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Classical tests for contemporaneously uncorrelated disturbances in the linear simultaneous equations model
Journal
of Econometrics
42 (1989) 299-317.
North-Holland
CLASSICAL TESTS FOR CONTEMPORANEOUSLY UNCORRELATED DISTURBANCES IN THE LINEAR SIMULTANEOUS EQUATIONS MODEL* Darrell A. TURKINGTON Unioersi!,~
of Wesrrm
Australia,
Perth, Austruliu
Received April 1988. final version received January
This paper obtains the the parameters of the construct the Lagrange that the disturbances procedure is given for
6009
1989
score vector, the information matrix, and the CramerrRao lower bound for linear simultaneous equations model. These concepts are then used to multiplier test statistic and the Wald test statistic for the null hypothesis of the model are contemporaneously uncorrelated. A straightforward forming the Lagrange multiplier test statistic.
1. Introduction Recent years have witnessed a proliferation in the application of classical test procedures to form test statistics for the hypotheses that are of interest in econometric models. For example, Godfrey and Breusch have worked out the Lagrange Multiplier (LM) test procedure for testing for autoregressive and moving average disturbances in dynamic economic models [see Godfrey (1978a, b), Breusch (1978) Godfrey and Breusch (1981)]; Breusch and Pagan obtained the LM test statistic for testing for uncorrelated disturbances in the seemingly unrelated regression equations (SURE) model and for testing hypotheses in the error components model [see Breusch and Pagan (1980)]; and Engle provides the LM test procedure for testing for block-recursive systems [see Engle (1982)]. Several authors have recommended the LM test procedure as a general diagnostic tool for econometric models [see Engle (1982) and *This paper was written while the author was a visiting lecturer at the Department of Economics, the University of Warwick. The author is grateful for useful comments from the participants in the Econometric Workshop at Warwick and at the University of Manchester. In particular comments from Mark Salmon and Mark Stewart of Warwick and Richard Smith. Len Gill, and Carry Phillips of Manchester were much appreciated. The author would also like to thank Jan Magnus for helpful discussions and for kindly providing him with access to his forthcoming book Lineur Strucfures. Thanks go also to an anonymous referee for useful comments. All errors remain the author’s responsibility.
0304-4076/89/$3.50’“1989.
Elsevier Science Publishers
B.V. (North-Holland)
300
D.A. Turkington, Classical tests for uncorrelated disturbances
Breusch and Pagan (1980)]. Other authors, in a limited-information framework, have studied the relationships between the classical test statistics for exogeneity and econometric test statistics such as the Wu-Hausman statistic [see Smith (1983, 1985) and Hwang (1985)]. One important hypothesis that appears to have been overlooked in this literature is the hypothesis that the disturbances in the linear simultaneous equations model are contemporaneously uncorrelated. We know that if this hypothesis is true, then asymptotically efficient estimators of the parameters of primary interest would be the two-stage least squares (2SLS) estimators or the limited-information maximum likelihood (LIML) estimators. However, if this hypothesis is false, asymptotically efficient estimators would be the three-stage least squares (3SLS) estimators or the full-information maximum likelihood (FIML) estimators. Given that the computational effort required in obtaining the former estimators is substantially less than that required in obtaining the latter estimators it is surprising that classical test procedures have not been applied to the hypothesis. This paper seeks to redress this situation. It presents a straightforward procedure for obtaining the LM test statistic for our hypothesis. Both the LM test and the Wald test are asymptotically equivalent in the sense that they have the same limiting distributions under the null hypothesis and under local alternatives. If one is content with first-order asymptotic approximations, it is hard to envisage situations where the Wald test statistic would be used in preference to the LM test statistic. The formation of the Wald test statistic requires that we use FIML estimation on our model whereas the LM test statistic only requires LIML. (Indeed, it is this relative computational ease that accounts for the popularity of the LM test statistic in econometrics.) Moreover, the main reason why we are interested in conducting a test for our hypothesis is to see whether on the grounds of asymptotic efficiency it is worthwhile computing the more burdensome FIML estimators! For these reasons we do not persevere with the Wald test statistic in the same way as we do for the LM test statistic. Instead we shall just outline the form of this test statistic. The remaining classical test statistic, namely the likelihood ratio test statistic, is not studied in this paper. The order of presentation is as follows: The model is given in section 2. Section 3 concerns itself with O-l matrices that are important to our analysis and gives a derivative and a selection matrix that are used extensively throughout the analysis. In section 4 we obtain the basic building blocks needed for classical hypothesis testing namely, the score vector, the information matrix, and the Cramer-Rao lower bound. The LM test statistic is derived in section 5 and a procedure for its application is given. Section 6 briefly concerns itself with the Wald test statistic. The final section is reserved for the conclusion.
D.A. Turkington,
Classical
tests for uncorreluted
disturbances
301
2. The model We consider a complete system of G linear stochastic structural equations in G jointly dependent current endogenous variables and k predetermined variables.’ The i th equation is written as y, = q’,p, + x,y, + 24,= Z+Y; + U,,
i=l
,...>
G,
where y, is a n x 1 vector of sample observations of one of the current endogenous variables, y. is an n x G, matrix of observations on the other G, current endogenous variables in the ith equation, X, is a n X k, matrix of observations on the k, predetermined variables in the equation, U, a n X 1 vector of random disturbances, ZZ, is the n x (G, + k;) matrix (r. X,) and 6, is the (G, + k,) X 1 vector (ply,‘)‘. The usual assumptions are placed on the disturbances. The expectation of ui is the null vector, and it is assumed that the disturbances are contemporaneously correlated so that E(%P,,)
= 0
for periods
=(J,,
for
s=t,
s # t, i,j=l,...,
G,
and E( u;u()
= u;,Z,,.
We write our model and assumptions
more succinctly
as
(1)
y=H6+u,
E(u)=O,
V(U)=S~JZ,
U-N(O,2@Z),
where y=(y;...y&
U=(U;...U;;)‘,
H is the block-diagonal
0
matrix, 0
Hl . . -HG
S=(6;...6&)‘,
1.
and 2 is the symmetric matrix, assumed to be positive definite, whose i, jth element is u,,. We assume that each equation in the model is identifiable by the a priori restrictions, that the II X k matrix X of all predetermined variables has rank k, and that plim( X’u,/n) = 0 for all i. Finally we assume that H, has full column rank and that the plim H,‘H,/n exists for all i and j. ‘The notation used in this paper is that used in Bowden and Turkington (1984, ch. 4).
302
D.A. Turkington,
A different YB+
way of writing Xr=
U,
Clussicul tests for uncorreluted
disturbances
our model is 0)
where Y is the n x G matrix of observations on the G current endogenous variables, X is the n X k matrix of observations on the k predetermined variables, B is a G X G matrix of coefficients of the endogenous variables in our equations, r is k x G matrix of the coefficients of the predetermined variables in our equations, and U is the n x G matrix (ut . . . uc). It follows that some of the elements of B are known a priori to be equal to one or zero, since y, has a coefficient of one in the ith equation and some endogenous variables are excluded from certain equations. Similarly, some of the elements of r are known a priori to be zero as certain predetermined variables are excluded from each equation. We assume B is nonsingular. The hypothesis that concerns us in this paper is that the disturbances of this model are contemporaneously uncorrelated. That is we wish to test the null hypothesis
Ho: against
u,]=O,
i#j,
the alternative H,:
0,/ # 0.
As mentioned in the introduction, the reason we wish to conduct such a test is that we are investigating whether estimators that are computational easy to compute, such as 2SLS estimators or LIML estimators, are as asymptotically efficient as estimators that require greater computational effort, such as 3SLS estimators or FIML estimators. If the null hypothesis is true this will be the case. Moreover, if the LM test is used, we need only compute the former estimators in our testing procedure. Should we accept the null hypothesis, then on the grounds of asymptotic efficiency these estimators suffice. There is no need to move to the more computationally difficult estimators. To illustrate these points consider the following trivial demand (A) and supply model (B): (A)
q=a,+b,p+c,Y+d,B’+u,,
03)
q=a,+b2p+cZR+d,T+u,.
Here the exogenous variables are income (Y), wealth (W), rainfall (R), and taxation (T), and we assume Eu,u, = ut2. If u,* = 0, then either equation can be estimated efficiently using 2SLS say. If utz # 0, then asymptotically efficient
D.A. Turkingion,
Clussicul tests for uncorreluted
disturhunces
303
estimation requires that we apply a full-information procedure, such as 3SLS, to the system made up of both equations. The LM test procedure based on the 2SLS estimators for H,: ur2 = 0 provides us with a relatively easy method to determine whether it is worthwhile moving to the more computationally difficult full-information estimator. Of course, in this simple example the additional computational effort in moving to the full-information estimation technique is not all that great, but one can imagine a more general system made up of many such demand and supply equations where full-information estimation places a considerable burden on our computational resources. One would then like a straight-forward test procedure based on limited-information estimation techniques to see if this extra computational effort is even required. The classical LM test procedure fits this bill. In deriving our classical test statistics certain O-l matrices will play an important role. We also need to obtain the relationship between the matrix B, when the model is written as in eq. (2) and the vector 8, used in eq. (1). This relationship enables us to write a vet B/d8 in terms of a selection matrix. These matters are taken up in the next section. 3. Preliminaries 3.1. O-I matrices Consider
a square
G x G matrix
where a, is the jth column the elements of A:
A given by
of A. Consider
1: IT al
vecA @Xl
=
a2
u(A)
=
the following
vectors formed
a11
021
a,,
QGl
a22
a32
aG2
a,2
from
jC(G+l)xl
a,
aa
a,,;-
I
304
D.A. Turkington, Classical tests for uncorrelated disturbances
The vector vet A is formed by stacking the columns of A one underneath the other, u(A) is formed stacking the elements of A on and below the main diagonal one underneath the other, and E(A) is formed by stacking the elements below the main diagonal one underneath the other. Notice that vet A contains all the elements in u(A) and in C(A). It follows then that there exist O-l matrices L (iG(G + 1) X G*) and z (fG(G - 1) x G2) with the properties LvecA
=u(A)
and
EvecA
=6(A).
These matrices are called elimination matrices. Notice also that, if A is symmetric, u(A) contains all the essential elements of A, so there exists a O-l matrix D (G2 x $G(G + 1)) with the property Do(A) Similarly, elements property
= vecA.
if A is strictly lower triangular, E(A) contains all the essential of A, so there exists a O-l matrix, z’ as it turns out, with the
L/U(A)
=vec(A).
The matrices D and z’ are called duplication matrices. Two other O-l matrices that are used in this paper matrix K ( G2 x G2) which has the property
are the commutation
K vecA = vec( A’), and
N = $( I + K ) which has the property NvecA
=vec$(A
+A’)
forallA.
These matrices along with other O-l matrices have been studied extensively by Magnus and Neudecker (1979, 1980, 1986) Henderson and Searle (1979, 1982) Tracy and Dwyer (1969) and others. For a comprehensive survey of their properties, see Magnus (1988). The results concerning these matrices that are used in this paper are presented in the appendix and were taken from Magnus (1988). These matrices arise in matrix differentiation as it is clear that ~3vecA/au(A)‘=
D
forAsymmetric
and C?vec( A’)/LI vet A’ = K.
D.A. Turkington,
Such derivatives tion matrices.
naturally
Clussical tests for uncorreloted disturbances
arise in the formation
305
of score vectors and informa-
3.2. The derivative a vet B/as’ The above derivative is also needed in forming the score vector and the information matrix. The easiest way of obtaining this derivative is to express B of eq. (2) in terms of 6 of eq. (1). To this end write the ith equation of our model as y, = YR,& + X$y, + u I) where R, and q are selection matrices with the properties that YR,r Y and X$ = X,. Then y, = YR,S, + XS,S, + ui with R, = (R, 0) and S, = (0 S,), so we can write
Y=(y,...yc)=Y(R,6,...R,6,)+X(S,6,...S,S,)+U. It follows
that
B=I,-
(~,a,...
~Jjo),
r= -(S,6,...S&,
and
(3)
vecB = vet IG - R6, where R is the block-diagonal
R=
From
L
‘.
.
*R,
eq. (3) we obtain
a vetB/W
matrix
the derivative
= -R.
(4)
4. The score vector, the information matrix, and the Cramer-Rao 4.1. The score vector: al/a0 The unknown
parameters
of our model are
lower bound
306
D.A. Turkington, Clussiml tests for uncorreluted disturhunces
where the u( .), vech, operator is as defined function is, apart from a constant, r(8)=
a1
as
Using obtain
=n
of the score vector is
a logldet BI
as
the chain
ai -= a6
- nR’vec
a/ -= au using
-1
rule for matrix
where R is the selection of the score vector is
Again
_
n 2
2
a tr
Z-‘U’U
a8
.
differentiation
B-” + H’(P
alog au
q(vecZ-lU’LiZ-‘-
and the result
(5)
in section
: atrz-WU au
3. The second component
f
differentiation
we obtain
n vecZ_‘),
(6)
where D is the duplication matrix defined in section (6) make up the score vector af/atI for our problem. 4.2. The information matrix:2 Z(O) = -plim Differentiating
-nR’K(B-“@
a21 = a6ad
-H’(Z-‘@
-
a21
auad
n-’
eqs. (5) and (6) we obtain
a21 = asas’
of eq. (4) we
Q I)&
matrix described
the chain rule of matrix
al - = au
-
3. The log likelihood
-tlogdetB+nlogldetBj-:trX-‘VU.
The first component -
in section
B-‘)R
3. Together
eqs. (5) and
( a21/atV19’)
the Hessian
- H’(Z-‘63
matrix
a’//a&YP:
I)H,
UZ--‘)D,
= zD’(,X-‘@Z-‘)D-D’[z-‘U’UE-‘OsZ“]D. 2
‘More correctly the information matrix I( 8) is defined as - EN ‘( a’//a&jtI’). But it is caxily shown that for the linear simultaneous equations model this is equnl to - plim II ‘( ~I’//iWW).
D.A. Turkington,
where K is the commutation asymptotic theory, and
I
=
defined
in section
3. From
standard
plim
n
= R’( z @ B-9).
matrix can be written
H’(Z-’ @Z)N
plim
Z(6)
matrix
H’(Zc9 u)
U’U plim = 1 n So the information
307
Clmsicui tests for uncorreluted disturbances
R’(P
n
8 BP’)D
+R’K(B-“@B-‘)R
1
I
D’(P
4.3. The Cramer-Rao
p(P
63 B_‘)R
ca2-‘)D
1
lower bound: Z-‘(e)
The final essential ingredient needed in forming classical test statistics is the Cramer-Rao lower bound for the asymptotic covariance matrix of a consistent estimator of 8, that is Z-‘(e). The work involved in inverting the information matrix is contained in appendix 2. Here we present the results. Write z-‘(e)
=
1;I; zs”1. Z
Then p
=
“0
plim H’(X’
l
@ P)H
n
-’ ’
1
(E@~)+2(Z@lYBp’)R
xR’(Z@
B-“z)
plim
1
NL’,
-l
H’(Z-’ 8 P)H Z8” = - 2 plim
n
R’( Z @ B-“2)
N = ( Ias)‘.
I
In these expressions L is the elimination matrix defined in section 3, N is the O-l matrix also defined in section 3, and P is the projection matrix P = X( X’X)-‘X’. Notice that 1’” gives us the usual lower bound on the asymp-
308
D.A. Turkington,
Clussical
tests for uncorrelated
disturbunces
totic covariance matrix of a consistent estimator of 6. The 3SLS and the FIML estimators for example reach this lower bound. Similarly I”” gives us a lower bound on the asymptotic covariance matrix of a consistent estimator of u, the vector that contains the variances and covariances of the disturbance terms. 5. The Lagrange multiplier test statistic for contemporareously uncorrelated disturbances Our model
is
y=m+u, u-
N(O,I@Z)
The null hypothesis
Ho: against
or
YB+XT=
U.
is
fJ,/ =O,
i#j,
the alternative Hi:
a,j # 0.
Notice that V = U(z) contains sis. Thus we can write H,:
U= 0
against
all the parameters
Hi:
specified by the null hypothe-
uz 0.
In this section we obtain an expression for the LM test statistic for H,. As mentioned in the introduction, this test statistic has been worked out for special cases. Breusch and Pagan (1980) give such a LM test statistic for the SURE model, that is the special case where our equations contain only right-hand predetermined variables. Engle (1982) gives the LM test statistic when the null hypothesis specifies a recursive system, that is, when the null He also gives the LM test hypothesis specifies H,. . U = 0, B is triangular. statistic when the null hypothesis specifies a block-recursive system. Smith (1983, 1985) obtains the LM test statistic in the limited-information framework, that is, when we have the structural form of one of the equations, say the first one, but only the reduced-form of the other current endogenous variables. We now wish to obtain the LM test statistic for the more general case. The test statistic is
D.A. Turkington, Classicul tests for uncorrelated disturhunces
309
where I”” refers to that part of the Cramer-Rao lower bound corresponding to the parameters V and 6 puts U equal to zero and evaluates all other parameters at the constrained MLE, i.e., at the LIML estimators. Under H,, Tl tends in distribution to a x2 random variable with degrees of freedom +G(G - l), so this distribution is used to find the appropriate upper-tail critical region. In forming Tl the first task is to obtain dl/dC and ZFa from al/au and I”” which we have in hand. We do this by means of an appropriate selection matrix S which has the property U = Sv. As v = L vet .I? and U = l vet 2, it must be true that SL = z. Clearly,
from eq. (6) and the property Dv( A) = vet Theorems 6 and 7 of appendix 1, SD’D = 2s - SLKL’= Moreover,
2s - zKL’=
if H, is true, Sv(2-‘) g
A for
symmetric
A. Using
2s.
= 0. Thus under
H,,
(7)
=Sv(A)=SLvec(A)=Lvec(A),
where A = ~-‘u’ZJ~P’ Moreover, 1”” =
SZ”“S’
=
jQ,y
@I,
(8)
where
H’(_r V(B)=2N(~~~)+4N(Z~~B~‘)R
plim i
xR’(Z@ Combining
B-“JY)N.
Finally,
using Theorem Tl = lvec n
-0) L’L vet A
9 of appendix
A’V(B)vecA
n
1 (9)
eqs. (7) (8) and (9) we get that -Tl = Lvec A’L’LV( n
-l
63 P)H
evaluated
1, we write
evaluated
at 6.
at 6.
310
D.A. Turkington,
Clussicul tests for uncorrehted
disturbances
To evaluate this test statistic further requires tedious algebra which is presented in appendix 3, where it is shown that
the outline
of
G-l
Tl = n c QjA,k,, !=I B, and A, are defined
and
Procedure
for forming
in the following
(2) Using
procedure.
the LM test statistic
(1) Apply LIML to each equation Form the LIML residual vectors ii,=y,--H,6”,
straightforward
for
i=l,...,
y, = H,8, + u, to get estimators
s’,, say.
G.
si and ii,, form
c,, =fi;ii,/n
for
i,j=l,...,
G,
and
j.
(3) Form
and
with P = X( X’X)-‘X’, matrix such that
YR,= r,. (4) Form
T, = n
c r=l
;;A$,.
P, = X,(X,‘X,)-‘X,‘,
and
xi
is the selection
matrix
D.A. Turkington,
Clussicul tests for uncorreluted
disrurhunce.s
311
Two points should be made about this test statistic. First, with no right-hand current endogenous variables our model becomes the SURE model. For this special case it is easily seen that
T, = ivecP[
N(l& 63 2’,)]vecA=
n
2 ‘c’r,:,
I=1 /=1
where r,;. = C,,/(~,,C?,~). This is the same LM test statistic obtained for this model by Breusch and Pagan (1980). Second, the 2SLS estimators, if,2” say, are asymptotically equivalent to the LIML estimators, 6,. Moreover, estimators of u,, based on the 2SLS residual vectors are asymptotically equivalent to the estimators of u,, based on the LIML residual vectors. Hence an asymptotically equivalent version of the LM test statistic may be obtained by using 2SLS in place of LIML in the above procedure.
6. The Wald test statistic Let s^, be the FIML estimator for S,, ii, =y, - q,t, be the MLE residual vector, and fi = (2,. . . iTic). Then the MLE of .Z is 2 = fi’fi/n, and we know that h(vece-vec2)
-f+N(O,V(B)),
where V(0) is as defined in eq. (9). The Wald Atest statistic for Ha: V = 0 is based on the asymptotic distribution of fi(U - U), where U contains the appropriate elements of vec(&. But as 0 = z vet(I), we have fi(i--iJ)=L&(vecZThe Wald
test statistic
vec_$):
N(O,LV(B)L').
is then
(11) where 4 signifies the MLE of 8. Under H,,, T2 has a limiting xz distribution with degrees of freedom :G(G - l), so this distribution is used to obtain the appropriate upper-tail critical region. A 3SLS version of the Wald test statistic is available. Let 6:, be the 3SLSE of 6;, b, = _y,- H,i, be the 3SLS rssidual vector, i? = (z?, . . . uc;), ’ and .Y$= b$/n. Then, as then 3SLS ,.estimators s^; and 2 are asymptotically equivalent to FIML estimators S and 2 [see Bowden and Turkington (1984, ch. 4)], the former can
312
D.A. Turkington, Classical tests for uncorrehted disturbunces
be used in place of the latter in forming a Wald-type test statistic. The statistic obtained in this way will be asymptotically equivalent to T2 of eq. (11). From a computational viewpoint this test statistic is more convenient than T,, though less convenient than LM test statistic T, or the 2SLS version of the LM statistic. 7. Conclusion In a very informative article Engle (1982) outlines a procedure for obtaining the LM test statistic (or a test statistic asymptotically equivalent to the LM test statistic) that avoids the necessity of first computing the information matrix Z(e) and the Cramer-Rao lower bound Z-‘(e). Moreover, he shows that this test statistic can be interpreted in terms of a R2 or a F test statistic of an underlying regression. Unfortunately his results require that the information matrix Z(0) be block-diagonal at least under the null hypothesis. Although this requirement is met in many econometric models and hypotheses, it does not hold for our case. In forming classical test statistics for the hypothesis in hand one is forced to first find Z(B) and Z-‘(e). The LM test statistic thus obtained does not appear to have a convenient intuitive interpretation in terms of R2 or F test statistics. However, the procedure outlined in section 5 is perfectly straightforward. We have also seen that a 3SLS version of the Wald test statistic is available. Although this eases the computational burden required in obtaining the Wald statistic, it is unlikely that this test statistic will be a serious contender for testing uncorrelated disturbances in the linear simultaneous equations model. Appendix 1:
Theorems on O-l matrices
The following used throughout Theorem A -*)NL’.
1.
results concerning the O-l matrices defined in section this paper and were obtained from Magnus (1988).
For
a nonsingular
matrix
A,
[D’(A @ A)D]-’
= LN(A-’
Theorem2.
FormatricesAandB,
Theorem 3.
DLN = N.
Theorem 4.
N=
Theorem 5.
For square G X G matrices A and B, K( A @ B) = (B 8 A)K.
Theorem (A@A)N.
6.
For
f(Z+
3 are
8
D’(A@B)D=D’(B@A)D.
K), N’= N.
square
G x G
matrix
A,
N( A @ A) = N( A @ A)N =
D.A. Turkington,
Theorem
7.
D’D = 2I-
Theorem
8.
EKL’ = 0.
Classical tests for uncorrelated disturbances
LKL’.
-where ith element is ai,. Then L’L vet(A)
Theorem 9. Let A be the matrix vet (A), where 0 azl
0 0
.f. ...
0 0
0 0
. . . . . . . . . . . . . . . . . . . , .
aGz
a,1
Appendix 2: Applying
“.
the formulae
0
aGG-1
The Cramer-Rao
=
1 .
lower bound Z -l(8)
for the inverse
H’(P I”’ =
313
of a partitioned
matrix,
we have
@I)H
plim
+ R’K(B-“8
n
BP’)R -1
-2R’(Z-‘@
B-‘)D{
D’(Z-‘8X’)D}
Applying Theorems 1 and 2 of appendix be written as, by Theorem 3, 2R’(/.Z-‘8 = R’(Z-‘8 Using
standard
H’(Y’
where
B-‘JW’)R
p
= R’(X’
8 B-“_-ZB-‘)R,
matrix
that plim H’(CI-‘69
= i
B-‘)R.
63 M)H
M is the projection
It follows
- R’K(B-“8
theory we note that
n
n
P)H
-’ i
8 B-‘)R
1, the third matrix in the bracket
B-“)N(I@EB-‘)R
asymptotic
plim
-lD’(B-’
.
I
can
.
314
D.A. Turkington,
By a similar
Ciussical
testsfor uncorreluted
disturbunces
analysis, (Z@~)+2(163~‘B-‘)R
plim i
1
xR’(I@B-“Z)
NL’
and I”” = - 2 plim
R’( I @ B-“2)
N.
G-l
Appendix 3.
T, = n c
i;Aihi
i=l
Using
Theorem
5 of appendix
1 and eqs. (9) and (10) of section
R’(I@B-“)
x R plim
x2[vecA’N(Z@32)]‘, where, it is understood, the parameters this expression under the assumption consider each component in turn: A.3.1.
2vecA’N(
Under
I @ Z) under Ho
H,,
Let e, be the jth
column
of IG. Then
5, we write
1 (A.11
are evaluated at 6 We need to evaluate that Ha: u,, = 0, i # j, is true. We
D.A. Turkington, Clussicui tests for uncorreluted disturbunces
[see Magnus
But N=
(1988)] and
f(I+K),
so
0
%l
I -.. (722
0
=-
315
1
0
DGG
1 2
0
011
1. I_ 022
0
0
.2u,,
(A.21
Now A = E-‘iJ’U,Y’,
so under
H,,
Pll
P12
. . .
Plc;
P21
P22
. . .
P2c
A=
p,,=-
with
4U, ~,,~,lJ
PGl
PC2
. . .
PGG 4
0
0
0
...
0
0
P21
0
0
...
0
0
P31
P32 . . .
0 . . .
. . . . . . . . .
0
0
Pa
PG2
PGG- 1
0
’
316
D.A. Turkington,
Classical
tests
for
uncorrelated
disturbances
and i
i.
Using
eqs. (A.2) and (A.3), it follows under 2vecA’N(I@Z)=
H,,
;(oou;u,... u;;u,)
~(0u~u,...u$4,) [
...
(A-3)
[email protected]~~%l) i)‘]. (A4
H’(F A.3.2.
(I@ B-‘)R
cd P)H
plim
n
-l
I
R’(I c3 B-l’) under H,
Using the formulae for a partitioned inverse and R, = ( Ri 0) we obtain, after a little work, that under H, this matrix is the block-diagonal matrix,
V(P-
P,)Y,
m1
R’B-1
plim
aI, Bm ‘R,
n
i
1
0
0
1
uGccB-‘R,
plim
i
A.3.3.
f(.Z @ 2-l)
Clearly
Yd(P- PG)Y, n
)
-Ix,
B-,,
G
under Ho
this matrix
is the block-diagonal
matrix
(A-6)
D.A. Turkington, Classical testsfor uncorrelated disturbances
317
Substituting (A.4), (A.5), and (A.6) in (A.l), ignoring the plims, and replacing unknown parameters by their constrained ML estimators, we obtain the expression G-1
Tl = n
c
k:A,i,,
r=l
as required. References Bowden. R.J. and D.A. Turkington, 1984. Instrumental variables, Econometric Society monograph in quantitative economics, No. 8 (Cambridge University Press, New York, NY). Breusch, T.S. and A.R. Pagan, 1980, The lagrange multiplier test and its applications to model specification in econometrics, Review of Economic Studies 47, 239-254. Engle, R.F., 1982, A general approach to Lagrange multiplier model diagnostics, Journal of Econometrics 20, 83-104. Godfrey. L.G., 1978a. Testing against general autoregressive and moving average error models when the regressors include lagged dependent variables, Econometrica 46, 1293-1302. Godfrey, L.G., 1978b, Testing for higher order serial correlation in regression equations when the regressors include lagged dependent variables, Econometrica 46, 1303-1310.* Godfrev. L.G. and T.D. Breusch, 1981, A review of recent work on testing for autocorrelation in dynamic economic models, in: D. Currie, R. Nobay, and D. Peel,-eds., Macro-economic analysis (Croom-Helm, Beckenham) 63-105. Hausman, J.A., 1979, Specification tests in econometrics, Econometrica 46, 1251-1272. Henderson, H.V. and S.R. Searle, 1979, Vet and vech with some uses in Jacobians and multivariate statistics, Canadian Journal of Statistics 7, 65-81. Henderson, H.V. and S.R. Searle, 1981, The vet-permutation matrix, the vet operator and Kronecker products: A review, Linear and Multilinear Algebra 9, 271-288. Hwang, H., 1985, The equivalence of Hausman and Lagrange multiplier tests of independence between disturbance and a subset of stochastic regressors, Economics Letters 17, 83-86. Magnus, J., 1988, Linear structures (Oxford University Press, New York, NY). Magnus, J.R. and H. Neudecker, 1979, The commutation matrix: Some properties and applications, Annals of Statistics 7, 381-394. Magnus. J.R. and H. Neudecker, 1980, The elimination matrix: Some lemmas and applications, SIAM Journal on Algebraic and Discrete Methods 1. 422-449. Magnus. J.R. and H. Neudecker, 1986, Symmetry, O-l matrices and Jacobians: A review, Econometric Theory 2, 157-190. Smith, R.J., 1983, On the classical nature of the Wu-Hausman statistics for the independence of stochastic regressors and disturbance, Economics Letters 11, 357-364. Smith, R.J., 1985, Wald tests for the independence of stochastic variables and disturbance of a single linear stochastic simultaneous equation, Economics Letters 17, 87-90. Tracy, D.S. and P.S. Dwyer, 1969, Multivariate maxima and minima with matrix derivatives, Journal of the American Statistical Association 64, 1576-1594. Wu, Demin, 1973, Alternative tests of independence between stochastic regressors and disturbances, Econometrica 41, 733-750.
of Econometrics
42 (1989) 299-317.
North-Holland
CLASSICAL TESTS FOR CONTEMPORANEOUSLY UNCORRELATED DISTURBANCES IN THE LINEAR SIMULTANEOUS EQUATIONS MODEL* Darrell A. TURKINGTON Unioersi!,~
of Wesrrm
Australia,
Perth, Austruliu
Received April 1988. final version received January
This paper obtains the the parameters of the construct the Lagrange that the disturbances procedure is given for
6009
1989
score vector, the information matrix, and the CramerrRao lower bound for linear simultaneous equations model. These concepts are then used to multiplier test statistic and the Wald test statistic for the null hypothesis of the model are contemporaneously uncorrelated. A straightforward forming the Lagrange multiplier test statistic.
1. Introduction Recent years have witnessed a proliferation in the application of classical test procedures to form test statistics for the hypotheses that are of interest in econometric models. For example, Godfrey and Breusch have worked out the Lagrange Multiplier (LM) test procedure for testing for autoregressive and moving average disturbances in dynamic economic models [see Godfrey (1978a, b), Breusch (1978) Godfrey and Breusch (1981)]; Breusch and Pagan obtained the LM test statistic for testing for uncorrelated disturbances in the seemingly unrelated regression equations (SURE) model and for testing hypotheses in the error components model [see Breusch and Pagan (1980)]; and Engle provides the LM test procedure for testing for block-recursive systems [see Engle (1982)]. Several authors have recommended the LM test procedure as a general diagnostic tool for econometric models [see Engle (1982) and *This paper was written while the author was a visiting lecturer at the Department of Economics, the University of Warwick. The author is grateful for useful comments from the participants in the Econometric Workshop at Warwick and at the University of Manchester. In particular comments from Mark Salmon and Mark Stewart of Warwick and Richard Smith. Len Gill, and Carry Phillips of Manchester were much appreciated. The author would also like to thank Jan Magnus for helpful discussions and for kindly providing him with access to his forthcoming book Lineur Strucfures. Thanks go also to an anonymous referee for useful comments. All errors remain the author’s responsibility.
0304-4076/89/$3.50’“1989.
Elsevier Science Publishers
B.V. (North-Holland)
300
D.A. Turkington, Classical tests for uncorrelated disturbances
Breusch and Pagan (1980)]. Other authors, in a limited-information framework, have studied the relationships between the classical test statistics for exogeneity and econometric test statistics such as the Wu-Hausman statistic [see Smith (1983, 1985) and Hwang (1985)]. One important hypothesis that appears to have been overlooked in this literature is the hypothesis that the disturbances in the linear simultaneous equations model are contemporaneously uncorrelated. We know that if this hypothesis is true, then asymptotically efficient estimators of the parameters of primary interest would be the two-stage least squares (2SLS) estimators or the limited-information maximum likelihood (LIML) estimators. However, if this hypothesis is false, asymptotically efficient estimators would be the three-stage least squares (3SLS) estimators or the full-information maximum likelihood (FIML) estimators. Given that the computational effort required in obtaining the former estimators is substantially less than that required in obtaining the latter estimators it is surprising that classical test procedures have not been applied to the hypothesis. This paper seeks to redress this situation. It presents a straightforward procedure for obtaining the LM test statistic for our hypothesis. Both the LM test and the Wald test are asymptotically equivalent in the sense that they have the same limiting distributions under the null hypothesis and under local alternatives. If one is content with first-order asymptotic approximations, it is hard to envisage situations where the Wald test statistic would be used in preference to the LM test statistic. The formation of the Wald test statistic requires that we use FIML estimation on our model whereas the LM test statistic only requires LIML. (Indeed, it is this relative computational ease that accounts for the popularity of the LM test statistic in econometrics.) Moreover, the main reason why we are interested in conducting a test for our hypothesis is to see whether on the grounds of asymptotic efficiency it is worthwhile computing the more burdensome FIML estimators! For these reasons we do not persevere with the Wald test statistic in the same way as we do for the LM test statistic. Instead we shall just outline the form of this test statistic. The remaining classical test statistic, namely the likelihood ratio test statistic, is not studied in this paper. The order of presentation is as follows: The model is given in section 2. Section 3 concerns itself with O-l matrices that are important to our analysis and gives a derivative and a selection matrix that are used extensively throughout the analysis. In section 4 we obtain the basic building blocks needed for classical hypothesis testing namely, the score vector, the information matrix, and the Cramer-Rao lower bound. The LM test statistic is derived in section 5 and a procedure for its application is given. Section 6 briefly concerns itself with the Wald test statistic. The final section is reserved for the conclusion.
D.A. Turkington,
Classical
tests for uncorreluted
disturbances
301
2. The model We consider a complete system of G linear stochastic structural equations in G jointly dependent current endogenous variables and k predetermined variables.’ The i th equation is written as y, = q’,p, + x,y, + 24,= Z+Y; + U,,
i=l
,...>
G,
where y, is a n x 1 vector of sample observations of one of the current endogenous variables, y. is an n x G, matrix of observations on the other G, current endogenous variables in the ith equation, X, is a n X k, matrix of observations on the k, predetermined variables in the equation, U, a n X 1 vector of random disturbances, ZZ, is the n x (G, + k;) matrix (r. X,) and 6, is the (G, + k,) X 1 vector (ply,‘)‘. The usual assumptions are placed on the disturbances. The expectation of ui is the null vector, and it is assumed that the disturbances are contemporaneously correlated so that E(%P,,)
= 0
for periods
=(J,,
for
s=t,
s # t, i,j=l,...,
G,
and E( u;u()
= u;,Z,,.
We write our model and assumptions
more succinctly
as
(1)
y=H6+u,
E(u)=O,
V(U)=S~JZ,
U-N(O,2@Z),
where y=(y;...y&
U=(U;...U;;)‘,
H is the block-diagonal
0
matrix, 0
Hl . . -HG
S=(6;...6&)‘,
1.
and 2 is the symmetric matrix, assumed to be positive definite, whose i, jth element is u,,. We assume that each equation in the model is identifiable by the a priori restrictions, that the II X k matrix X of all predetermined variables has rank k, and that plim( X’u,/n) = 0 for all i. Finally we assume that H, has full column rank and that the plim H,‘H,/n exists for all i and j. ‘The notation used in this paper is that used in Bowden and Turkington (1984, ch. 4).
302
D.A. Turkington,
A different YB+
way of writing Xr=
U,
Clussicul tests for uncorreluted
disturbances
our model is 0)
where Y is the n x G matrix of observations on the G current endogenous variables, X is the n X k matrix of observations on the k predetermined variables, B is a G X G matrix of coefficients of the endogenous variables in our equations, r is k x G matrix of the coefficients of the predetermined variables in our equations, and U is the n x G matrix (ut . . . uc). It follows that some of the elements of B are known a priori to be equal to one or zero, since y, has a coefficient of one in the ith equation and some endogenous variables are excluded from certain equations. Similarly, some of the elements of r are known a priori to be zero as certain predetermined variables are excluded from each equation. We assume B is nonsingular. The hypothesis that concerns us in this paper is that the disturbances of this model are contemporaneously uncorrelated. That is we wish to test the null hypothesis
Ho: against
u,]=O,
i#j,
the alternative H,:
0,/ # 0.
As mentioned in the introduction, the reason we wish to conduct such a test is that we are investigating whether estimators that are computational easy to compute, such as 2SLS estimators or LIML estimators, are as asymptotically efficient as estimators that require greater computational effort, such as 3SLS estimators or FIML estimators. If the null hypothesis is true this will be the case. Moreover, if the LM test is used, we need only compute the former estimators in our testing procedure. Should we accept the null hypothesis, then on the grounds of asymptotic efficiency these estimators suffice. There is no need to move to the more computationally difficult estimators. To illustrate these points consider the following trivial demand (A) and supply model (B): (A)
q=a,+b,p+c,Y+d,B’+u,,
03)
q=a,+b2p+cZR+d,T+u,.
Here the exogenous variables are income (Y), wealth (W), rainfall (R), and taxation (T), and we assume Eu,u, = ut2. If u,* = 0, then either equation can be estimated efficiently using 2SLS say. If utz # 0, then asymptotically efficient
D.A. Turkingion,
Clussicul tests for uncorreluted
disturhunces
303
estimation requires that we apply a full-information procedure, such as 3SLS, to the system made up of both equations. The LM test procedure based on the 2SLS estimators for H,: ur2 = 0 provides us with a relatively easy method to determine whether it is worthwhile moving to the more computationally difficult full-information estimator. Of course, in this simple example the additional computational effort in moving to the full-information estimation technique is not all that great, but one can imagine a more general system made up of many such demand and supply equations where full-information estimation places a considerable burden on our computational resources. One would then like a straight-forward test procedure based on limited-information estimation techniques to see if this extra computational effort is even required. The classical LM test procedure fits this bill. In deriving our classical test statistics certain O-l matrices will play an important role. We also need to obtain the relationship between the matrix B, when the model is written as in eq. (2) and the vector 8, used in eq. (1). This relationship enables us to write a vet B/d8 in terms of a selection matrix. These matters are taken up in the next section. 3. Preliminaries 3.1. O-I matrices Consider
a square
G x G matrix
where a, is the jth column the elements of A:
A given by
of A. Consider
1: IT al
vecA @Xl
=
a2
u(A)
=
the following
vectors formed
a11
021
a,,
QGl
a22
a32
aG2
a,2
from
jC(G+l)xl
a,
aa
a,,;-
I
304
D.A. Turkington, Classical tests for uncorrelated disturbances
The vector vet A is formed by stacking the columns of A one underneath the other, u(A) is formed stacking the elements of A on and below the main diagonal one underneath the other, and E(A) is formed by stacking the elements below the main diagonal one underneath the other. Notice that vet A contains all the elements in u(A) and in C(A). It follows then that there exist O-l matrices L (iG(G + 1) X G*) and z (fG(G - 1) x G2) with the properties LvecA
=u(A)
and
EvecA
=6(A).
These matrices are called elimination matrices. Notice also that, if A is symmetric, u(A) contains all the essential elements of A, so there exists a O-l matrix D (G2 x $G(G + 1)) with the property Do(A) Similarly, elements property
= vecA.
if A is strictly lower triangular, E(A) contains all the essential of A, so there exists a O-l matrix, z’ as it turns out, with the
L/U(A)
=vec(A).
The matrices D and z’ are called duplication matrices. Two other O-l matrices that are used in this paper matrix K ( G2 x G2) which has the property
are the commutation
K vecA = vec( A’), and
N = $( I + K ) which has the property NvecA
=vec$(A
+A’)
forallA.
These matrices along with other O-l matrices have been studied extensively by Magnus and Neudecker (1979, 1980, 1986) Henderson and Searle (1979, 1982) Tracy and Dwyer (1969) and others. For a comprehensive survey of their properties, see Magnus (1988). The results concerning these matrices that are used in this paper are presented in the appendix and were taken from Magnus (1988). These matrices arise in matrix differentiation as it is clear that ~3vecA/au(A)‘=
D
forAsymmetric
and C?vec( A’)/LI vet A’ = K.
D.A. Turkington,
Such derivatives tion matrices.
naturally
Clussical tests for uncorreloted disturbances
arise in the formation
305
of score vectors and informa-
3.2. The derivative a vet B/as’ The above derivative is also needed in forming the score vector and the information matrix. The easiest way of obtaining this derivative is to express B of eq. (2) in terms of 6 of eq. (1). To this end write the ith equation of our model as y, = YR,& + X$y, + u I) where R, and q are selection matrices with the properties that YR,r Y and X$ = X,. Then y, = YR,S, + XS,S, + ui with R, = (R, 0) and S, = (0 S,), so we can write
Y=(y,...yc)=Y(R,6,...R,6,)+X(S,6,...S,S,)+U. It follows
that
B=I,-
(~,a,...
~Jjo),
r= -(S,6,...S&,
and
(3)
vecB = vet IG - R6, where R is the block-diagonal
R=
From
L
‘.
.
*R,
eq. (3) we obtain
a vetB/W
matrix
the derivative
= -R.
(4)
4. The score vector, the information matrix, and the Cramer-Rao 4.1. The score vector: al/a0 The unknown
parameters
of our model are
lower bound
306
D.A. Turkington, Clussiml tests for uncorreluted disturhunces
where the u( .), vech, operator is as defined function is, apart from a constant, r(8)=
a1
as
Using obtain
=n
of the score vector is
a logldet BI
as
the chain
ai -= a6
- nR’vec
a/ -= au using
-1
rule for matrix
where R is the selection of the score vector is
Again
_
n 2
2
a tr
Z-‘U’U
a8
.
differentiation
B-” + H’(P
alog au
q(vecZ-lU’LiZ-‘-
and the result
(5)
in section
: atrz-WU au
3. The second component
f
differentiation
we obtain
n vecZ_‘),
(6)
where D is the duplication matrix defined in section (6) make up the score vector af/atI for our problem. 4.2. The information matrix:2 Z(O) = -plim Differentiating
-nR’K(B-“@
a21 = a6ad
-H’(Z-‘@
-
a21
auad
n-’
eqs. (5) and (6) we obtain
a21 = asas’
of eq. (4) we
Q I)&
matrix described
the chain rule of matrix
al - = au
-
3. The log likelihood
-tlogdetB+nlogldetBj-:trX-‘VU.
The first component -
in section
B-‘)R
3. Together
eqs. (5) and
( a21/atV19’)
the Hessian
- H’(Z-‘63
matrix
a’//a&YP:
I)H,
UZ--‘)D,
= zD’(,X-‘@Z-‘)D-D’[z-‘U’UE-‘OsZ“]D. 2
‘More correctly the information matrix I( 8) is defined as - EN ‘( a’//a&jtI’). But it is caxily shown that for the linear simultaneous equations model this is equnl to - plim II ‘( ~I’//iWW).
D.A. Turkington,
where K is the commutation asymptotic theory, and
I
=
defined
in section
3. From
standard
plim
n
= R’( z @ B-9).
matrix can be written
H’(Z-’ @Z)N
plim
Z(6)
matrix
H’(Zc9 u)
U’U plim = 1 n So the information
307
Clmsicui tests for uncorreluted disturbances
R’(P
n
8 BP’)D
+R’K(B-“@B-‘)R
1
I
D’(P
4.3. The Cramer-Rao
p(P
63 B_‘)R
ca2-‘)D
1
lower bound: Z-‘(e)
The final essential ingredient needed in forming classical test statistics is the Cramer-Rao lower bound for the asymptotic covariance matrix of a consistent estimator of 8, that is Z-‘(e). The work involved in inverting the information matrix is contained in appendix 2. Here we present the results. Write z-‘(e)
=
1;I; zs”1. Z
Then p
=
“0
plim H’(X’
l
@ P)H
n
-’ ’
1
(E@~)+2(Z@lYBp’)R
xR’(Z@
B-“z)
plim
1
NL’,
-l
H’(Z-’ 8 P)H Z8” = - 2 plim
n
R’( Z @ B-“2)
N = ( Ias)‘.
I
In these expressions L is the elimination matrix defined in section 3, N is the O-l matrix also defined in section 3, and P is the projection matrix P = X( X’X)-‘X’. Notice that 1’” gives us the usual lower bound on the asymp-
308
D.A. Turkington,
Clussical
tests for uncorrelated
disturbunces
totic covariance matrix of a consistent estimator of 6. The 3SLS and the FIML estimators for example reach this lower bound. Similarly I”” gives us a lower bound on the asymptotic covariance matrix of a consistent estimator of u, the vector that contains the variances and covariances of the disturbance terms. 5. The Lagrange multiplier test statistic for contemporareously uncorrelated disturbances Our model
is
y=m+u, u-
N(O,I@Z)
The null hypothesis
Ho: against
or
YB+XT=
U.
is
fJ,/ =O,
i#j,
the alternative Hi:
a,j # 0.
Notice that V = U(z) contains sis. Thus we can write H,:
U= 0
against
all the parameters
Hi:
specified by the null hypothe-
uz 0.
In this section we obtain an expression for the LM test statistic for H,. As mentioned in the introduction, this test statistic has been worked out for special cases. Breusch and Pagan (1980) give such a LM test statistic for the SURE model, that is the special case where our equations contain only right-hand predetermined variables. Engle (1982) gives the LM test statistic when the null hypothesis specifies a recursive system, that is, when the null He also gives the LM test hypothesis specifies H,. . U = 0, B is triangular. statistic when the null hypothesis specifies a block-recursive system. Smith (1983, 1985) obtains the LM test statistic in the limited-information framework, that is, when we have the structural form of one of the equations, say the first one, but only the reduced-form of the other current endogenous variables. We now wish to obtain the LM test statistic for the more general case. The test statistic is
D.A. Turkington, Classicul tests for uncorrelated disturhunces
309
where I”” refers to that part of the Cramer-Rao lower bound corresponding to the parameters V and 6 puts U equal to zero and evaluates all other parameters at the constrained MLE, i.e., at the LIML estimators. Under H,, Tl tends in distribution to a x2 random variable with degrees of freedom +G(G - l), so this distribution is used to find the appropriate upper-tail critical region. In forming Tl the first task is to obtain dl/dC and ZFa from al/au and I”” which we have in hand. We do this by means of an appropriate selection matrix S which has the property U = Sv. As v = L vet .I? and U = l vet 2, it must be true that SL = z. Clearly,
from eq. (6) and the property Dv( A) = vet Theorems 6 and 7 of appendix 1, SD’D = 2s - SLKL’= Moreover,
2s - zKL’=
if H, is true, Sv(2-‘) g
A for
symmetric
A. Using
2s.
= 0. Thus under
H,,
(7)
=Sv(A)=SLvec(A)=Lvec(A),
where A = ~-‘u’ZJ~P’ Moreover, 1”” =
SZ”“S’
=
jQ,y
@I,
(8)
where
H’(_r V(B)=2N(~~~)+4N(Z~~B~‘)R
plim i
xR’(Z@ Combining
B-“JY)N.
Finally,
using Theorem Tl = lvec n
-0) L’L vet A
9 of appendix
A’V(B)vecA
n
1 (9)
eqs. (7) (8) and (9) we get that -Tl = Lvec A’L’LV( n
-l
63 P)H
evaluated
1, we write
evaluated
at 6.
at 6.
310
D.A. Turkington,
Clussicul tests for uncorrehted
disturbances
To evaluate this test statistic further requires tedious algebra which is presented in appendix 3, where it is shown that
the outline
of
G-l
Tl = n c QjA,k,, !=I B, and A, are defined
and
Procedure
for forming
in the following
(2) Using
procedure.
the LM test statistic
(1) Apply LIML to each equation Form the LIML residual vectors ii,=y,--H,6”,
straightforward
for
i=l,...,
y, = H,8, + u, to get estimators
s’,, say.
G.
si and ii,, form
c,, =fi;ii,/n
for
i,j=l,...,
G,
and
j.
(3) Form
and
with P = X( X’X)-‘X’, matrix such that
YR,= r,. (4) Form
T, = n
c r=l
;;A$,.
P, = X,(X,‘X,)-‘X,‘,
and
xi
is the selection
matrix
D.A. Turkington,
Clussicul tests for uncorreluted
disrurhunce.s
311
Two points should be made about this test statistic. First, with no right-hand current endogenous variables our model becomes the SURE model. For this special case it is easily seen that
T, = ivecP[
N(l& 63 2’,)]vecA=
n
2 ‘c’r,:,
I=1 /=1
where r,;. = C,,/(~,,C?,~). This is the same LM test statistic obtained for this model by Breusch and Pagan (1980). Second, the 2SLS estimators, if,2” say, are asymptotically equivalent to the LIML estimators, 6,. Moreover, estimators of u,, based on the 2SLS residual vectors are asymptotically equivalent to the estimators of u,, based on the LIML residual vectors. Hence an asymptotically equivalent version of the LM test statistic may be obtained by using 2SLS in place of LIML in the above procedure.
6. The Wald test statistic Let s^, be the FIML estimator for S,, ii, =y, - q,t, be the MLE residual vector, and fi = (2,. . . iTic). Then the MLE of .Z is 2 = fi’fi/n, and we know that h(vece-vec2)
-f+N(O,V(B)),
where V(0) is as defined in eq. (9). The Wald Atest statistic for Ha: V = 0 is based on the asymptotic distribution of fi(U - U), where U contains the appropriate elements of vec(&. But as 0 = z vet(I), we have fi(i--iJ)=L&(vecZThe Wald
test statistic
vec_$):
N(O,LV(B)L').
is then
(11) where 4 signifies the MLE of 8. Under H,,, T2 has a limiting xz distribution with degrees of freedom :G(G - l), so this distribution is used to obtain the appropriate upper-tail critical region. A 3SLS version of the Wald test statistic is available. Let 6:, be the 3SLSE of 6;, b, = _y,- H,i, be the 3SLS rssidual vector, i? = (z?, . . . uc;), ’ and .Y$= b$/n. Then, as then 3SLS ,.estimators s^; and 2 are asymptotically equivalent to FIML estimators S and 2 [see Bowden and Turkington (1984, ch. 4)], the former can
312
D.A. Turkington, Classical tests for uncorrehted disturbunces
be used in place of the latter in forming a Wald-type test statistic. The statistic obtained in this way will be asymptotically equivalent to T2 of eq. (11). From a computational viewpoint this test statistic is more convenient than T,, though less convenient than LM test statistic T, or the 2SLS version of the LM statistic. 7. Conclusion In a very informative article Engle (1982) outlines a procedure for obtaining the LM test statistic (or a test statistic asymptotically equivalent to the LM test statistic) that avoids the necessity of first computing the information matrix Z(e) and the Cramer-Rao lower bound Z-‘(e). Moreover, he shows that this test statistic can be interpreted in terms of a R2 or a F test statistic of an underlying regression. Unfortunately his results require that the information matrix Z(0) be block-diagonal at least under the null hypothesis. Although this requirement is met in many econometric models and hypotheses, it does not hold for our case. In forming classical test statistics for the hypothesis in hand one is forced to first find Z(B) and Z-‘(e). The LM test statistic thus obtained does not appear to have a convenient intuitive interpretation in terms of R2 or F test statistics. However, the procedure outlined in section 5 is perfectly straightforward. We have also seen that a 3SLS version of the Wald test statistic is available. Although this eases the computational burden required in obtaining the Wald statistic, it is unlikely that this test statistic will be a serious contender for testing uncorrelated disturbances in the linear simultaneous equations model. Appendix 1:
Theorems on O-l matrices
The following used throughout Theorem A -*)NL’.
1.
results concerning the O-l matrices defined in section this paper and were obtained from Magnus (1988).
For
a nonsingular
matrix
A,
[D’(A @ A)D]-’
= LN(A-’
Theorem2.
FormatricesAandB,
Theorem 3.
DLN = N.
Theorem 4.
N=
Theorem 5.
For square G X G matrices A and B, K( A @ B) = (B 8 A)K.
Theorem (A@A)N.
6.
For
f(Z+
3 are
8
D’(A@B)D=D’(B@A)D.
K), N’= N.
square
G x G
matrix
A,
N( A @ A) = N( A @ A)N =
D.A. Turkington,
Theorem
7.
D’D = 2I-
Theorem
8.
EKL’ = 0.
Classical tests for uncorrelated disturbances
LKL’.
-where ith element is ai,. Then L’L vet(A)
Theorem 9. Let A be the matrix vet (A), where 0 azl
0 0
.f. ...
0 0
0 0
. . . . . . . . . . . . . . . . . . . , .
aGz
a,1
Appendix 2: Applying
“.
the formulae
0
aGG-1
The Cramer-Rao
=
1 .
lower bound Z -l(8)
for the inverse
H’(P I”’ =
313
of a partitioned
matrix,
we have
@I)H
plim
+ R’K(B-“8
n
BP’)R -1
-2R’(Z-‘@
B-‘)D{
D’(Z-‘8X’)D}
Applying Theorems 1 and 2 of appendix be written as, by Theorem 3, 2R’(/.Z-‘8 = R’(Z-‘8 Using
standard
H’(Y’
where
B-‘JW’)R
p
= R’(X’
8 B-“_-ZB-‘)R,
matrix
that plim H’(CI-‘69
= i
B-‘)R.
63 M)H
M is the projection
It follows
- R’K(B-“8
theory we note that
n
n
P)H
-’ i
8 B-‘)R
1, the third matrix in the bracket
B-“)N(I@EB-‘)R
asymptotic
plim
-lD’(B-’
.
I
can
.
314
D.A. Turkington,
By a similar
Ciussical
testsfor uncorreluted
disturbunces
analysis, (Z@~)+2(163~‘B-‘)R
plim i
1
xR’(I@B-“Z)
NL’
and I”” = - 2 plim
R’( I @ B-“2)
N.
G-l
Appendix 3.
T, = n c
i;Aihi
i=l
Using
Theorem
5 of appendix
1 and eqs. (9) and (10) of section
R’(I@B-“)
x R plim
x2[vecA’N(Z@32)]‘, where, it is understood, the parameters this expression under the assumption consider each component in turn: A.3.1.
2vecA’N(
Under
I @ Z) under Ho
H,,
Let e, be the jth
column
of IG. Then
5, we write
1 (A.11
are evaluated at 6 We need to evaluate that Ha: u,, = 0, i # j, is true. We
D.A. Turkington, Clussicui tests for uncorreluted disturbunces
[see Magnus
But N=
(1988)] and
f(I+K),
so
0
%l
I -.. (722
0
=-
315
1
0
DGG
1 2
0
011
1. I_ 022
0
0
.2u,,
(A.21
Now A = E-‘iJ’U,Y’,
so under
H,,
Pll
P12
. . .
Plc;
P21
P22
. . .
P2c
A=
p,,=-
with
4U, ~,,~,lJ
PGl
PC2
. . .
PGG 4
0
0
0
...
0
0
P21
0
0
...
0
0
P31
P32 . . .
0 . . .
. . . . . . . . .
0
0
Pa
PG2
PGG- 1
0
’
316
D.A. Turkington,
Classical
tests
for
uncorrelated
disturbances
and i
i.
Using
eqs. (A.2) and (A.3), it follows under 2vecA’N(I@Z)=
H,,
;(oou;u,... u;;u,)
~(0u~u,...u$4,) [
...
(A-3)
[email protected]~~%l) i)‘]. (A4
H’(F A.3.2.
(I@ B-‘)R
cd P)H
plim
n
-l
I
R’(I c3 B-l’) under H,
Using the formulae for a partitioned inverse and R, = ( Ri 0) we obtain, after a little work, that under H, this matrix is the block-diagonal matrix,
V(P-
P,)Y,
m1
R’B-1
plim
aI, Bm ‘R,
n
i
1
0
0
1
uGccB-‘R,
plim
i
A.3.3.
f(.Z @ 2-l)
Clearly
Yd(P- PG)Y, n
)
-Ix,
B-,,
G
under Ho
this matrix
is the block-diagonal
matrix
(A-6)
D.A. Turkington, Classical testsfor uncorrelated disturbances
317
Substituting (A.4), (A.5), and (A.6) in (A.l), ignoring the plims, and replacing unknown parameters by their constrained ML estimators, we obtain the expression G-1
Tl = n
c
k:A,i,,
r=l
as required. References Bowden. R.J. and D.A. Turkington, 1984. Instrumental variables, Econometric Society monograph in quantitative economics, No. 8 (Cambridge University Press, New York, NY). Breusch, T.S. and A.R. Pagan, 1980, The lagrange multiplier test and its applications to model specification in econometrics, Review of Economic Studies 47, 239-254. Engle, R.F., 1982, A general approach to Lagrange multiplier model diagnostics, Journal of Econometrics 20, 83-104. Godfrey. L.G., 1978a. Testing against general autoregressive and moving average error models when the regressors include lagged dependent variables, Econometrica 46, 1293-1302. Godfrey, L.G., 1978b, Testing for higher order serial correlation in regression equations when the regressors include lagged dependent variables, Econometrica 46, 1303-1310.* Godfrev. L.G. and T.D. Breusch, 1981, A review of recent work on testing for autocorrelation in dynamic economic models, in: D. Currie, R. Nobay, and D. Peel,-eds., Macro-economic analysis (Croom-Helm, Beckenham) 63-105. Hausman, J.A., 1979, Specification tests in econometrics, Econometrica 46, 1251-1272. Henderson, H.V. and S.R. Searle, 1979, Vet and vech with some uses in Jacobians and multivariate statistics, Canadian Journal of Statistics 7, 65-81. Henderson, H.V. and S.R. Searle, 1981, The vet-permutation matrix, the vet operator and Kronecker products: A review, Linear and Multilinear Algebra 9, 271-288. Hwang, H., 1985, The equivalence of Hausman and Lagrange multiplier tests of independence between disturbance and a subset of stochastic regressors, Economics Letters 17, 83-86. Magnus, J., 1988, Linear structures (Oxford University Press, New York, NY). Magnus, J.R. and H. Neudecker, 1979, The commutation matrix: Some properties and applications, Annals of Statistics 7, 381-394. Magnus. J.R. and H. Neudecker, 1980, The elimination matrix: Some lemmas and applications, SIAM Journal on Algebraic and Discrete Methods 1. 422-449. Magnus. J.R. and H. Neudecker, 1986, Symmetry, O-l matrices and Jacobians: A review, Econometric Theory 2, 157-190. Smith, R.J., 1983, On the classical nature of the Wu-Hausman statistics for the independence of stochastic regressors and disturbance, Economics Letters 11, 357-364. Smith, R.J., 1985, Wald tests for the independence of stochastic variables and disturbance of a single linear stochastic simultaneous equation, Economics Letters 17, 87-90. Tracy, D.S. and P.S. Dwyer, 1969, Multivariate maxima and minima with matrix derivatives, Journal of the American Statistical Association 64, 1576-1594. Wu, Demin, 1973, Alternative tests of independence between stochastic regressors and disturbances, Econometrica 41, 733-750.