class of exible parametric densities
Table 1. Results of the Simulation Experiment With Bivariate Normal
Distributed Disturbances
Figure 3. Bivariate Flexible Parametric Density, 01 D .1, 10 D - .1,
and 11 D .2.
next section we also discuss the performance of the exible
parametric density in case the error terms are heteroscedastic. In particular, we focus on heteroscedasticity in ˜1 when
the correlation between ˜1 and ˜2 is constant. Note that the
variance of ˜2 is generally unidentied. Thus many types of
heteroscedasticity in the distribution of ˜2 could be solved by
generalizing the single index in (2).
3.
SIMULATION STUDY
In this section we perform a limited simulation exercise to
investigate the performance of the exible parametric density
and the power of the normality test discussed in Section 2.
We focus two main sources of inconsistency, departures from
normality and heteroscedasticity in the error terms.
We consider the simulation experiment
yi D ‚10 C ‚11 xi C ‚12 wi C ˜1i
Flexible parametric
N D 1000
‚ 10
‚ 11
‚ 12
‚ 20
‚ 21
‚ 22
‘1
‘ 12
00
01
02
10
11
12
20
21
22
1000
050
ƒ051
1000
ƒ099
099
2000
1000
(0072)
(0025)
(0051)
(0060)
(0045)
(0054)
(0060)
(0034)
1000
050
ƒ051
1000
ƒ099
099
2000
1000
1
0
0
ƒ00002
sdev(˜1 )
sdev(˜2 )
cov(˜1 1 ˜2 )
Log-likelihood
Rejections
NOTE:
K D1
Normal
ƒ1412.71
(0073)
(0024)
(0049)
(0062)
(0043)
(0054)
(0059)
(0036)
(00064)
2000
(0061)
1000
(0013)
1000
(0087)
ƒ1412.35
Standard errors are given in parentheses.
0
K D2
1000
050
ƒ051
1000
ƒ099
099
2000
1000
1
00001
00026
00003
00034
00006
ƒ00004
ƒ00005
ƒ00014
(0074)
(0025)
(0049)
(0061)
(0043)
(0054)
(0059)
(0036)
(0019)
(0012)
(0010)
(0017)
(0017)
(00051)
(00067)
(00050)
2000
(0070)
1000
(0020)
1001
(010)
ƒ1411.18
1
36
Journal of Business & Economic Statistics, January 2003
Table 2. Results of the Simulation Experiment With Bivariate t
Distributed Disturbances
Table 4. Results of the Simulation Experiment, Where the Error Terms
are Distributed According to a Bivariate Normal Distribution With
Heteroscedasticity Depending on the Explanatory Variable x
Flexible parametric
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
N D 1000
‚ 10
‚ 11
‚ 12
‚ 20
‚ 21
‚ 22
‘1
‘ 12
00
01
02
10
11
12
20
21
22
099
050
ƒ051
1006
ƒ1005
1007
1092
095
(0076)
(0028)
(0055)
(011)
(0096)
(011)
(022)
(022)
1003
050
ƒ051
1005
ƒ1002
1004
2003
1018
1
0
0
ƒ012
sdev(˜1 )
sdev(˜2 )
cov(˜1 1 ˜2 )
Log-likelihood
Rejections
NOTE:
KD1
Normal
ƒ1371.30
(0064)
(0029)
(0044)
(0071)
(0065)
(0076)
(019)
(022)
(0082)
1088 (018)
093 (0059)
072 (032)
ƒ1348.14
69
K D2
(0040)
(0019)
(0026)
(0055)
(0059)
(0065)
(0096)
(0053)
098
050
ƒ051
1008
ƒ1008
1009
2005
097
1
0012
ƒ0078
00025
ƒ0014
00099
ƒ0047
ƒ00068
0013
(0019)
(0062)
(0018)
(0037)
(0028)
(0012)
(0013)
(00049)
2000
(0098)
1000
(0030)
094
(016)
ƒ1331.18
97
Flexible parametric
N D 1000
K D1
Normal
‚10
1004 (0066)
‚11
055 (0068)
ƒ051 (0063)
‚12
‚20
099 (0059)
ƒ1000 (0047)
‚21
‚22
1000 (0050)
‘1
2010 (039)
‘ 12
058 (0071)
00
01
02
10
11
12
20
21
22
sdev(˜1 )
sdev(˜2 )
corr(˜1 1 ˜2 )
029 (0064)
cov(˜1 1 ˜2 )
ƒ1436.58
Log-likelihood
Rejections
1002
050
ƒ051
1001
ƒ092
093
2021
1051
1
0
0
ƒ017
(0068)
(0051)
(0047)
(0086)
(0063)
(0072)
(038)
(031)
(0032)
2008
(033)
094
(0044)
069
(0084)
1006
(039)
ƒ1362.86
100
K D2
1000
(0058)
051
(0051)
ƒ050
(0044)
1028
(010)
ƒ1026
(0097)
1026
(0094)
2017
(044)
095
(040)
1
0012
(0021)
014
(015)
0016
(0033)
ƒ0054
(012)
ƒ00027 (0013)
ƒ0086
(0035)
ƒ00023 (00077)
00096 (0013)
2025
(064)
1029
(018)
044
(014)
1040
(1024)
ƒ1315.80
100
Standard errors are given in parentheses.
NOTE:
with K D 1 and K D 2 (denoted by ¨1ü and ¨2ü ). The case
of the normal disturbances is presented in detail in Table 1;
the case of the t disturbances, in Table 2; and the case of the
chi-squared disturbances, in Table 3. The simulation results
from the heteroscedastic error terms are presented in Table 4
for the covariance matrix dependent on the observed variable
Table 3. Results of the Simulation Experiment With Bivariate
Chi-Squared Distributed Disturbances
Standard errors are given in parentheses.
x and in Table 5 for the covariance matrix dependent on an
unobserved variable.
It is remarkable how well standard maximum likelihood
under the assumption of normally distributed disturbances performs. Departures from normality do not cause serious bias
Table 5. Results of the Simulation Experiment, Where the Error Terms
are Distributed According to a Bivariate Normal Distribution With
Heteroscedasticity Depending on anpUnobservedpVariable r Taking
Three Possible Values, .5, 1, and 1.5
Flexible parametric
Flexible parametric
N D 1000
‚ 10
094 (0089)
‚ 11
050 (0029)
ƒ050 (0054)
‚ 12
‚ 20
1009 (0075)
ƒ1004 (0076)
‚ 21
‚ 22
1003 (0083)
‘1
2003 (0084)
‘ 12
1028 (015)
00
01
02
10
11
12
20
21
22
sdev(˜1 )
sdev(˜2 )
cov(˜1 1 ˜2 )
ƒ1403.24
Log-likelihood
Rejections
NOTE:
K D1
Normal
093
050
ƒ050
1005
ƒ1004
1004
2001
1027
1
0
0
ƒ0018
(0047)
(0027)
(0036)
(0039)
(0037)
(0037)
(0053)
(0053)
(0032)
1094
(0074)
097
(0038)
1012
(017)
ƒ1391.81
89
Standard errors are given in parentheses.
K D2
095
(0061)
050
(0018)
ƒ049
(0021)
1012
(0041)
ƒ1006
(0028)
1006
(0024)
1089
(0045)
1019
(0020)
1
ƒ011
(0021)
ƒ0039
(0030)
ƒ0088
(0014)
0066
(0031)
ƒ00067 (0032)
ƒ0046
(0019)
0037
(0017)
00034 (00037)
1061
(0042)
085
(0012)
056
(0066)
ƒ1345.13
100
N D 1000
‚10
1001 (0060)
‚11
050 (0027)
ƒ050 (0047)
‚12
‚20
099 (0054)
ƒ1000 (0047)
‚21
‚22
1000 (0046)
‘1
1099 (0057)
‘ 12
1000 (0098)
00
01
02
10
11
12
20
21
22
sdev(˜1 )
sdev(˜1 )
050 (0050)
corr(˜1 1 ˜2 )
cov(˜1 1 ˜2 )
ƒ1405.10
Log-likelihood
Rejections
NOTE:
K D1
Normal
1001
050
ƒ050
1000
ƒ1000
1000
2000
1000
1
0
(0035)
(0026)
(0037)
(0032)
(0033)
(0031)
(0036)
(0023)
0
ƒ00013 (0014)
1099
(0052)
1000
(0028)
050
(0020)
099
(015)
ƒ1405.09
0
Standard errors are given in parentheses.
K D2
1000
(0031)
050
(0027)
ƒ050
(0036)
1000
(0027)
ƒ1000
(0029)
1000
(0026)
2000
(0012)
1000
(0013)
1
00013 (0020)
ƒ0015
(0032)
ƒ00004 (0011)
ƒ0015
(0031)
00063 (0026)
ƒ00074 (00096)
ƒ00030 (0012)
00027 (00067)
2001
(025)
1002
(013)
050
(00091)
1009
(025)
ƒ1402.10
12
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
van der Klaauw and Koning: Testing the Normality Assumption
in the parameter estimates. In case the true disturbances follow a t distribution, all estimated parameters are within two
standard deviations of their true values. This is the case even
when the disturbances follow a transformed chi-squared distribution that is nonsymmetric. This suggests that if obtaining estimates for the parameters ‚ is the only interest, than it
is sufcient to assume normality and obtain estimates. However, in the presence of heteroscedasticity in the error terms
that is dependent on one of the observed exogenous variables,
the parameter estimates under normality are somewhat biased
(see Table 4). In particular, the parameter estimate of ‚11 differs from the true value. This bias disappears when estimating
the model using the exible parametric specication. When
the heteroscedasticity does not depend on exogenous variables
but the covariance matrix differs between groups, the parameter estimates obtained under normality are again close to their
true value.
Tables 1–5 also report the number of rejections of normality
for each simulation experiment. The simulation results indicate that the normality test against both the class ¨1ü and ¨2ü
performs well. The number of incorrect rejections is small
when compared to the level of signicance. The number of
correct rejections is very high except in the simulation experiment, where the distribution of the error terms differs between
groups. The test is very powerful when the error terms are
not bivariate normal distributed, and it is successful in detecting heteroscedasticity depending on observed exogenous variables. The simulation results indicate that the test has more
power when the tted distribution belongs to ¨2ü than when
it belongs to ¨1ü . Because departures from normality do not
cause serious bias in the parameter estimates of the regression equation, Hausman tests for normality against a semiparametric alternative probably will not be as powerful as the
likelihood-based tests discussed in this article.
The results from the simulation study suggest that the estimates for the ‚1 parameters (derived under the assumption
of normality) are reasonably robust to misspecication of the
error term. This is not true only if the covariance matrix of the
error terms depends on observed explanatory variables. In this
case, using the exible parametric density can help improve
the parameter estimates. Furthermore, if one is interested not
only in the parameter estimates of ‚1 , but also in the complete distribution function of outcome variable y conditional
on explanatory variables x, then using the exible parametric
specication is worthwhile.
4.
AN APPLICATION TO A MODEL OF
TRAVEL DEMAND
In this section we apply the exible parametric estimation
procedure and the normality tests to a model describing travel
demand. This section consists of three parts: rst, we present
a structural model, then we discuss the data, and nally, we
present the empirical results.
The model assumes that in a given year households need to
travel a stochastic number of kilometers and that these households minimize the costs of traveling. At the beginning of the
year, a household makes a prediction of the number of kilometers to be traveled, which is denoted by y ü . The actual number
37
of kilometers traveled in this year is related to the prediction
by y D y ü ¢ ˜, where ˜ is a random prediction error with a positive support and a nite mean Œ. The household can choose
to travel either by car or by other means. The latter is most
likely public transportation; other possibilities may be bikes,
car pooling, and so forth. The cost of using a car consists of
xed costs, cc , and variable costs, vc . Fixed costs are depreciation of the car, maintenance, insurance, and taxes for car
ownership. Variable costs are the costs of fuel. The costs of
other transportation do not contain any xed costs. The variable costs, denoted by vo , are associated with the costs of
public transportation. However, opportunity costs may also be
important, because traveling with public transportation usually
takes more time than traveling by car.
The household decision problem now involves choosing
whether or not to use a car. Assuming that (risk-neutral)
households minimize their expected costs, the decision rule
implies that a car is used if
E6cc C vc y7 µ E6vo y70
(6)
Now assume that cc , vc , and vo are known by the household.
This means that the household decision simplies to using a
car if
cc
0
E6y7 ¶
(7)
vo ƒ vc
If we substitute the relation between the actual number of
kilometers driven and the predicted number of kilometers, then
this implies that a household chooses to use a car if
yü ¶
cc
0
Œ4vo ƒ vc 5
Now we parameterize the unknown components of the
model as
y ü D exp4x 0 ‚ C v1 51
which is the distance function and a costs function
cc
D exp4z0 ƒ C v2 51
vo ƒ vc
where v1 and v2 denote unobserved individual specic effects.
We normalize ΠD 1.
The model reduces to a regression equation, which has the
structure
log4y5 D x 0 ‚ C ˜1
with ˜1 D v1 C ln4˜5. A selection equation indicates whether
the household owns a car,
x 0 ‚ ƒ z0 ƒ C ˜2 ¶ 01
with the household-specic term ˜2 equal to v1 ƒ v2 , which is
known to the household but unknown to the econometrician.
Because v1 is included in both ˜1 and ˜2 , these disturbances
are not a priori independent.
To improve the identication of the model, we impose an
exclusion restriction, that is, nd a variable that is included in
z and excluded from x. According to the model, the predicted
number of kilometers affects both the actual number of kilometers and the decision to buy a car, whereas the xed and
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
38
Journal of Business & Economic Statistics, January 2003
variable costs of using a car and the variable costs of other
transportation affect only the decision of whether to use a car.
Thus the exclusion restriction should be a variable that affects
only the costs functions. Because the Dutch government provides free public transportation passes to some students and
some civil servants, we use the dummy variable that indicates
whether the household has such a free public transportation
pass as a variable which is included in z but not in x.
The data we use is a subset from the Dutch database
on transportation behavior of Statistics Netherlands. This
database contains 34,454 households. To avoid complications of households owning more than one car, we focus on
single-person households. Furthermore, we exclude individuals younger than age 18 years, the legal age for obtaining a
drivers license in The Netherlands. This restricts the database
to 7404 individuals. To construct our nal dataset, we also
exclude 130 individuals who own a car but for whom the number of kilometers driven in the past year is unknown and also
756 individuals for whom one or more explanatory variables
are missing. Our nal dataset consists of 6518 observations.
Table 6 provides some characteristics of the dataset. The
data contain 3110 individuals who own a car and 3408 who
do not own a car. Except for region, all variables display differences in car ownership rates. Although 58% of the men are
car owners, only 41% of the women have a car. Until people
reach age 65, the car ownership rate increases with age; after
that, there is a large drop. Furthermore, car ownership rates
increase with income and level of education. Finally, individuals living in areas with a low degree of urbanization, fulltime employed workers, and individuals who do not have a
government-provided free public transportation pass are more
likely to own a car than their counterparts.
Using these data, we estimate the structural model discussed
previously. In the rst step, we estimate the model under the
assumption that the disturbances ˜1 and ˜2 follow a bivariate
normal distribution. Next, we relax this assumption by assuming that the density of the disturbances belongs to either ¨1ü
or ¨2ü . For these two cases, we test whether the restriction of
normality of the disturbances can be imposed.
All parameter estimates are presented in Tables 7 and 8. The
parameter estimates of the distance function (Table 7) and of
the costs function (Table 8) are not very sensitive to the specication of the distribution function of the error term, which
is in line with the results from the simulation study. The most
important covariates in the distance function are gender, age,
income, level of education, and individual labor market status;
those in the costs function are age, degree of urbanization, and
income. Although the availability of a free public transportation pass provided by the government has the expected effect
on the costs function, the corresponding parameter estimate is
not signicantly different from 0.
The only individual characteristics important in both the distance and costs functions are age and income. Age affects both
the distance function and the costs function negatively. Young
people travel more kilometers than older people, whereas the
costs of car use relative to other transportation modes decrease
with age. By comparing the coefcients, we can see that persons age 50–64 are most likely to own a car. Income has an
opposite effect on the distance and costs functions. Persons
Table 6. Some Characteristics of the Dataset
Car owner
Gender
Male
Female
Age
18–24
25–29
30–39
40–49
50–64
65+
Region
West
North
East
South
Degree of urbanization
Very high
High
Average
Low
Very low
Net income (guilders)
0–15,000
15,000–23,000
23,000–30,000
30,000–38,000
38,000–52,000
52,000+
Level of education
Primary
Lower secondary
Higher secondary
University
Labor market status
Full-time work
Part-time work
Student
Unemployed
Nonparticipant
Free public transport pass
No
Yes
Average number of kilometers
Observations
Yes
No
1523
1587
1102
2306
106
416
746
467
692
683
518
415
485
296
451
1243
1588
211
895
416
1881
207
959
361
764
786
694
563
303
1329
970
497
397
215
158
411
539
695
773
534
999
1046
574
402
278
109
237
712
1022
1139
688
856
1075
789
1626
178
37
105
1164
771
245
406
250
1736
3074
36
151093
3110
2965
443
3408
with greater income travel more kilometers, whereas the costs
of car use relative to other transportation decrease. Considering that using public transportation takes more time, this latter
effect can be explained by differences in opportunity costs.
Time is more costly for individuals with high incomes. The
degree of urbanization affects only the costs function. Because
less public transportation is available in areas with low degrees
of urbanization, the costs of using public transportation are
higher in these areas (e.g., waiting times are longer). In contrast, car use in areas with a high degree of urbanization is
more costly, because individuals often incur additional costs,
such as parking costs. Employed individuals, both full-time
and part-time, travel more kilometers. Because we do not distinguish between traveling for private purposes and traveling
for professional purposes, this may be caused by commuting
or because their work requires them to travel. Finally, persons
with a higher level of education travel more than those with
less education.
van der Klaauw and Koning: Testing the Normality Assumption
39
Table 8. Estimation Results of the Travel Demand Model:
Costs Function
Table 7. Estimation Results of the Travel Demand Model:
Distance Function
Distance function (‚)
Costs function (ƒ)
Flexible parametric
KD1
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
Normal
Intercept
9041
Gender
ƒ029
Female
Age
ƒ013
25–29
ƒ024
30–39
ƒ028
40–49
ƒ033
50–64
ƒ059
65+
Region
North
00027
ƒ00007
East
ƒ0054
South
Degree of urbanization
High
0025
Average
0064
Low
0077
Very low
0064
Net income (guilders)
15,000–23,000
0086
23,000–30,000
023
30,000–38,000
030
38,000–52,000
037
52,000+
050
Level of education
Lower secondary
010
Higher secondary
017
University
028
Labor market status
Part-time work
0017
ƒ028
Student
ƒ029
Unemployed
ƒ025
Nonparticipant
‘1
063
‘ 12
0041
00
01
02
10
11
12
20
21
22
NOTE:
(032)
9077
(0038) ƒ024
(0088)
(0084)
(0089)
(010)
(0086)
ƒ014
ƒ023
ƒ030
ƒ035
ƒ050
(011)
K D2
9087
(015)
(0022) ƒ023
(0026)
ƒ015
ƒ023
ƒ030
ƒ036
ƒ050
(0066)
(0064)
(0067)
(0071)
(0073)
(0061)
(0059)
(0062)
(0064)
(0067)
(0050)
00066 (0045) 00010 (0047)
(0031)
0013 (0027) 0016 (0028)
(0040) ƒ0057 (0034) ƒ0061 (0034)
(0032)
(0035)
(0038)
(0044)
ƒ0012
ƒ0035
ƒ0033
ƒ0037
(0033)
(0039)
(0043)
(0050)
(0075) ƒ012
(011)
0089
(014)
012
(016)
017
(018)
030
(0061)
(0065)
(0066)
(0069)
(0072)
0022
0093
011
015
027
(0067)
(0075)
(0081)
(0086)
(0090)
(0051)
(0062)
(0064)
(0040)
(0041)
(0043)
0035
0085
020
(0044)
(0046)
(0047)
(0042)
(0066)
(0073)
(0082)
(0058)
(011)
(0069)
(0046)
(00088)
(017)
ƒ00070
ƒ00088
ƒ00043
ƒ0011
0043
0097
021
0023
ƒ019
ƒ018
ƒ023
071
ƒ045
1
0
0
049
(0050)
(010)
(0059)
(0037)
(0018)
(0034)
Flexible parametric
0035
ƒ016
ƒ017
ƒ022
076
ƒ054
1
ƒ00035
0038
ƒ0012
(0031) 038
015
ƒ0052
016
00005
(0052)
(012)
(0063)
(0039)
(023)
(031)
(0049)
(012)
(0045)
(041)
(012)
(014)
(0079)
(0098)
Standard errors are in parentheses.
The likelihood ratio test statistics of normality against the
exible parametric density with K D 1 and K D 2 equal 283.7
and 299.6, implying that we must reject the hypothesis that
the disturbances follow a normal distribution. The main difference with the estimates under normality is that we do not nd
any correlation between ˜1 and ˜2 under the assumption of
normality. This suggests that there is no unobserved selection
between car ownership and car use, which implies that both
the regression and selection equations can be estimated consistently separately of each other by, for example, OLS and
probit. The exible parametric densities show a correlation
between both disturbance terms that could not be captured by
a normal density. The estimated covariance between ˜1 and ˜2
is ƒ023 for K D 1 and ƒ030 for K D 2. In the context of our
theoretical model, this implies a positive correlation between
Normal
Intercept
10081 (035)
Gender
Female
0046 (0053)
Age
ƒ049 (013)
25–29
ƒ056 (013)
30–39
ƒ065 (013)
40–49
ƒ1000 (015)
50–64
ƒ076 (014)
65+
Region
ƒ011 (0092)
North
East
0049 (0058)
ƒ016 (0072)
South
Degree of urbanization
ƒ018 (0068)
High
ƒ051 (0086)
Average
ƒ057 (0096)
Low
ƒ065 (011)
Very low
Net income (guilders)
ƒ021 (010)
15,000–23,000
ƒ048 (014)
23,000–30,000
ƒ072 (016)
30,000–38,000
ƒ090 (018)
38,000–52,000
ƒ1004 (020)
52,000+
Level of education
ƒ016 (0079)
Lower secondary
ƒ026 (0089)
Higher secondary
ƒ015 (0095)
University
Labor market status
Part-time work
015 (0095)
Student
014 (019)
Unemployed
0058 (011)
ƒ0023 (0082)
Nonparticipant
Free public transport pass
Yes
021 (016)
ƒ6386.34
Log-likelihood
NOTE:
KD1
10086
(014)
0017 (0038)
ƒ041
ƒ046
ƒ058
ƒ085
ƒ061
(010)
(010)
(011)
(011)
(012)
K D2
10091
(029)
0047 (0095)
ƒ040
ƒ046
ƒ058
ƒ086
ƒ059
(013)
(012)
(013)
(018)
(013)
ƒ0075 (0079) ƒ0077 (0088)
0060 (0050)
0065 (0055)
ƒ013 (0062) ƒ013 (0070)
ƒ018
ƒ047
ƒ053
ƒ059
(0056)
(0057)
(0064)
(0078)
ƒ018
ƒ049
ƒ056
ƒ061
(0076)
(015)
(017)
(019)
ƒ026
ƒ048
ƒ068
ƒ083
ƒ090
(0085)
(0089)
(0090)
(0093)
(010)
ƒ020
ƒ046
ƒ069
ƒ086
ƒ095
(011)
(017)
(024)
(030)
(037)
ƒ016
ƒ024
ƒ012
(0066) ƒ016
(0068) ƒ025
(0075) ƒ013
(0089)
(012)
(012)
013
013
011
ƒ0054
(0080)
015
(016)
016
(0095)
013
(0068) ƒ0040
(0094)
(019)
(014)
(0094)
014 (012)
ƒ6244.49
015 (013)
ƒ6236.54
Standard errors are in parentheses.
v1 and v2 , because cov4˜1 1 ˜2 5 D var4v1 5 ƒ cov4v1 1 v2 5. Obviously, there are some unobserved covariates, which increase
both the costs of car use relative to other transportation and the
expected number of kilometers traveled. Thus such a covariate
has similar effects to, for example, age. Note that we maintained the assumption that the prediction error ˜ is independent of the individual specic effects v1 and v2 .
Figures 4 and 5 show the marginal densities of the disturbances ˜1 and ˜2 estimated under the exible parametric densities and normality. Both exible parametric densities have
more mass close to the mode and slightly fatter tails than the
normal density. The estimated standard deviation of ˜1 is .64
in both the K D 1 and K D 2 specications, almost similar to
the estimate under normality. For the standard deviation of ˜2 ,
we nd .89 and .90 for ˜2 for the K D 1 and the K D 2 density.
Figures 6 and 7 graph the estimated densities of the normal
model and the model estimated using an exible parametric
density (K D 2). We see that the estimated exible parametric density is a bit more spread out than the normal density,
and that the estimated contour lines in Figure 7 are not the
ellipsoids of Figure 6.
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
40
Journal of Business & Economic Statistics, January 2003
Figure 4. The Estimated Marginal Density Function of ˜1 .
5.
CONCLUSION
In this article we have derived a test for normality in sample selection models. To our knowledge, no such test has been
derived previously. The test is motivated on the seminonparametric maximum likelihood method introduced by Gallant and
Nychka (1987). A exible parametric density function is used
to approximate the density function of the error terms. The
bivariate normal density is a special case of this exible parametric density function that allows us to test for normality.
The test also provides an alternative density function in the
event that normality is rejected.
Although the simulation study provided in this article is
limited, we believe that this test is promising. The test performed well in almost all simulation experiments, the percentage of incorrect rejections is below the signicance level,
whereas the power is high. This is not true only when the
Figure 6. Estimated Density Under the Normality Assumption.
Figure 5. The Estimated Marginal Density Function of ˜2 .
distribution of the error terms is heteroscedastic depending on
an unobserved variable. The test is powerful against departures
from normality and against heteroscedasticity depending on
an observed exogenous variable. However, it should be noted
that in most cases the parameter estimates do not seem very
sensitive to the distributional assumptions of the disturbances.
Even in the event that the disturbances do not follow a normal
distribution, maximum likelihood under the assumption of normality provides estimates close to the true values. Only if the
covariance matrix of the error terms depends on an observed
exogenous variable are the parameter estimates biased. The
simulation study indicates that the bias disappears when using
the exible parametric density. The exible parametric density
is thus useful in cases where not only parameter estimates, but
also the complete distributional structure are of interest, and
in the case of heteroscedasticity.
van der Klaauw and Koning: Testing the Normality Assumption
41
APPENDIX A: RELEVANT INTEGRALS OF THE
FLEXIBLE PARAMETRIC DENSITY IN
THE SAMPLE SELECTION MODEL
In the sample selection model, all relevant integrals are of
the form
Zˆ
hü 4˜5d˜2
a
and
Z
b
Z
ˆ
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
ƒˆ
Z
ˆ
ƒˆ
hü 4˜5d˜1 d˜ 2
This latter integral also appears in S when b D ˆ and in the
mean restrictions where also b D ˆ and hü 4˜5 is replaced by
˜i hü 4˜5 for i D 11 2. (Because of the structure of the Hermite
series, this solves similarly.)
Substituting for hü 4˜5, we obtain integrals of the type
and
Z
b
ƒˆ
Z
a
ˆ
ƒˆ
j
˜i1 ˜2 ”4˜1 1 ˜2 5d˜2
j
˜i1 ˜2 ”4˜1 1 ˜2 5d˜ 1 d˜2 1
where ”4˜1 1 ˜2 5 is the bivariate normal density function.
Because
”4˜1 1 ˜2 5 D ”4˜2 —˜1 5”4˜1 51
we can rewrite these integrals as
˜i1 ”4˜1 5
Z
ˆ
a
j
˜2 ”4˜2 —˜1 5d˜ 2
and
Z
b
ƒˆ
j
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Testing the Normality Assumption in the Sample
Selection Model With an Application to Travel
Demand
Bas van der Klaauw & Ruud H Koning
To cite this article: Bas van der Klaauw & Ruud H Koning (2003) Testing the Normality
Assumption in the Sample Selection Model With an Application to Travel Demand, Journal of
Business & Economic Statistics, 21:1, 31-42, DOI: 10.1198/073500102288618739
To link to this article: http://dx.doi.org/10.1198/073500102288618739
Published online: 01 Jan 2012.
Submit your article to this journal
Article views: 58
View related articles
Citing articles: 1 View citing articles
Full Terms & Conditions of access and use can be found at
http://www.tandfonline.com/action/journalInformation?journalCode=ubes20
Download by: [Universitas Maritim Raja Ali Haji]
Date: 13 January 2016, At: 00:38
Testing the Normality Assumption in the
Sample Selection Model With an
Application to Travel Demand
Bas van der Klaauw
Department of Economics, Free University Amsterdam, De Boelelaan 1105, NL-1081 HV Amsterdam,
The Netherlands and Scholar (klaauw@tinbergen.nl)
Ruud H. Koning
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
Department of Econometrics, University of Groningen, P.O. Box 800, NL-9700 AV Groningen,
The Netherlands (r.h.koning@eco.rug.nl)
In this article we introduce a test for the normality assumption in the sample selection model. The test is
based on a exible parametric specication of the density function of the error terms in the model. This
specication follows a Hermite series with bivariate normality as a special case. All parameters of the
model are estimated both under normality and under the more general exible parametric specication,
which enables testing for normality using a standard likelihood ratio test. If normality is rejected, then
the exible parametric specication provides consistent parameter estimates. The test has reasonable
power, as is shown by a simulation study. The test also detects some types of ignored heteroscedasticity.
Finally, we apply the exible specication of the density to a travel demand model and test for normality
in this model.
KEY WORDS:
1.
Flexible parametric density estimation; Hermite series; Heteroscedasticity; Sample
selection.
INTRODUCTION
eduction (Heckman, Tobias, and Vytlacil 2000), and to estimate the effect of union membership on earnings (Lee 1978).
In the switching regression model, many of the interesting
policy parameters, such as the treatment effect on the treated,
depend not only on the functional form of the relationship, but
also on the joint distribution of the error terms. Therefore, for
policy evaluation and also for simulations, both the parameters
of the functional form of the relationship and the joint distribution of the error terms must be known. Of course, this does
not hold for either the switching regression model, or also for
other models, like the sample selection model.
Usually, the sample selection model is estimated either by
maximum likelihood or by Heckman’s two-step procedure
(Heckman 1979), under the assumption that the error term
in the regression equation and the error term in the selection
equation follow a bivariate normal distribution. However, it
is often argued that the parameter estimates are sensitive to
the distributional assumptions (Manski 1989). In fact, Newey
(1999a) proved that only under very restrictive assumptions,
the parameter estimates are consistent against misspecication
of the distribution of the error terms. In many cases, the normal distribution might be overly restrictive. As mentioned earlier, the sample selection model is often used to model earnings, and it is well known from the economic literature that
the distribution of (the logarithm of) earnings in the population has fatter tails than the normal distribution function (e.g.,
Heckman et al. 2000).
Maximum likelihood is the most popular estimation method
in microeconometrics. The method yields consistent (in fact,
asymptotically efcient) estimators if the model is specied
correctly. However, correct specication may not be known
beforehand. Two major sources of misspecication are incorrect specication of the functional form of the relationship
under study (e.g., omitting exogenous variables or inappropriately assuming linearity) and misspecication of the stochastic
structure of the model (e.g., neglecting heteroscedasticity or
misspecication of the distribution of the random variables).
The maximum likelihood estimator is generally inconsistent
in these cases. This is particularly true in the case of limited
dependent variable models (e.g., Hurd 1979; Smith 1989). In
this article we focus on one particular form of misspecication: misspecication of the distribution of the error terms. We
retain the assumption of correct specication of the functional
form of the relationship.
The model that we study is the sample selection model
introduced by Heckman (1979). This type of model accounts
for problems arising because the outcome of the endogenous
variable is observed only for a (selective) part of the sample. For example, sample selection models are used to study
wage equations, where only the wages of employed workers
are observed (e.g., Buchinsky 1998; Melenberg and Van Soest
1993). This model is not only very useful when the outcome
variable is only observed for a (nonrandom) part of the population, but it can easily be extended to the switching regression
model. The switching regression model is the point of departure for many interesting problems in economics. For example, it is used to evaluate active labor market programs (Heckman, LaLonde, and Smith 1999), to investigate the effect of
migration on earnings (Tunalõ 2000), to study the returns to
© 2003 American Statistical Association
Journal of Business & Economic Statistics
January 2003, Vol. 21, No. 1
DOI 10.1198/073500102288618739
31
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
32
Journal of Business & Economic Statistics, January 2003
We introduce a formal test for the normality assumption
within the sample selection model. The test is motivated by the
seminonparametric maximum likelihood estimation method of
Gallant and Nychka (1987) in which the true distribution of
the error terms is approximated by a Hermite series. The
density of the error terms follows a very exible parametric
specication, which is estimated using maximum likelihood
together with all other parameters of the model. The test thus
provides an alternative distribution of the error terms in the
event that normality is rejected. Because the parameters are
estimated
p using maximum likelihood, the estimators are efcient, N consistent, and asymptotically normal. To examine the power of the normality test, we perform a simulation
study, which will also provide practical experience about the
actual exibility of the parametric specication of the density. For the binary choice model and duration models, Smith
(1989) proposed a similar type of polynomial specication to
test for normality. Gabler, Laisney, and Lechner (1993) provided Monte Carlo evidence on this test and applied it to a
model of female labor force participation.
So far we did not discuss heteroscedasticity of the error
terms. Ignored heteroscedasticity causes the usual estimators in limited dependent-variable models to be inconsistent.
For example, Hurd (1979) showed that the bias caused by
ignored heteroscedasticity is severe in the truncated regression model. Not many estimation procedures are robust against
heteroscedasticity. An exception is the quantile regression
approach of Buchinsky (1998). The disadvantage of quantile
regression methods is that the complete distribution of the
error terms is not estimated. This complicates performing simulations, and because the mean is not estimated, it is difcult
to dene relevant policy parameters (see Abadie, Angrist, and
Imbens 1998, who applied the quantile regression estimator to
the switching regression model). In our simulation study we
also focus on the robustness of the exible parametric density
against heteroscedastic error terms.
An advantage of the exible parametric density used in this
article is its general applicability; it is not specic to one particular econometric model. The approach taken herein can be
used to examine the sensitivity of estimation results to the
assumption of normality in other microeconometric models
where no formal test of normality is available. We illustrate
both the exible parametric density and the normality test in
a model of travel demand and car ownership.
The article is organized as follows. Section 2 discusses the
sample selection model and the exible parametric density
that naturally leads to a normality test for the sample selection
model. Section 3 reports some simulations to investigate the
performance of the exible parametric density and the power
of the normality test. Section 4 discusses an application of
the estimation method and the normality test to the model of
travel demand. Finally, Section 5 concludes.
2.
IDENTIFICATION AND ESTIMATION OF THE
SAMPLE SELECTION MODEL
In this section we discuss some identication and estimation issues associated with the sample selection model. In particular, we focus on the exible parametric density that leads
to a test for bivariate normality.
Within a population, we are interested in the distribution
of an outcome variable y conditional on a vector of exogenous variables x1 . Therefore, we specify for individual i 4i D
11 : : : 1 N 5 the regression equation
yi D ‚01 x1i C ˜1i 0
(1)
The dependent variable yi is observed for a subsample only. A
binary variable zi is introduced that equals 1 if the realization
of yi is observed and otherwise equals 0. The selection rule
depends on a set of exogenous variables x2 and is given by
züi D ‚02 x2i C ˜2i
and
zi D
(
11
01
züi > 0
züi µ 00
(2)
As mentioned in Section 1, we focus only on the joint distribution of the error terms 4˜1i 1 ˜2i 5, and thus we simply assume
linear specications in both the regression and the selection
equation. If the conditional expectation of ˜1i given zi D 1
does not equal 0, ordinary least squares (OLS) estimation of
(1) will not yield consistent estimates for the parameters of
interest, ‚1 .
If one is willing to assume that the error terms 4˜1i 1 ˜2i 5 follow a bivariate normal distribution, then the model can be estimated using Heckman’s two-stage procedure (see Heckman
1979). In the rst step, a probit model is estimated for the
selection equation to obtain ‚O 2 , in the second step, the regression equation
yi D
‚01 x1i
0
”4‚O 2 x2i 5
C ‡i
C ‘ 1
0
ê4‚O x 5
2
(3)
2i
is estimated (by OLS) for the subsample for which a realization of yi is actually observed 4zi D 15. The parameter
denotes the correlation between ˜1 and ˜2 , and ‘ 12 is the
variance of ˜1 (the variance of ˜2 is normalized to 1). If
the vector x1i equals the vector x2i , then identication of the
model hinges entirely on the functional form and distributional
assumptions, which makes the parameter estimates very sensitive to the distributional assumptions (see Manski 1989 for
an extensive overview). Therefore, in most economic applications, an exclusion restriction is imposed—at least one variable enters only the selection equation and not the regression
equation.
The bivariate normality assumption may be overly restrictive. Therefore, starting with Lee (1982), who modied Heckman’s two-step procedure to allow in a general way for
departures from normality, a number of estimation methods
have been introduced that relax the distributional assumptions. These (semiparametric) estimation methods generalize the probit estimation in the rst step to, for example,
kernel-based estimation and use more exible error correction terms in the regression equation (e.g., Ahn and Powell
1993; Cosslett 1991; Ichimura and Lee 1991; Newey 1999b).
This provides consistent estimators for the parameters of interest ‚1 , but computation of the corrected standard errors of
these parameters is quite cumbersome (e.g., Ahn and Powell
van der Klaauw and Koning: Testing the Normality Assumption
33
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
1993; Cosslet 1991). A main disadvantage of most semiparametric estimation methods is that the intercept in the outcome
equation remains unidentied (and thus cannot be estimated).
As stressed by Heckman (1990), in many economic applications the intercept is an important parameter of interest.
The alternative to two-step estimation is maximum likelihood estimation. This requires full specication of the joint
density function of the error terms, which is denoted by
f 4˜1 1 ˜2 5. Under the assumption of correct specication, maximum likelihood estimation yields consistent estimates of all
parameters of interest and straightforward computation of the
asymptotic distribution. The log-likelihood function of the
sample selection model is
log ¬ D
N
X
iD1
³Z
zi log
ˆ
ƒ‚02 x2i
f 4yi ƒ ‚01 x1i 1 ˜2 5d˜2
³Z
C 41 ƒ zi 5 log
ƒ‚02 x2i
ƒˆ
Z
ˆ
ƒˆ
´
´
f 4˜1 1 ˜2 5d˜ 1 d˜2 0
(4)
In maximum likelihood estimation, consistency depends
crucially on the correct specication of the joint density of the
error terms. Therefore, it is convenient to specify this density
so that it is exible. We use exible parametric specications
that are inspired by seminonparametric estimation as introduced by Gallant and Nychka (1987). The unknown density
function f 4˜1 1 ˜2 5 is approximated by a Hermite series.
We consider densities h4˜5 in a subclass ¨K that are of the
Hermite form
hü 4˜5 D PK2 4˜5”2 4˜—è51
(5)
where PK 4˜5 is a polynomial of degree K and ”4˜—è5 is the
normal density function with mean 0 and covariance matrix è.
In fact, Gallant and Nychka (1987) restricted è to being a
diagonal matrix. Because of the squaring, no restrictions are
necessary to ensure that hü 4˜5 is nonnegative. Some restrictions are required to ensure that the density integrates to 1 and
for identication. We return to these restrictions later.
Gallant and Nychka (1987) showed that a large class ¨
of density functions can be approximated arbitrarily well by
increasing the number of terms K of the polynomial PK 4˜5.
They gave the precise conditions dening ¨ . For our purposes, it sufces to note that the fattest tails allowed are t-like
tails, and the thinnest tails allowed are thinner than normal-like
tails. Any sort of skewness and kurtosis (especially in that part
of the distribution where most probability mass is observed)
is allowed, and only very violently oscillatory densities are
excluded from ¨ . Gallant and Nychka (1987) proved that the
densities in ¨ can be estimated consistently by increasing K
with the number of observations. In (5), the normal density
is used as the base class for ¨K , but this is not necessary;
any density with a moment-generating function could be used
(see, e.g., Cameron and Johansson 1997).
It is also possible to assume that the true density is a member of ¨K and hence to interpret ¨K as a exible class
of density functions. This is the interpretation that we follow in this article. This interpretation is especially appealing
if one wants to examine the sensitivity of estimation results
obtained by assuming normality to this distributional assumption, because it allows one to use the standard framework
of inference. Because we want to test for bivariate normality
(with unrestricted correlation) in the sample selection model,
we do not restrict the covariance matrix, unlike Gallant and
Nychka (1987). This provides a formal test for normality in
the sample selection model; that is, PK 4˜5 is constant for all
values of ˜. Such a test for normality has not yet been derived
in the literature. An alternative could be to perform Hausman
tests on the slope coefcients of the regressions equation. One
could either use Hausman tests to test normality against a
semiparametric alternative or also against the exible parametric alternative discussed in this article. The rst alternative
may be somewhat cumbersome, because it requires computing standard errors in the semiparametric estimation, which is
not always straightforward. The second alternative is easy to
implement, but these tests are not expected to be as powerful
as the tests derived in this article. The simulations in Section 3
indicate that the slope coefcients are not very sensitive to
departures from normality. Hausman tests leave out any information on the distribution of the error terms.
In line with Gallant and Nychka (1987), we parameterize
hü 4˜5 as
´2
³ K
X
ij ˜i1 ˜j2 exp4ƒ˜0 èƒ1 ˜5
hü 4˜5 D
i1 jD0
D
K
X
jCl
i1 j1 k1 lD0
ƒ 0 ƒ1
ij kl ˜iCk
1 ˜2 exp4 ˜ è ˜50
This bivariate density can be easily generalized to a density function for more variables, which is important when,
for example, studying switching regression models containing
three error terms. To ensure that this density integrates to 1,
a restriction on the parameters is necessary. This restriction
can take the form of an explicit restriction on the parameters
of the density. However, for computational convenience, we
follow Gabler et al. (1993), who ensured integration to 1 by
scaling the density. Dene S by
SD
D
Z
K
X
R2 i1 j1 k1 lD0
K
X
i1 j1 k1 lD0
jCl
ij kl ˜1iCk ˜2 exp4ƒ˜0 èƒ1 ˜5d˜
ij kl
Z
ˆ
ƒˆ
jCl
˜2
Z
ˆ
ƒˆ
˜iCk
exp4ƒ˜0 èƒ1 ˜5d˜ 1 d˜2 0
1
Because of the denition of S, the following density integrates
to 1:
h4˜5 D hü 4˜5=S0
It is clear that the parameters are identied up to a scale
only, so normalization is necessary. Therefore, we impose
00 D 1. For K D 0, h4˜5 now reduces to the bivariate normal
density. The exibility of this family of densities is illustrated
in Figures 1–3 (K D 2). It is clear that the contour lines differ
from the usual ellipsoids of the bivariate normal density.
In the sample
p selection model, identication is achieved by
setting è22 D 2, to ensure identication of the scale of (2).
To x the location of the distribution function, we optimize
the log-likelihood function (4) under the restrictions that the
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
34
Journal of Business & Economic Statistics, January 2003
Figure 1. Bivariate Flexible Parametric Density, 01 D .1, 10 D - .1,
and 11 D - .2.
Figure 2. Bivariate Flexible Parametric Density, 01 D .1, 10 D .1,
and 11 D 0.
means of ˜1 and ˜2 equal 0. Alternatively, one could impose
explicit restrictions on the parameters to ensure that the
means of ˜1 and ˜2 are 0. However, for K ¶ 2, these are rather
complex nonlinear restrictions. Another alternative was proposed by Melenberg and Van Soest (1993), who suggested not
imposing restrictions on the parameters of the density function
of ˜ to ensure a zero mean, but rather restricting the intercepts
of (1) and (2). For the purpose of this article, such a strategy is
not very useful. Our approach allows testing the null hypothesis that 4˜1 1 ˜2 5 is distributed according to a normal distribution function against the alternative hypothesis that 4˜1 1 ˜2 5
has some other mean zero bivariate distribution function in the
class of distribution functions ¨K for any xed K. In other
words, for some xed K, we test for joint signicance of all
ij with i C j ¶ 1. The additional number of parameters under
the alternative hypotheses compared to the null hypotheses
is 4K C 152 ƒ 1. Note that there are two restrictions on these
parameters to x the location of the distribution. Therefore,
the likelihood ratio test statistic is distributed according to a
chi-squared distribution with 4K C 152 ƒ 3 degrees of freedom.
Optimizing the log-likelihood function subject to the means
of ˜1 and ˜2 being equal to 0 requires evaluating the relevant
integrals in the log-likelihood function and the expressions for
the means. Because of the general covariance matrix in the
base function ”4˜—è5, these integrals do not have the same
computationally attractive properties as they do when using
the base function proposed by Gallant and Nychka (1987).
However, because of the special structure of the sample selection model, we are able to avoid evaluations of bivariate normal integrals, so the computational cost of this generalization
is limited. Appendix A gives the recursion formulas that can
be used to determine the relevant integrals explicitly.
Finally, the exible parametric density discussed earlier
assumes homoscedasticity of the error terms. As is clear from
(3), the estimators are not robust against ignored heteroscedasticity (i.e., ‘ 1 is not the same for all observations). In the
van der Klaauw and Koning: Testing the Normality Assumption
35
and
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
züi D ‚20 C ‚21 vi C ‚22 wi C ˜2i 1
i D 11 : : : 1 N 1
with true parameters ‚10 D 1, ‚11 D 05, ‚12 D ƒ05, ‚20 D 1,
‚21 D ƒ1, and ‚22 D 1. The exogenous variables xi and vi
are independently ® 401 35 distributed, and wi is distributed
uniformly on 6ƒ31 37. We perform ve experiments in which
we vary the distribution of ˜. The sample size N is 1000.
(Similar simulation experiments with a sample size N equal
to 500 gave the same results and are available on request.)
Within each experiment, we draw 100 samples. In the rst
three experiments, we draw ˜ from either a bivariate normal distribution with mean 0, a bivariate t distribution, or
a centered chi-squared distribution. The t distribution has
fatter tails than the normal distribution and the chi-squared
distribution is asymmetric, so both these cases are a deviation from normality. For all experiments, we set var4˜1 5 D 4,
var4˜2 5 D 1, and cov4˜1 1 ˜2 5 D 1. In the last two experiments, the error terms are bivariate normal distributed but
heteroscedastic. First, the distribution of ˜ depends on the
explanatory variable x. Second, we distinguish three groups
of individuals who have different covariance matrices. (This
simulation experiment has some similarities with experiments
in Hurd 1979.) The details of the simulation experiments are
given in Appendix B. The simulations were performed on Pentium workstations using the constrained maximum likelihood
(CML) library of GAUSS.
Because the computer time required for optimization of the
log-likelihood function increases quickly with K, we estimate
only the model for K D 1 and K D 2 and for the bivariate
normal distributed disturbances. Hence we consider the normality test against the class of exible parametric densities
Table 1. Results of the Simulation Experiment With Bivariate Normal
Distributed Disturbances
Figure 3. Bivariate Flexible Parametric Density, 01 D .1, 10 D - .1,
and 11 D .2.
next section we also discuss the performance of the exible
parametric density in case the error terms are heteroscedastic. In particular, we focus on heteroscedasticity in ˜1 when
the correlation between ˜1 and ˜2 is constant. Note that the
variance of ˜2 is generally unidentied. Thus many types of
heteroscedasticity in the distribution of ˜2 could be solved by
generalizing the single index in (2).
3.
SIMULATION STUDY
In this section we perform a limited simulation exercise to
investigate the performance of the exible parametric density
and the power of the normality test discussed in Section 2.
We focus two main sources of inconsistency, departures from
normality and heteroscedasticity in the error terms.
We consider the simulation experiment
yi D ‚10 C ‚11 xi C ‚12 wi C ˜1i
Flexible parametric
N D 1000
‚ 10
‚ 11
‚ 12
‚ 20
‚ 21
‚ 22
‘1
‘ 12
00
01
02
10
11
12
20
21
22
1000
050
ƒ051
1000
ƒ099
099
2000
1000
(0072)
(0025)
(0051)
(0060)
(0045)
(0054)
(0060)
(0034)
1000
050
ƒ051
1000
ƒ099
099
2000
1000
1
0
0
ƒ00002
sdev(˜1 )
sdev(˜2 )
cov(˜1 1 ˜2 )
Log-likelihood
Rejections
NOTE:
K D1
Normal
ƒ1412.71
(0073)
(0024)
(0049)
(0062)
(0043)
(0054)
(0059)
(0036)
(00064)
2000
(0061)
1000
(0013)
1000
(0087)
ƒ1412.35
Standard errors are given in parentheses.
0
K D2
1000
050
ƒ051
1000
ƒ099
099
2000
1000
1
00001
00026
00003
00034
00006
ƒ00004
ƒ00005
ƒ00014
(0074)
(0025)
(0049)
(0061)
(0043)
(0054)
(0059)
(0036)
(0019)
(0012)
(0010)
(0017)
(0017)
(00051)
(00067)
(00050)
2000
(0070)
1000
(0020)
1001
(010)
ƒ1411.18
1
36
Journal of Business & Economic Statistics, January 2003
Table 2. Results of the Simulation Experiment With Bivariate t
Distributed Disturbances
Table 4. Results of the Simulation Experiment, Where the Error Terms
are Distributed According to a Bivariate Normal Distribution With
Heteroscedasticity Depending on the Explanatory Variable x
Flexible parametric
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
N D 1000
‚ 10
‚ 11
‚ 12
‚ 20
‚ 21
‚ 22
‘1
‘ 12
00
01
02
10
11
12
20
21
22
099
050
ƒ051
1006
ƒ1005
1007
1092
095
(0076)
(0028)
(0055)
(011)
(0096)
(011)
(022)
(022)
1003
050
ƒ051
1005
ƒ1002
1004
2003
1018
1
0
0
ƒ012
sdev(˜1 )
sdev(˜2 )
cov(˜1 1 ˜2 )
Log-likelihood
Rejections
NOTE:
KD1
Normal
ƒ1371.30
(0064)
(0029)
(0044)
(0071)
(0065)
(0076)
(019)
(022)
(0082)
1088 (018)
093 (0059)
072 (032)
ƒ1348.14
69
K D2
(0040)
(0019)
(0026)
(0055)
(0059)
(0065)
(0096)
(0053)
098
050
ƒ051
1008
ƒ1008
1009
2005
097
1
0012
ƒ0078
00025
ƒ0014
00099
ƒ0047
ƒ00068
0013
(0019)
(0062)
(0018)
(0037)
(0028)
(0012)
(0013)
(00049)
2000
(0098)
1000
(0030)
094
(016)
ƒ1331.18
97
Flexible parametric
N D 1000
K D1
Normal
‚10
1004 (0066)
‚11
055 (0068)
ƒ051 (0063)
‚12
‚20
099 (0059)
ƒ1000 (0047)
‚21
‚22
1000 (0050)
‘1
2010 (039)
‘ 12
058 (0071)
00
01
02
10
11
12
20
21
22
sdev(˜1 )
sdev(˜2 )
corr(˜1 1 ˜2 )
029 (0064)
cov(˜1 1 ˜2 )
ƒ1436.58
Log-likelihood
Rejections
1002
050
ƒ051
1001
ƒ092
093
2021
1051
1
0
0
ƒ017
(0068)
(0051)
(0047)
(0086)
(0063)
(0072)
(038)
(031)
(0032)
2008
(033)
094
(0044)
069
(0084)
1006
(039)
ƒ1362.86
100
K D2
1000
(0058)
051
(0051)
ƒ050
(0044)
1028
(010)
ƒ1026
(0097)
1026
(0094)
2017
(044)
095
(040)
1
0012
(0021)
014
(015)
0016
(0033)
ƒ0054
(012)
ƒ00027 (0013)
ƒ0086
(0035)
ƒ00023 (00077)
00096 (0013)
2025
(064)
1029
(018)
044
(014)
1040
(1024)
ƒ1315.80
100
Standard errors are given in parentheses.
NOTE:
with K D 1 and K D 2 (denoted by ¨1ü and ¨2ü ). The case
of the normal disturbances is presented in detail in Table 1;
the case of the t disturbances, in Table 2; and the case of the
chi-squared disturbances, in Table 3. The simulation results
from the heteroscedastic error terms are presented in Table 4
for the covariance matrix dependent on the observed variable
Table 3. Results of the Simulation Experiment With Bivariate
Chi-Squared Distributed Disturbances
Standard errors are given in parentheses.
x and in Table 5 for the covariance matrix dependent on an
unobserved variable.
It is remarkable how well standard maximum likelihood
under the assumption of normally distributed disturbances performs. Departures from normality do not cause serious bias
Table 5. Results of the Simulation Experiment, Where the Error Terms
are Distributed According to a Bivariate Normal Distribution With
Heteroscedasticity Depending on anpUnobservedpVariable r Taking
Three Possible Values, .5, 1, and 1.5
Flexible parametric
Flexible parametric
N D 1000
‚ 10
094 (0089)
‚ 11
050 (0029)
ƒ050 (0054)
‚ 12
‚ 20
1009 (0075)
ƒ1004 (0076)
‚ 21
‚ 22
1003 (0083)
‘1
2003 (0084)
‘ 12
1028 (015)
00
01
02
10
11
12
20
21
22
sdev(˜1 )
sdev(˜2 )
cov(˜1 1 ˜2 )
ƒ1403.24
Log-likelihood
Rejections
NOTE:
K D1
Normal
093
050
ƒ050
1005
ƒ1004
1004
2001
1027
1
0
0
ƒ0018
(0047)
(0027)
(0036)
(0039)
(0037)
(0037)
(0053)
(0053)
(0032)
1094
(0074)
097
(0038)
1012
(017)
ƒ1391.81
89
Standard errors are given in parentheses.
K D2
095
(0061)
050
(0018)
ƒ049
(0021)
1012
(0041)
ƒ1006
(0028)
1006
(0024)
1089
(0045)
1019
(0020)
1
ƒ011
(0021)
ƒ0039
(0030)
ƒ0088
(0014)
0066
(0031)
ƒ00067 (0032)
ƒ0046
(0019)
0037
(0017)
00034 (00037)
1061
(0042)
085
(0012)
056
(0066)
ƒ1345.13
100
N D 1000
‚10
1001 (0060)
‚11
050 (0027)
ƒ050 (0047)
‚12
‚20
099 (0054)
ƒ1000 (0047)
‚21
‚22
1000 (0046)
‘1
1099 (0057)
‘ 12
1000 (0098)
00
01
02
10
11
12
20
21
22
sdev(˜1 )
sdev(˜1 )
050 (0050)
corr(˜1 1 ˜2 )
cov(˜1 1 ˜2 )
ƒ1405.10
Log-likelihood
Rejections
NOTE:
K D1
Normal
1001
050
ƒ050
1000
ƒ1000
1000
2000
1000
1
0
(0035)
(0026)
(0037)
(0032)
(0033)
(0031)
(0036)
(0023)
0
ƒ00013 (0014)
1099
(0052)
1000
(0028)
050
(0020)
099
(015)
ƒ1405.09
0
Standard errors are given in parentheses.
K D2
1000
(0031)
050
(0027)
ƒ050
(0036)
1000
(0027)
ƒ1000
(0029)
1000
(0026)
2000
(0012)
1000
(0013)
1
00013 (0020)
ƒ0015
(0032)
ƒ00004 (0011)
ƒ0015
(0031)
00063 (0026)
ƒ00074 (00096)
ƒ00030 (0012)
00027 (00067)
2001
(025)
1002
(013)
050
(00091)
1009
(025)
ƒ1402.10
12
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
van der Klaauw and Koning: Testing the Normality Assumption
in the parameter estimates. In case the true disturbances follow a t distribution, all estimated parameters are within two
standard deviations of their true values. This is the case even
when the disturbances follow a transformed chi-squared distribution that is nonsymmetric. This suggests that if obtaining estimates for the parameters ‚ is the only interest, than it
is sufcient to assume normality and obtain estimates. However, in the presence of heteroscedasticity in the error terms
that is dependent on one of the observed exogenous variables,
the parameter estimates under normality are somewhat biased
(see Table 4). In particular, the parameter estimate of ‚11 differs from the true value. This bias disappears when estimating
the model using the exible parametric specication. When
the heteroscedasticity does not depend on exogenous variables
but the covariance matrix differs between groups, the parameter estimates obtained under normality are again close to their
true value.
Tables 1–5 also report the number of rejections of normality
for each simulation experiment. The simulation results indicate that the normality test against both the class ¨1ü and ¨2ü
performs well. The number of incorrect rejections is small
when compared to the level of signicance. The number of
correct rejections is very high except in the simulation experiment, where the distribution of the error terms differs between
groups. The test is very powerful when the error terms are
not bivariate normal distributed, and it is successful in detecting heteroscedasticity depending on observed exogenous variables. The simulation results indicate that the test has more
power when the tted distribution belongs to ¨2ü than when
it belongs to ¨1ü . Because departures from normality do not
cause serious bias in the parameter estimates of the regression equation, Hausman tests for normality against a semiparametric alternative probably will not be as powerful as the
likelihood-based tests discussed in this article.
The results from the simulation study suggest that the estimates for the ‚1 parameters (derived under the assumption
of normality) are reasonably robust to misspecication of the
error term. This is not true only if the covariance matrix of the
error terms depends on observed explanatory variables. In this
case, using the exible parametric density can help improve
the parameter estimates. Furthermore, if one is interested not
only in the parameter estimates of ‚1 , but also in the complete distribution function of outcome variable y conditional
on explanatory variables x, then using the exible parametric
specication is worthwhile.
4.
AN APPLICATION TO A MODEL OF
TRAVEL DEMAND
In this section we apply the exible parametric estimation
procedure and the normality tests to a model describing travel
demand. This section consists of three parts: rst, we present
a structural model, then we discuss the data, and nally, we
present the empirical results.
The model assumes that in a given year households need to
travel a stochastic number of kilometers and that these households minimize the costs of traveling. At the beginning of the
year, a household makes a prediction of the number of kilometers to be traveled, which is denoted by y ü . The actual number
37
of kilometers traveled in this year is related to the prediction
by y D y ü ¢ ˜, where ˜ is a random prediction error with a positive support and a nite mean Œ. The household can choose
to travel either by car or by other means. The latter is most
likely public transportation; other possibilities may be bikes,
car pooling, and so forth. The cost of using a car consists of
xed costs, cc , and variable costs, vc . Fixed costs are depreciation of the car, maintenance, insurance, and taxes for car
ownership. Variable costs are the costs of fuel. The costs of
other transportation do not contain any xed costs. The variable costs, denoted by vo , are associated with the costs of
public transportation. However, opportunity costs may also be
important, because traveling with public transportation usually
takes more time than traveling by car.
The household decision problem now involves choosing
whether or not to use a car. Assuming that (risk-neutral)
households minimize their expected costs, the decision rule
implies that a car is used if
E6cc C vc y7 µ E6vo y70
(6)
Now assume that cc , vc , and vo are known by the household.
This means that the household decision simplies to using a
car if
cc
0
E6y7 ¶
(7)
vo ƒ vc
If we substitute the relation between the actual number of
kilometers driven and the predicted number of kilometers, then
this implies that a household chooses to use a car if
yü ¶
cc
0
Œ4vo ƒ vc 5
Now we parameterize the unknown components of the
model as
y ü D exp4x 0 ‚ C v1 51
which is the distance function and a costs function
cc
D exp4z0 ƒ C v2 51
vo ƒ vc
where v1 and v2 denote unobserved individual specic effects.
We normalize ΠD 1.
The model reduces to a regression equation, which has the
structure
log4y5 D x 0 ‚ C ˜1
with ˜1 D v1 C ln4˜5. A selection equation indicates whether
the household owns a car,
x 0 ‚ ƒ z0 ƒ C ˜2 ¶ 01
with the household-specic term ˜2 equal to v1 ƒ v2 , which is
known to the household but unknown to the econometrician.
Because v1 is included in both ˜1 and ˜2 , these disturbances
are not a priori independent.
To improve the identication of the model, we impose an
exclusion restriction, that is, nd a variable that is included in
z and excluded from x. According to the model, the predicted
number of kilometers affects both the actual number of kilometers and the decision to buy a car, whereas the xed and
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
38
Journal of Business & Economic Statistics, January 2003
variable costs of using a car and the variable costs of other
transportation affect only the decision of whether to use a car.
Thus the exclusion restriction should be a variable that affects
only the costs functions. Because the Dutch government provides free public transportation passes to some students and
some civil servants, we use the dummy variable that indicates
whether the household has such a free public transportation
pass as a variable which is included in z but not in x.
The data we use is a subset from the Dutch database
on transportation behavior of Statistics Netherlands. This
database contains 34,454 households. To avoid complications of households owning more than one car, we focus on
single-person households. Furthermore, we exclude individuals younger than age 18 years, the legal age for obtaining a
drivers license in The Netherlands. This restricts the database
to 7404 individuals. To construct our nal dataset, we also
exclude 130 individuals who own a car but for whom the number of kilometers driven in the past year is unknown and also
756 individuals for whom one or more explanatory variables
are missing. Our nal dataset consists of 6518 observations.
Table 6 provides some characteristics of the dataset. The
data contain 3110 individuals who own a car and 3408 who
do not own a car. Except for region, all variables display differences in car ownership rates. Although 58% of the men are
car owners, only 41% of the women have a car. Until people
reach age 65, the car ownership rate increases with age; after
that, there is a large drop. Furthermore, car ownership rates
increase with income and level of education. Finally, individuals living in areas with a low degree of urbanization, fulltime employed workers, and individuals who do not have a
government-provided free public transportation pass are more
likely to own a car than their counterparts.
Using these data, we estimate the structural model discussed
previously. In the rst step, we estimate the model under the
assumption that the disturbances ˜1 and ˜2 follow a bivariate
normal distribution. Next, we relax this assumption by assuming that the density of the disturbances belongs to either ¨1ü
or ¨2ü . For these two cases, we test whether the restriction of
normality of the disturbances can be imposed.
All parameter estimates are presented in Tables 7 and 8. The
parameter estimates of the distance function (Table 7) and of
the costs function (Table 8) are not very sensitive to the specication of the distribution function of the error term, which
is in line with the results from the simulation study. The most
important covariates in the distance function are gender, age,
income, level of education, and individual labor market status;
those in the costs function are age, degree of urbanization, and
income. Although the availability of a free public transportation pass provided by the government has the expected effect
on the costs function, the corresponding parameter estimate is
not signicantly different from 0.
The only individual characteristics important in both the distance and costs functions are age and income. Age affects both
the distance function and the costs function negatively. Young
people travel more kilometers than older people, whereas the
costs of car use relative to other transportation modes decrease
with age. By comparing the coefcients, we can see that persons age 50–64 are most likely to own a car. Income has an
opposite effect on the distance and costs functions. Persons
Table 6. Some Characteristics of the Dataset
Car owner
Gender
Male
Female
Age
18–24
25–29
30–39
40–49
50–64
65+
Region
West
North
East
South
Degree of urbanization
Very high
High
Average
Low
Very low
Net income (guilders)
0–15,000
15,000–23,000
23,000–30,000
30,000–38,000
38,000–52,000
52,000+
Level of education
Primary
Lower secondary
Higher secondary
University
Labor market status
Full-time work
Part-time work
Student
Unemployed
Nonparticipant
Free public transport pass
No
Yes
Average number of kilometers
Observations
Yes
No
1523
1587
1102
2306
106
416
746
467
692
683
518
415
485
296
451
1243
1588
211
895
416
1881
207
959
361
764
786
694
563
303
1329
970
497
397
215
158
411
539
695
773
534
999
1046
574
402
278
109
237
712
1022
1139
688
856
1075
789
1626
178
37
105
1164
771
245
406
250
1736
3074
36
151093
3110
2965
443
3408
with greater income travel more kilometers, whereas the costs
of car use relative to other transportation decrease. Considering that using public transportation takes more time, this latter
effect can be explained by differences in opportunity costs.
Time is more costly for individuals with high incomes. The
degree of urbanization affects only the costs function. Because
less public transportation is available in areas with low degrees
of urbanization, the costs of using public transportation are
higher in these areas (e.g., waiting times are longer). In contrast, car use in areas with a high degree of urbanization is
more costly, because individuals often incur additional costs,
such as parking costs. Employed individuals, both full-time
and part-time, travel more kilometers. Because we do not distinguish between traveling for private purposes and traveling
for professional purposes, this may be caused by commuting
or because their work requires them to travel. Finally, persons
with a higher level of education travel more than those with
less education.
van der Klaauw and Koning: Testing the Normality Assumption
39
Table 8. Estimation Results of the Travel Demand Model:
Costs Function
Table 7. Estimation Results of the Travel Demand Model:
Distance Function
Distance function (‚)
Costs function (ƒ)
Flexible parametric
KD1
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
Normal
Intercept
9041
Gender
ƒ029
Female
Age
ƒ013
25–29
ƒ024
30–39
ƒ028
40–49
ƒ033
50–64
ƒ059
65+
Region
North
00027
ƒ00007
East
ƒ0054
South
Degree of urbanization
High
0025
Average
0064
Low
0077
Very low
0064
Net income (guilders)
15,000–23,000
0086
23,000–30,000
023
30,000–38,000
030
38,000–52,000
037
52,000+
050
Level of education
Lower secondary
010
Higher secondary
017
University
028
Labor market status
Part-time work
0017
ƒ028
Student
ƒ029
Unemployed
ƒ025
Nonparticipant
‘1
063
‘ 12
0041
00
01
02
10
11
12
20
21
22
NOTE:
(032)
9077
(0038) ƒ024
(0088)
(0084)
(0089)
(010)
(0086)
ƒ014
ƒ023
ƒ030
ƒ035
ƒ050
(011)
K D2
9087
(015)
(0022) ƒ023
(0026)
ƒ015
ƒ023
ƒ030
ƒ036
ƒ050
(0066)
(0064)
(0067)
(0071)
(0073)
(0061)
(0059)
(0062)
(0064)
(0067)
(0050)
00066 (0045) 00010 (0047)
(0031)
0013 (0027) 0016 (0028)
(0040) ƒ0057 (0034) ƒ0061 (0034)
(0032)
(0035)
(0038)
(0044)
ƒ0012
ƒ0035
ƒ0033
ƒ0037
(0033)
(0039)
(0043)
(0050)
(0075) ƒ012
(011)
0089
(014)
012
(016)
017
(018)
030
(0061)
(0065)
(0066)
(0069)
(0072)
0022
0093
011
015
027
(0067)
(0075)
(0081)
(0086)
(0090)
(0051)
(0062)
(0064)
(0040)
(0041)
(0043)
0035
0085
020
(0044)
(0046)
(0047)
(0042)
(0066)
(0073)
(0082)
(0058)
(011)
(0069)
(0046)
(00088)
(017)
ƒ00070
ƒ00088
ƒ00043
ƒ0011
0043
0097
021
0023
ƒ019
ƒ018
ƒ023
071
ƒ045
1
0
0
049
(0050)
(010)
(0059)
(0037)
(0018)
(0034)
Flexible parametric
0035
ƒ016
ƒ017
ƒ022
076
ƒ054
1
ƒ00035
0038
ƒ0012
(0031) 038
015
ƒ0052
016
00005
(0052)
(012)
(0063)
(0039)
(023)
(031)
(0049)
(012)
(0045)
(041)
(012)
(014)
(0079)
(0098)
Standard errors are in parentheses.
The likelihood ratio test statistics of normality against the
exible parametric density with K D 1 and K D 2 equal 283.7
and 299.6, implying that we must reject the hypothesis that
the disturbances follow a normal distribution. The main difference with the estimates under normality is that we do not nd
any correlation between ˜1 and ˜2 under the assumption of
normality. This suggests that there is no unobserved selection
between car ownership and car use, which implies that both
the regression and selection equations can be estimated consistently separately of each other by, for example, OLS and
probit. The exible parametric densities show a correlation
between both disturbance terms that could not be captured by
a normal density. The estimated covariance between ˜1 and ˜2
is ƒ023 for K D 1 and ƒ030 for K D 2. In the context of our
theoretical model, this implies a positive correlation between
Normal
Intercept
10081 (035)
Gender
Female
0046 (0053)
Age
ƒ049 (013)
25–29
ƒ056 (013)
30–39
ƒ065 (013)
40–49
ƒ1000 (015)
50–64
ƒ076 (014)
65+
Region
ƒ011 (0092)
North
East
0049 (0058)
ƒ016 (0072)
South
Degree of urbanization
ƒ018 (0068)
High
ƒ051 (0086)
Average
ƒ057 (0096)
Low
ƒ065 (011)
Very low
Net income (guilders)
ƒ021 (010)
15,000–23,000
ƒ048 (014)
23,000–30,000
ƒ072 (016)
30,000–38,000
ƒ090 (018)
38,000–52,000
ƒ1004 (020)
52,000+
Level of education
ƒ016 (0079)
Lower secondary
ƒ026 (0089)
Higher secondary
ƒ015 (0095)
University
Labor market status
Part-time work
015 (0095)
Student
014 (019)
Unemployed
0058 (011)
ƒ0023 (0082)
Nonparticipant
Free public transport pass
Yes
021 (016)
ƒ6386.34
Log-likelihood
NOTE:
KD1
10086
(014)
0017 (0038)
ƒ041
ƒ046
ƒ058
ƒ085
ƒ061
(010)
(010)
(011)
(011)
(012)
K D2
10091
(029)
0047 (0095)
ƒ040
ƒ046
ƒ058
ƒ086
ƒ059
(013)
(012)
(013)
(018)
(013)
ƒ0075 (0079) ƒ0077 (0088)
0060 (0050)
0065 (0055)
ƒ013 (0062) ƒ013 (0070)
ƒ018
ƒ047
ƒ053
ƒ059
(0056)
(0057)
(0064)
(0078)
ƒ018
ƒ049
ƒ056
ƒ061
(0076)
(015)
(017)
(019)
ƒ026
ƒ048
ƒ068
ƒ083
ƒ090
(0085)
(0089)
(0090)
(0093)
(010)
ƒ020
ƒ046
ƒ069
ƒ086
ƒ095
(011)
(017)
(024)
(030)
(037)
ƒ016
ƒ024
ƒ012
(0066) ƒ016
(0068) ƒ025
(0075) ƒ013
(0089)
(012)
(012)
013
013
011
ƒ0054
(0080)
015
(016)
016
(0095)
013
(0068) ƒ0040
(0094)
(019)
(014)
(0094)
014 (012)
ƒ6244.49
015 (013)
ƒ6236.54
Standard errors are in parentheses.
v1 and v2 , because cov4˜1 1 ˜2 5 D var4v1 5 ƒ cov4v1 1 v2 5. Obviously, there are some unobserved covariates, which increase
both the costs of car use relative to other transportation and the
expected number of kilometers traveled. Thus such a covariate
has similar effects to, for example, age. Note that we maintained the assumption that the prediction error ˜ is independent of the individual specic effects v1 and v2 .
Figures 4 and 5 show the marginal densities of the disturbances ˜1 and ˜2 estimated under the exible parametric densities and normality. Both exible parametric densities have
more mass close to the mode and slightly fatter tails than the
normal density. The estimated standard deviation of ˜1 is .64
in both the K D 1 and K D 2 specications, almost similar to
the estimate under normality. For the standard deviation of ˜2 ,
we nd .89 and .90 for ˜2 for the K D 1 and the K D 2 density.
Figures 6 and 7 graph the estimated densities of the normal
model and the model estimated using an exible parametric
density (K D 2). We see that the estimated exible parametric density is a bit more spread out than the normal density,
and that the estimated contour lines in Figure 7 are not the
ellipsoids of Figure 6.
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
40
Journal of Business & Economic Statistics, January 2003
Figure 4. The Estimated Marginal Density Function of ˜1 .
5.
CONCLUSION
In this article we have derived a test for normality in sample selection models. To our knowledge, no such test has been
derived previously. The test is motivated on the seminonparametric maximum likelihood method introduced by Gallant and
Nychka (1987). A exible parametric density function is used
to approximate the density function of the error terms. The
bivariate normal density is a special case of this exible parametric density function that allows us to test for normality.
The test also provides an alternative density function in the
event that normality is rejected.
Although the simulation study provided in this article is
limited, we believe that this test is promising. The test performed well in almost all simulation experiments, the percentage of incorrect rejections is below the signicance level,
whereas the power is high. This is not true only when the
Figure 6. Estimated Density Under the Normality Assumption.
Figure 5. The Estimated Marginal Density Function of ˜2 .
distribution of the error terms is heteroscedastic depending on
an unobserved variable. The test is powerful against departures
from normality and against heteroscedasticity depending on
an observed exogenous variable. However, it should be noted
that in most cases the parameter estimates do not seem very
sensitive to the distributional assumptions of the disturbances.
Even in the event that the disturbances do not follow a normal
distribution, maximum likelihood under the assumption of normality provides estimates close to the true values. Only if the
covariance matrix of the error terms depends on an observed
exogenous variable are the parameter estimates biased. The
simulation study indicates that the bias disappears when using
the exible parametric density. The exible parametric density
is thus useful in cases where not only parameter estimates, but
also the complete distributional structure are of interest, and
in the case of heteroscedasticity.
van der Klaauw and Koning: Testing the Normality Assumption
41
APPENDIX A: RELEVANT INTEGRALS OF THE
FLEXIBLE PARAMETRIC DENSITY IN
THE SAMPLE SELECTION MODEL
In the sample selection model, all relevant integrals are of
the form
Zˆ
hü 4˜5d˜2
a
and
Z
b
Z
ˆ
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:38 13 January 2016
ƒˆ
Z
ˆ
ƒˆ
hü 4˜5d˜1 d˜ 2
This latter integral also appears in S when b D ˆ and in the
mean restrictions where also b D ˆ and hü 4˜5 is replaced by
˜i hü 4˜5 for i D 11 2. (Because of the structure of the Hermite
series, this solves similarly.)
Substituting for hü 4˜5, we obtain integrals of the type
and
Z
b
ƒˆ
Z
a
ˆ
ƒˆ
j
˜i1 ˜2 ”4˜1 1 ˜2 5d˜2
j
˜i1 ˜2 ”4˜1 1 ˜2 5d˜ 1 d˜2 1
where ”4˜1 1 ˜2 5 is the bivariate normal density function.
Because
”4˜1 1 ˜2 5 D ”4˜2 —˜1 5”4˜1 51
we can rewrite these integrals as
˜i1 ”4˜1 5
Z
ˆ
a
j
˜2 ”4˜2 —˜1 5d˜ 2
and
Z
b
ƒˆ
j