Directory UMM :Data Elmu:jurnal:J-a:Journal of Econometrics:Vol97.Issue2.Aug2000:
Journal of Econometrics 97 (2000) 345}364
An alternative approach to obtaining
Nagar-type moment approximations in
simultaneous equation models
Garry D.A. Phillips*
School of Business and Economics, University of Exeter, Streatham Court, Exeter EX4 4PU, UK
Received 1 December 1997; received in revised form 1 September 1999; accepted 1 September 1999
Abstract
This paper examines asymptotic expansions for estimation errors expressed explicitly
as functions of underlying random variables. Taylor series expansions are obtained from
which "rst and second moment approximations are derived. While the expansions are
essentially equivalent to the traditional Nagar type, the terms are expressed in a form
which enables moment approximations to be obtained in a particularly straightforward
way once the partial derivatives have been found. The approach is illustrated by
considering the k-class estimators in a static simultaneous equation model where the
disturbances are non-spherical. ( 2000 Elsevier Science S.A. All rights reserved.
JEL classixcation: C13; C22
Keywords: Simultaneous equation models; Non-spherical disturbances; Nagar expansions; Bias and second moment approximations
1. Introduction
In an important paper Nagar (1959) analysed the small sample properties
of the general k-class of simultaneous equation estimators and, in particular,
found expressions for the bias to the order of ¹~1 where ¹ is the number of
* Tel.: #44-1392-263241; fax: #44-1392-263242.
E-mail address: [email protected] (G.D.A. Phillips).
0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 9 9 ) 0 0 0 7 5 - 5
346
G.D.A. Phillips / Journal of Econometrics 97 (2000) 345}364
observations, and for the second moment matrix to the order of ¹~2. In
obtaining these results Nagar used an asymptotic expansion for the estimation
error of the form:
p~1
(a!a)" + ¹(~1@2)se #¹(~1@2)pr ,
s
p
s/1
(1)
where a is an estimator for a and the e , s"1,2, p!1, and r , the remainder
s
p
term, are all of stochastic order unity as ¹PR. The sum of the retained terms
are then assumed to mimic the behaviour of the estimator and the moments of
this sum are used to approximate the moments of the estimator. The approach
followed by Nagar was not entirely new; see, for example, Kendall (1954), but it
was through Nagar's work that econometricians became aware of the potentiality of the methodology. However, while the approach is of considerable interest,
it is not especially easy to understand what is happening within it. In fact, it uses
an implicit Taylor series expansion but this is not obvious.1 Another characteristic of the method is that it often involves evaluating the expectations of many,
often complex, stochastic terms and derivations of moment approximations can
be lengthy and tedious.
In this paper an alternative, but equivalent, method of expanding the k-class
estimators is presented which "rst expresses the estimator as a function of a set
of underlying random variables, e.g. reduced form coe$cient estimates. This is
a natural way to proceed and it was the approach adopted by Sargan (1976) who
noted that a wide variety of econometric estimators can be regarded as functions
of the sample data "rst and second moments. Thus, the problem of approximating the estimation error by means of a Taylor series expansion is placed in
a particularly familiar framework whereby the resulting methodology is better
understood. Sargan also gave conditions under which the moment approximations are valid. Although the resulting expansion is essentially the same as
Nagar's (in fact, Sargan referred to his expansion as a Nagar expansion), it is
easier to interpret and its form is such that the task of deriving the moment
approximations in the general case is considerably changed. In fact, with this
approach, the major part of the analysis is concerned with deriving the relevant
partial derivatives; only minimal evaluation of expectations is required. Our
interest here is to examine this method of "nding moment approximations
Nagar's results were obtained in the context of a simultaneous equation
model where all the predetermined variables are exogenous and where all the
structural disturbances are normally distributed, serially independent and
homoscedastic. In much econometric work, however, it may be unrealistic to
suppose that disturbances have these desirable properties and in single equation
1 This work was originally undertaken to clarify this point.
G.D.A. Phillips / Journal of Econometrics 97 (2000) 345}364
347
regression models considerable e!ort has been devoted to analysing the implications for econometric estimation of departures from standard assumptions. It is
equally important that a similar consideration be given to simultaneous equation models and analysing the "rst and second moments will make a contribution towards this. Hence, the analysis in this paper, as well as expositing an
alternative approach to obtaining moment approximations, will considerably
extend the Nagar results to models in which both serial correlation and heteroscedasticity are permitted. It is shown that the bias approximation for this
general case may be obtained without assuming that the disturbances are
normally distributed; however, normality is required in "nding an approximation to the second moment.
2. Model, notation and general assumptions
We shall consider a simultaneous equation model which includes as its "rst
equation
y "> b#Z c#u ,
(2)
1
2
1
1
where y and > are, respectively, a ¹]1 vector and a ¹]g matrix of
1
2
observations on g#1 endogenous variables and Z is a ¹]k matrix of
1
observations on k exogenous variables. b and c are, respectively, g]1 and k]1
vectors of unknown parameters and u is a ¹]1vector of stationary distur1
bances with positive-de"nite covariance matrix E(u u@ )"R and "nite mo1 1
1
ments up to fourth order. The complete reduced form of the system includes
> "ZP #< ,
(3)
1
1
1
where > "(y : > ) and Z"(Z : Z ) is a ¹]K matrix of observations on
1
1
2
1
2
K exogenous variables, P "(n : P ) is a K](g#1) matrix of reduced form
1
1
2
parameters and < "(v :< ) is a ¹](g#1) matrix of reduced form distur1
1 2
bances. The transpose of each row of < has zero mean vector and
1
(g#1)](g#1) positive-de"nite covariance matrix X "(u ) while the
1
ij
(¹](g#1)]1 vector, vec < , has a positive-de"nite covariance matrix of
1
dimension (¹](g#1))](¹](g#1)) given by Cov(vec< )"Xvec and "nite
1
1
moments up to fourth order. It is further assumed that: (i) The ¹]K matrix Z is
non-stochastic and of rank K and lim
¹~1Z@Z"R , a K]K positiveT?=
ZZ
de"nite limit matrix.
(ii) Eq. (2) is over-identi"ed so that K'g#k, i.e. the number of excluded
exogenous variables exceeds the number required for the equation to be just
identi"ed. In cases where second moments are analysed we shall assume that
K exceeds g#k by at least two. These over-identifying restrictions are su$cient
to ensure that the Nagar expansion is valid in the case considered by Nagar; see
348
G.D.A. Phillips / Journal of Econometrics 97 (2000) 345}364
Sargan (1974) who showed the validity when moments exist,2 and we shall
employ them here also even though a formal extension of Sargans results to the
more general case considered here has yet to be made. However, when `momentsa do not exist it does not follow that `momenta approximations are
worthless. They may be viewed as pseudo-moments, i.e moments of distributions that approximate the one of interest; see, for example, Phillips (1983).
3. Preliminary results
For ease of exposition we shall introduce the proposed approach by "rst
considering the two-stage least-squares (2SLS) estimators of the unknown
parameters of (2) which are given by
bK
>@ > ! !
Z@ Z
Z@
1 2
1 1
1
with "ZP #< into (2) and rearrang2
2
2
ing, we may write
y "(ZP )b#Z c#< b#u
1
2
1
2
1
"Xa#< b#u ,
2
1
where
a"(b@, c@)@ and X"(ZP :Z ).
2 1
(8)
(9)
2 Later, Sargan (1976) extended his analysis to demonstrate the validity under more general
conditions than Nagar's (but including the existence of the corresponding moments).
G.D.A. Phillips / Journal of Econometrics 97 (2000) 345}364
349
Consider the hypothetical ordinary least-squares (OLS) estimator in (8)
a8"(X@X)~1X@y .
1
This estimator in (10) is clearly unbiased and so E(a8)"a where
E(a8)"(X@X)~1X@E(y )"(X@X)~1X@Zn
1
1
P@ Z@Z ~1 P@ Z@Zn
P@ Z@ZP
2
1
2
2
1 ,
" 2
Z@ Z
Z@ ZP
Z@ Zn
1 1
1
2
1 1
on using (9). It follows that
A
B A
B
(10)
(11)
E(a8)"a"f (vec P ).
1
We have thus established the important result that a( , the 2SLS estimator of a,
can be written in the form a( "f (vec PK ) where a"f (vec P ).
1
1
4. The proposed approach
For simplicity we shall examine the ith component of (6) which we write in the
form
a( "f (vec PK ), i"1, 2,2, g#k.
(12)
i
i
1
It is assumed that, conditional on the exogenous variables, the function f (.) is
i
di!erentiable with uniformly bounded derivatives up to fourth order in a neighbourhood of vec P as ¹PR, and that the components of vec PK have "nite
1
1
moments up to, at least, fourth order. This ensures that the approximations,
based on the Taylor expansion, will have an error with an order of magnitude
which is well determined. These assumptions are similar to those proposed by
Sargan (1976, p. 430), but see also Shao (1988) who gives conditions for
determining the order of the error term when approximating the moments of
real-valued smooth functions of regression coe$cient estimates,3 a situation
which has much in common with that examined here. Expanding the function in
a Taylor series expansion about the point vec P yields the result
1
f (vec PK )"f (vec P )#(vec(PK !P ))@f (1)
i
1
i
1
1
1 i
1
# (vec(PK !P ))@f (2) (vec(PK !P ))
1
1
1
1 i
2!
3 Shao employs more restrictive assumptions than is done here. Heteroscedasticity of disturbances
is allowed but not dependence.
350
G.D.A. Phillips / Journal of Econometrics 97 (2000) 345}364
1 K g`1
# + + (n( !n )(vec(PK !P ))@f (3) (vec(PK !P ))
1
1
rs
rs
1
1 i,rs
3!
r/1 s/1
#o (¹~3@2),
(13)
1
where f (1) is a K(g#1)]1 vector of "rst-order partial derivatives,
i
Rf /Rvec PK ; f (2) is a (K(g#1))](K(g#1)) matrix of second-order partial derivi
1 i
atives, R2f /R vec PK (Lvec PK )@; and f (3) is a (K(g#1))](K(g#1)) matrix of
i,rs
i
1
1
third-order partial derivatives de"ned as
Lf (2)
f (3) " i , r"1,2, K, s"1,2, g#1.
i,rs Ln
rs
All the derivatives are evaluated at PK "P . The term vec(PK !P ) is
1
1
1
1
a K(g#1)]1 vector of random variables with mean vector 0 and is such that
vec(PK !P )"(I?(Z@Z)~1Z@)vec < , Cov(vec < )"Xvec,
1
1
1
1
1
(14)
Cov(vec(PK !P ))"(I?(Z@Z)~1Z@)Xvec(I?Z(Z@Z)~1).
1
1
1
It is of interest to compare (13) with the counterpart employed by Nagar.
Since (13) is an asymptotic expansion in which the estimation error is expanded
in terms of stochastic order ¹~1@2, ¹~1 and ¹~3@2, with a remainder term which
is o (¹~3@2), it must be equivalent to the Nagar expansion in which successive
1
terms are of the same stochastic order. The corresponding terms will be the same
even though they look quite di!erent. To see this, consider the estimation error
in (4) which can be written as
>@ > ! ! ,
Z@
1 1
1 2
1
Nagar's analysis commences by putting
A
a( !a"
B A
B
(15)
a( !a"[Q~1#X@< #
An alternative approach to obtaining
Nagar-type moment approximations in
simultaneous equation models
Garry D.A. Phillips*
School of Business and Economics, University of Exeter, Streatham Court, Exeter EX4 4PU, UK
Received 1 December 1997; received in revised form 1 September 1999; accepted 1 September 1999
Abstract
This paper examines asymptotic expansions for estimation errors expressed explicitly
as functions of underlying random variables. Taylor series expansions are obtained from
which "rst and second moment approximations are derived. While the expansions are
essentially equivalent to the traditional Nagar type, the terms are expressed in a form
which enables moment approximations to be obtained in a particularly straightforward
way once the partial derivatives have been found. The approach is illustrated by
considering the k-class estimators in a static simultaneous equation model where the
disturbances are non-spherical. ( 2000 Elsevier Science S.A. All rights reserved.
JEL classixcation: C13; C22
Keywords: Simultaneous equation models; Non-spherical disturbances; Nagar expansions; Bias and second moment approximations
1. Introduction
In an important paper Nagar (1959) analysed the small sample properties
of the general k-class of simultaneous equation estimators and, in particular,
found expressions for the bias to the order of ¹~1 where ¹ is the number of
* Tel.: #44-1392-263241; fax: #44-1392-263242.
E-mail address: [email protected] (G.D.A. Phillips).
0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 9 9 ) 0 0 0 7 5 - 5
346
G.D.A. Phillips / Journal of Econometrics 97 (2000) 345}364
observations, and for the second moment matrix to the order of ¹~2. In
obtaining these results Nagar used an asymptotic expansion for the estimation
error of the form:
p~1
(a!a)" + ¹(~1@2)se #¹(~1@2)pr ,
s
p
s/1
(1)
where a is an estimator for a and the e , s"1,2, p!1, and r , the remainder
s
p
term, are all of stochastic order unity as ¹PR. The sum of the retained terms
are then assumed to mimic the behaviour of the estimator and the moments of
this sum are used to approximate the moments of the estimator. The approach
followed by Nagar was not entirely new; see, for example, Kendall (1954), but it
was through Nagar's work that econometricians became aware of the potentiality of the methodology. However, while the approach is of considerable interest,
it is not especially easy to understand what is happening within it. In fact, it uses
an implicit Taylor series expansion but this is not obvious.1 Another characteristic of the method is that it often involves evaluating the expectations of many,
often complex, stochastic terms and derivations of moment approximations can
be lengthy and tedious.
In this paper an alternative, but equivalent, method of expanding the k-class
estimators is presented which "rst expresses the estimator as a function of a set
of underlying random variables, e.g. reduced form coe$cient estimates. This is
a natural way to proceed and it was the approach adopted by Sargan (1976) who
noted that a wide variety of econometric estimators can be regarded as functions
of the sample data "rst and second moments. Thus, the problem of approximating the estimation error by means of a Taylor series expansion is placed in
a particularly familiar framework whereby the resulting methodology is better
understood. Sargan also gave conditions under which the moment approximations are valid. Although the resulting expansion is essentially the same as
Nagar's (in fact, Sargan referred to his expansion as a Nagar expansion), it is
easier to interpret and its form is such that the task of deriving the moment
approximations in the general case is considerably changed. In fact, with this
approach, the major part of the analysis is concerned with deriving the relevant
partial derivatives; only minimal evaluation of expectations is required. Our
interest here is to examine this method of "nding moment approximations
Nagar's results were obtained in the context of a simultaneous equation
model where all the predetermined variables are exogenous and where all the
structural disturbances are normally distributed, serially independent and
homoscedastic. In much econometric work, however, it may be unrealistic to
suppose that disturbances have these desirable properties and in single equation
1 This work was originally undertaken to clarify this point.
G.D.A. Phillips / Journal of Econometrics 97 (2000) 345}364
347
regression models considerable e!ort has been devoted to analysing the implications for econometric estimation of departures from standard assumptions. It is
equally important that a similar consideration be given to simultaneous equation models and analysing the "rst and second moments will make a contribution towards this. Hence, the analysis in this paper, as well as expositing an
alternative approach to obtaining moment approximations, will considerably
extend the Nagar results to models in which both serial correlation and heteroscedasticity are permitted. It is shown that the bias approximation for this
general case may be obtained without assuming that the disturbances are
normally distributed; however, normality is required in "nding an approximation to the second moment.
2. Model, notation and general assumptions
We shall consider a simultaneous equation model which includes as its "rst
equation
y "> b#Z c#u ,
(2)
1
2
1
1
where y and > are, respectively, a ¹]1 vector and a ¹]g matrix of
1
2
observations on g#1 endogenous variables and Z is a ¹]k matrix of
1
observations on k exogenous variables. b and c are, respectively, g]1 and k]1
vectors of unknown parameters and u is a ¹]1vector of stationary distur1
bances with positive-de"nite covariance matrix E(u u@ )"R and "nite mo1 1
1
ments up to fourth order. The complete reduced form of the system includes
> "ZP #< ,
(3)
1
1
1
where > "(y : > ) and Z"(Z : Z ) is a ¹]K matrix of observations on
1
1
2
1
2
K exogenous variables, P "(n : P ) is a K](g#1) matrix of reduced form
1
1
2
parameters and < "(v :< ) is a ¹](g#1) matrix of reduced form distur1
1 2
bances. The transpose of each row of < has zero mean vector and
1
(g#1)](g#1) positive-de"nite covariance matrix X "(u ) while the
1
ij
(¹](g#1)]1 vector, vec < , has a positive-de"nite covariance matrix of
1
dimension (¹](g#1))](¹](g#1)) given by Cov(vec< )"Xvec and "nite
1
1
moments up to fourth order. It is further assumed that: (i) The ¹]K matrix Z is
non-stochastic and of rank K and lim
¹~1Z@Z"R , a K]K positiveT?=
ZZ
de"nite limit matrix.
(ii) Eq. (2) is over-identi"ed so that K'g#k, i.e. the number of excluded
exogenous variables exceeds the number required for the equation to be just
identi"ed. In cases where second moments are analysed we shall assume that
K exceeds g#k by at least two. These over-identifying restrictions are su$cient
to ensure that the Nagar expansion is valid in the case considered by Nagar; see
348
G.D.A. Phillips / Journal of Econometrics 97 (2000) 345}364
Sargan (1974) who showed the validity when moments exist,2 and we shall
employ them here also even though a formal extension of Sargans results to the
more general case considered here has yet to be made. However, when `momentsa do not exist it does not follow that `momenta approximations are
worthless. They may be viewed as pseudo-moments, i.e moments of distributions that approximate the one of interest; see, for example, Phillips (1983).
3. Preliminary results
For ease of exposition we shall introduce the proposed approach by "rst
considering the two-stage least-squares (2SLS) estimators of the unknown
parameters of (2) which are given by
bK
>@ > ! !
Z@ Z
Z@
1 2
1 1
1
with "ZP #< into (2) and rearrang2
2
2
ing, we may write
y "(ZP )b#Z c#< b#u
1
2
1
2
1
"Xa#< b#u ,
2
1
where
a"(b@, c@)@ and X"(ZP :Z ).
2 1
(8)
(9)
2 Later, Sargan (1976) extended his analysis to demonstrate the validity under more general
conditions than Nagar's (but including the existence of the corresponding moments).
G.D.A. Phillips / Journal of Econometrics 97 (2000) 345}364
349
Consider the hypothetical ordinary least-squares (OLS) estimator in (8)
a8"(X@X)~1X@y .
1
This estimator in (10) is clearly unbiased and so E(a8)"a where
E(a8)"(X@X)~1X@E(y )"(X@X)~1X@Zn
1
1
P@ Z@Z ~1 P@ Z@Zn
P@ Z@ZP
2
1
2
2
1 ,
" 2
Z@ Z
Z@ ZP
Z@ Zn
1 1
1
2
1 1
on using (9). It follows that
A
B A
B
(10)
(11)
E(a8)"a"f (vec P ).
1
We have thus established the important result that a( , the 2SLS estimator of a,
can be written in the form a( "f (vec PK ) where a"f (vec P ).
1
1
4. The proposed approach
For simplicity we shall examine the ith component of (6) which we write in the
form
a( "f (vec PK ), i"1, 2,2, g#k.
(12)
i
i
1
It is assumed that, conditional on the exogenous variables, the function f (.) is
i
di!erentiable with uniformly bounded derivatives up to fourth order in a neighbourhood of vec P as ¹PR, and that the components of vec PK have "nite
1
1
moments up to, at least, fourth order. This ensures that the approximations,
based on the Taylor expansion, will have an error with an order of magnitude
which is well determined. These assumptions are similar to those proposed by
Sargan (1976, p. 430), but see also Shao (1988) who gives conditions for
determining the order of the error term when approximating the moments of
real-valued smooth functions of regression coe$cient estimates,3 a situation
which has much in common with that examined here. Expanding the function in
a Taylor series expansion about the point vec P yields the result
1
f (vec PK )"f (vec P )#(vec(PK !P ))@f (1)
i
1
i
1
1
1 i
1
# (vec(PK !P ))@f (2) (vec(PK !P ))
1
1
1
1 i
2!
3 Shao employs more restrictive assumptions than is done here. Heteroscedasticity of disturbances
is allowed but not dependence.
350
G.D.A. Phillips / Journal of Econometrics 97 (2000) 345}364
1 K g`1
# + + (n( !n )(vec(PK !P ))@f (3) (vec(PK !P ))
1
1
rs
rs
1
1 i,rs
3!
r/1 s/1
#o (¹~3@2),
(13)
1
where f (1) is a K(g#1)]1 vector of "rst-order partial derivatives,
i
Rf /Rvec PK ; f (2) is a (K(g#1))](K(g#1)) matrix of second-order partial derivi
1 i
atives, R2f /R vec PK (Lvec PK )@; and f (3) is a (K(g#1))](K(g#1)) matrix of
i,rs
i
1
1
third-order partial derivatives de"ned as
Lf (2)
f (3) " i , r"1,2, K, s"1,2, g#1.
i,rs Ln
rs
All the derivatives are evaluated at PK "P . The term vec(PK !P ) is
1
1
1
1
a K(g#1)]1 vector of random variables with mean vector 0 and is such that
vec(PK !P )"(I?(Z@Z)~1Z@)vec < , Cov(vec < )"Xvec,
1
1
1
1
1
(14)
Cov(vec(PK !P ))"(I?(Z@Z)~1Z@)Xvec(I?Z(Z@Z)~1).
1
1
1
It is of interest to compare (13) with the counterpart employed by Nagar.
Since (13) is an asymptotic expansion in which the estimation error is expanded
in terms of stochastic order ¹~1@2, ¹~1 and ¹~3@2, with a remainder term which
is o (¹~3@2), it must be equivalent to the Nagar expansion in which successive
1
terms are of the same stochastic order. The corresponding terms will be the same
even though they look quite di!erent. To see this, consider the estimation error
in (4) which can be written as
>@ > ! ! ,
Z@
1 1
1 2
1
Nagar's analysis commences by putting
A
a( !a"
B A
B
(15)
a( !a"[Q~1#X@< #