Manajemen | Fakultas Ekonomi Universitas Maritim Raja Ali Haji 073500103288619368
Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
On the Performance of Some Robust Instrumental
Variables Estimators
Bo E Honoré & Luojia Hu
To cite this article: Bo E Honoré & Luojia Hu (2004) On the Performance of Some Robust
Instrumental Variables Estimators, Journal of Business & Economic Statistics, 22:1, 30-39, DOI:
10.1198/073500103288619368
To link to this article: http://dx.doi.org/10.1198/073500103288619368
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: 00:20
On the Performance of Some Robust
Instrumental Variables Estimators
Bo E. H ONORÉ
Department of Economics, Princeton University, Princeton, NJ 08544 ( honore@princeton.edu)
Luojia H U
Department of Economics, Northwestern University, Evanston, IL 60208 ( luojiahu@northwestern.edu)
This article considers instrumental variables versions of the quantile and rank regression estimators. The
asymptotic properties of the estimators are discussed, and a small-scale Monte Carlo study is used to
illustrate the potential advantages of the approach. Finally, the proposed methods are implemented for
two empirical examples.
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:20 13 January 2016
KEY WORDS: Endogeneity; Quantile regression; Rank regression.
1.
INTRODUCTION
2.
Some attention has recently been paid to alternatives to least
squares estimators of the linear regression model. These include quantile regression estimators due to Koenker and Bassett
(1978), which have recently become quite popular (see e.g.,
the recent survey by Buchinsky 1998), and rank regression estimators, which have been used much less often. Although it
is difcult to rank estimators that make different identifying
assumptions, these alternative estimators may be preferable to
more conventionalestimators in models where they all estimate
the same parameter. In particular, quantile and rank regression
estimators are robust (in a formal sense) and hence much less
sensitive to outliers. Moreover, in situations where the estimators do not estimate the same parameter, it can be interesting to
compare the estimates. For example, a comparison of the coefcient in quantile regressions for different quantiles can lead to
insights that could not be easily obtained from standard instrumental variables (IV) estimation.
The contribution of this article is to consider standard
method-of-moments estimators that can be interpreted as
IV versions of the quantile and rank regression estimators. We
briey review how the asymptotic distributions for estimators
dened by quantiles and ranks can be derived, and how one can
estimate the asymptotic variance of such estimators. We also
note that like limited information maximum likelihood (LIML),
but unlike two-stage least squares (2SLS), the proposed estimators are invariant to normalizations. Like their counterparts
for linear regression models, these estimators are robust, and a
Monte Carlo study conrms that they can be superior to 2SLS
and LIML under nonnormality. The Monte Carlo study also
suggests that the asymptotic distribution can be used to construct fairly reliable tests and condence intervals.
Besides the Monte Carlo study, we also illustrate the use of
the estimators in two empirical examples and discuss how the
estimators can be calculated in practice, as well as how one can
do inference by applying a method similar to that proposed by
Andersen and Rubin (1950).
The rest of the article is organized as follows. Section 2 denes the estimators, and Section 3 discusses their large-sample
properties. Section 4 presents the results of a small Monte Carlo
study designed to investigate the small-sample properties of the
estimators, and considers empirical examples. Section 5 concludes the article. Throughout this article, we assume random
sampling.
GENERAL IDEA
Consider a simple linear regression model,
yi D x0i ¯ C "i ;
(1)
where yi is the dependent variable, xi is a vector of explanatory
variables, and "i is the error term. 2SLS and LIML estimation
of ¯ rest on the assumption that
E["i zi ] D 0
(2)
for some vector of variables, zi , whose dimension is at least as
great as the dimension of xi .
Although economic theory often leads to Euler equations
which can be used to motivate conditions like (2), most applications of 2SLS and LIML are based on more heuristic arguments for (2). These arguments typically are not specic to
the moments of "i zi . Instead, it is argued that "i and zi are unrelated in some unspecied sense, which is then used to motivate (2). It is clear that (statistical) independence of "i and zi
can be used to form many moment conditions other than (2).
This article focuses on the moment conditions that correspond
to those that yield the quantile regression and rank regression
estimators in the linear regression model. Because the estimators that we consider do not exploit all the moment conditions
implied by independence between "i and zi , it is clear that the
estimators generally will not, be asymptotically efcient. Also,
because the moment conditions that we use are nonlinear, it is
not obvious that they actually identify the parameter of interest. The moment condition certainly will be satised at the true
value of the parameter, but it might also be satised at other
points. For consistency of the estimators, it is thus necessary to
assume identication. Hong and Tamer (2003) presented a detailed discussion of conditions under which quantile regression
models with endogeneity are identied. As we discuss later,
lack of identication does not rule out using the methods to
conduct (asymptotically)valid inference about the true, but possibly unidentied, parameter vector.
© 2004 American Statistical Association
Journal of Business & Economic Statistics
January 2004, Vol. 22, No. 1
DOI 10.1198/073500103288619368
30
Honoré and Hu: Robust Instrumental Variables Estimators
31
Independence between "i and zi implies that all quantiles
of "i given zi are constant. For any ®, there thus exists a constant c® such that P."i · c® jzi j/ D ®. Focusing on the median
(® D 21 ), we then have P."i ¡ c1=2 · 0jzi j/ D 12 . If xi contains an
intercept, then this can be rewritten in such a way that c1=2 D 0
(by adding c1=2 to the intercept and subtracting it from "i ). We
then have
med["i jzi j] D 0;
which implies that
E[sgn. yi ¡ x0i ¯/ ¢ zi ] D 0:
(3)
iD1
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:20 13 January 2016
A generalized method-of-moments (GMM) estimator, ¯OMIV ,
can then be dened as the minimizer of some norm of its sample
analog,
n
1X
n
iD1
. fyi ¡ x0i b > 0g ¡ fyi ¡ x0i b < 0g/ ¢ zi :
(4)
n
(5)
iD1
As mentioned, we consider method-of-moments estimation
based on (5), and, following Abadie (1995), who proposed the
same set of moment conditions, we proceed under the assumption that ¯ is identied from the moment conditions. Taking
a much more general approach, Hong and Tamer (2003) recently characterized the set of parameters that are consistent
with the moment conditions and provided an estimator of that
set. A different approach to IV estimation of quantile regression
models was proposed by Abadie, Angrist, and Imbens (2002).
Their model is different from the model considered here, and
thus the two approaches are not easily comparable.
Independence between "i and zi also implies that the ranks of
the "i ’s are independent of the zi ’s, which in turn implies that
cov[R."i /; zi ] D 0;
and
³ ´¡1 X
¡
¢
n
sgn . yi ¡ x0i b/ ¡ . yj ¡ x0j b/ sgn.zi ¡ zj /;
2
(9)
i 0g ¡ .1 ¡ ®/ ¢ fyi ¡ x0i b < 0g ¢ zi :
n
JureÏcková 1971). In the econometrics literature, a similar idea
has been used by Han (1987), Cavanagh and Sherman (1998),
Abrevaya (1999, 2000), and Honoré and Powell (1994) to construct estimators of transformation models and of censored and
truncated regression models. Other rank IV estimators can be
dened as well. For example, one could base estimators on the
observation that Spearman’s rank correlation and Kendall’s tau
must both equal 0. This leads to estimators dened by minimizing the norms of
´
n ³
nC1
1X
0
R. yi ¡ xi b/ ¡
¢ R.zi /
(8)
n
2
(6)
where R."i / is the rank of "i . Note
Pvariables,
P that for n scalar
a1 ; a2 ; : : : ; an , we dene R.ai / D j .aj · ai /¡ 12 j6Di .aj D
ai /. In words, ties are dealt with by assigning all of the tied
observations the average rank. For vectors, a1 ; a2 ; : : : ; an , we
.1/
.2/
.m/
dene R.ai / D .R.ai /; R.ai /; : : : ; R.ai //, where m is the
dimension of the vector a.
We can then dene a “rank IV” estimator, ¯QRIV , by making
the sample analog of
´
n ³
nC1
1X
0
R. yi ¡ xi b/ ¡
¢ zi
(7)
n
2
iD1
as close to 0 as possible, in some norm. In the case where
xi D zi , the estimator based on (7) could also be dened by minimizing
³ ´¡1 X
n
j. yi ¡ yj / ¡ .x0i b ¡ x0j b/j:
2
i
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
On the Performance of Some Robust Instrumental
Variables Estimators
Bo E Honoré & Luojia Hu
To cite this article: Bo E Honoré & Luojia Hu (2004) On the Performance of Some Robust
Instrumental Variables Estimators, Journal of Business & Economic Statistics, 22:1, 30-39, DOI:
10.1198/073500103288619368
To link to this article: http://dx.doi.org/10.1198/073500103288619368
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: 00:20
On the Performance of Some Robust
Instrumental Variables Estimators
Bo E. H ONORÉ
Department of Economics, Princeton University, Princeton, NJ 08544 ( honore@princeton.edu)
Luojia H U
Department of Economics, Northwestern University, Evanston, IL 60208 ( luojiahu@northwestern.edu)
This article considers instrumental variables versions of the quantile and rank regression estimators. The
asymptotic properties of the estimators are discussed, and a small-scale Monte Carlo study is used to
illustrate the potential advantages of the approach. Finally, the proposed methods are implemented for
two empirical examples.
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:20 13 January 2016
KEY WORDS: Endogeneity; Quantile regression; Rank regression.
1.
INTRODUCTION
2.
Some attention has recently been paid to alternatives to least
squares estimators of the linear regression model. These include quantile regression estimators due to Koenker and Bassett
(1978), which have recently become quite popular (see e.g.,
the recent survey by Buchinsky 1998), and rank regression estimators, which have been used much less often. Although it
is difcult to rank estimators that make different identifying
assumptions, these alternative estimators may be preferable to
more conventionalestimators in models where they all estimate
the same parameter. In particular, quantile and rank regression
estimators are robust (in a formal sense) and hence much less
sensitive to outliers. Moreover, in situations where the estimators do not estimate the same parameter, it can be interesting to
compare the estimates. For example, a comparison of the coefcient in quantile regressions for different quantiles can lead to
insights that could not be easily obtained from standard instrumental variables (IV) estimation.
The contribution of this article is to consider standard
method-of-moments estimators that can be interpreted as
IV versions of the quantile and rank regression estimators. We
briey review how the asymptotic distributions for estimators
dened by quantiles and ranks can be derived, and how one can
estimate the asymptotic variance of such estimators. We also
note that like limited information maximum likelihood (LIML),
but unlike two-stage least squares (2SLS), the proposed estimators are invariant to normalizations. Like their counterparts
for linear regression models, these estimators are robust, and a
Monte Carlo study conrms that they can be superior to 2SLS
and LIML under nonnormality. The Monte Carlo study also
suggests that the asymptotic distribution can be used to construct fairly reliable tests and condence intervals.
Besides the Monte Carlo study, we also illustrate the use of
the estimators in two empirical examples and discuss how the
estimators can be calculated in practice, as well as how one can
do inference by applying a method similar to that proposed by
Andersen and Rubin (1950).
The rest of the article is organized as follows. Section 2 denes the estimators, and Section 3 discusses their large-sample
properties. Section 4 presents the results of a small Monte Carlo
study designed to investigate the small-sample properties of the
estimators, and considers empirical examples. Section 5 concludes the article. Throughout this article, we assume random
sampling.
GENERAL IDEA
Consider a simple linear regression model,
yi D x0i ¯ C "i ;
(1)
where yi is the dependent variable, xi is a vector of explanatory
variables, and "i is the error term. 2SLS and LIML estimation
of ¯ rest on the assumption that
E["i zi ] D 0
(2)
for some vector of variables, zi , whose dimension is at least as
great as the dimension of xi .
Although economic theory often leads to Euler equations
which can be used to motivate conditions like (2), most applications of 2SLS and LIML are based on more heuristic arguments for (2). These arguments typically are not specic to
the moments of "i zi . Instead, it is argued that "i and zi are unrelated in some unspecied sense, which is then used to motivate (2). It is clear that (statistical) independence of "i and zi
can be used to form many moment conditions other than (2).
This article focuses on the moment conditions that correspond
to those that yield the quantile regression and rank regression
estimators in the linear regression model. Because the estimators that we consider do not exploit all the moment conditions
implied by independence between "i and zi , it is clear that the
estimators generally will not, be asymptotically efcient. Also,
because the moment conditions that we use are nonlinear, it is
not obvious that they actually identify the parameter of interest. The moment condition certainly will be satised at the true
value of the parameter, but it might also be satised at other
points. For consistency of the estimators, it is thus necessary to
assume identication. Hong and Tamer (2003) presented a detailed discussion of conditions under which quantile regression
models with endogeneity are identied. As we discuss later,
lack of identication does not rule out using the methods to
conduct (asymptotically)valid inference about the true, but possibly unidentied, parameter vector.
© 2004 American Statistical Association
Journal of Business & Economic Statistics
January 2004, Vol. 22, No. 1
DOI 10.1198/073500103288619368
30
Honoré and Hu: Robust Instrumental Variables Estimators
31
Independence between "i and zi implies that all quantiles
of "i given zi are constant. For any ®, there thus exists a constant c® such that P."i · c® jzi j/ D ®. Focusing on the median
(® D 21 ), we then have P."i ¡ c1=2 · 0jzi j/ D 12 . If xi contains an
intercept, then this can be rewritten in such a way that c1=2 D 0
(by adding c1=2 to the intercept and subtracting it from "i ). We
then have
med["i jzi j] D 0;
which implies that
E[sgn. yi ¡ x0i ¯/ ¢ zi ] D 0:
(3)
iD1
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:20 13 January 2016
A generalized method-of-moments (GMM) estimator, ¯OMIV ,
can then be dened as the minimizer of some norm of its sample
analog,
n
1X
n
iD1
. fyi ¡ x0i b > 0g ¡ fyi ¡ x0i b < 0g/ ¢ zi :
(4)
n
(5)
iD1
As mentioned, we consider method-of-moments estimation
based on (5), and, following Abadie (1995), who proposed the
same set of moment conditions, we proceed under the assumption that ¯ is identied from the moment conditions. Taking
a much more general approach, Hong and Tamer (2003) recently characterized the set of parameters that are consistent
with the moment conditions and provided an estimator of that
set. A different approach to IV estimation of quantile regression
models was proposed by Abadie, Angrist, and Imbens (2002).
Their model is different from the model considered here, and
thus the two approaches are not easily comparable.
Independence between "i and zi also implies that the ranks of
the "i ’s are independent of the zi ’s, which in turn implies that
cov[R."i /; zi ] D 0;
and
³ ´¡1 X
¡
¢
n
sgn . yi ¡ x0i b/ ¡ . yj ¡ x0j b/ sgn.zi ¡ zj /;
2
(9)
i 0g ¡ .1 ¡ ®/ ¢ fyi ¡ x0i b < 0g ¢ zi :
n
JureÏcková 1971). In the econometrics literature, a similar idea
has been used by Han (1987), Cavanagh and Sherman (1998),
Abrevaya (1999, 2000), and Honoré and Powell (1994) to construct estimators of transformation models and of censored and
truncated regression models. Other rank IV estimators can be
dened as well. For example, one could base estimators on the
observation that Spearman’s rank correlation and Kendall’s tau
must both equal 0. This leads to estimators dened by minimizing the norms of
´
n ³
nC1
1X
0
R. yi ¡ xi b/ ¡
¢ R.zi /
(8)
n
2
(6)
where R."i / is the rank of "i . Note
Pvariables,
P that for n scalar
a1 ; a2 ; : : : ; an , we dene R.ai / D j .aj · ai /¡ 12 j6Di .aj D
ai /. In words, ties are dealt with by assigning all of the tied
observations the average rank. For vectors, a1 ; a2 ; : : : ; an , we
.1/
.2/
.m/
dene R.ai / D .R.ai /; R.ai /; : : : ; R.ai //, where m is the
dimension of the vector a.
We can then dene a “rank IV” estimator, ¯QRIV , by making
the sample analog of
´
n ³
nC1
1X
0
R. yi ¡ xi b/ ¡
¢ zi
(7)
n
2
iD1
as close to 0 as possible, in some norm. In the case where
xi D zi , the estimator based on (7) could also be dened by minimizing
³ ´¡1 X
n
j. yi ¡ yj / ¡ .x0i b ¡ x0j b/j:
2
i