Journal
of Econometrics
16 (1981) 3755398.
IDENTIFICATION
NorthHolland
OF RATIONAL M.H.
Company
EXPECTATIONS
MODELS
PESARAN”
Trinity College, Cambridge Received February
Publishing
CL33 9DD, UK
1980, final version received June 1981
This paper deals with the problem of the identification of simultaneous Rational Expectations (RE) models. In the case of RE models with current expectations of the endogenous variables, the necessary and sufficient conditions for the global identification are derived explicitly in terms of the structural parameters and the linear homogenous identifying restrictions. It is shown that in the absence of a priori restrictions on the processes generating the exogenous variables and the disturbances, RE models and general distributed lag models are ‘observationally equivalent’. In the case of RE models with future expectations of the endogenous variables, a general solution that highlights the ‘nonuniqueness’ problem and from which other solutions such as forward or backward solutions can be obtained. is derived. It is shown that untestable and often quite arbitrary restrictions are needed if RF models with future expectations are to be identifiable. Certain order conditions similar to those obtained for the identification of RE models with current expectations are also derived for this case.
1. Introduction The problem of expectation formation is at the heart of dynamic macroeconomic model building and policy formulation, and any hypotheses concerning the way expectations are formed merit critical appraisal. Faced with unobservable expectations, economists have resorted to two different, but closely related, general methods of modelling expectations.’ Firstly, the method where expectations are extrapolative/adaptive/autoregressive specified to be determined by a weighted average of past observations of the
*If it had not been for Professor Hahn, this paper would probably not have been written. I am grateful to him for encouraging me to write it. I would also like to acknowledge Mr. T.K. Carne’s help with respect to some of the mathematical problems in the appendix, and Ken Wallis’s very helpful comments on previous versions of the paper. I am particularly grateful to two anonymous referees of the Journal for their constructive comments and criticisms. I have also benefited from comments I received when previous versions of the paper were presented at a Churchill College Seminar at Cambridge, the Social Science Research Council Study Group on Model Selection at Southampton University and the Econometric Society World Conference at AixenProvence. ‘Direct measurement of price expectations compiled from survey materials are, however, attempted in a number of countries and are studied, for example, by Turnovsky (1970), Turnovsky and Wachter (1972), Knol (1974) Carlson and Parkin (1975), Pesando (1975), Carlson (1977), Mullineaux (1978, 1980) and Figlewski and Wachtel (1981).
016557410/81/00000000/$02.75
Q 1981 NorthHolland
316
M.H. Pewrun,
Identijicution
of RE m&Is
variable under consideration. Secondly, the rational method, where expectations are formed as mathematical expectations of variables of a given structural mode1 of the economy conditional on all the available information at the time. Although the Rational Expectation (RE) hypothesis goes back to Muth (1961), it is only recently that it has come to play a central role in theoretical and applied economics. Different aspects of the literature on RE models are surveyed critically by Poole (1976) and Shiller (1978), and the econometric implications of the RE hypothesis have been discussed by Nelson (1975) McCallum (1976a, b) and, more recently. by Wallis (1980), Revankar (1980) and Hoffman and Schmidt (1981). In spite of voluminous literature on RE models, there are still major ambiguities that surround their use in applied economics. The possible nonuniqueness of their solution, the almost intractable problems that genuine nonlinearities create for their study, and the difficulties connected with their parameter identification have caused most practising econometricians, with some justification, to shy away from RE models. Our main aim in this paper is to investigate the third problem within the framework of a general linear RE model both with current and future expectations of the endogenous variables. The problem of the identification of the structural parameters of RE models with current expectations of the endogenous variables, assuming that expectations of the exogenous variables are given, has been discussed by Wallis (1980, pp. 62.64). The conditions obtained by Wallis even for this case are, however, of limited use, both because they are given in terms of the rank of reduced form matrices and because they are only applicable when identification restrictions take the form of knowing specific structural parameters and can not be used to deal with genera1 parameter restrictions explicitly. In this paper we shall start our investigation by reconsidering the identification problem discussed by Wallis. We point out the importance for the identification of RE models of distinguishing the exogenous variables such as seasonal dummies and time trends that can be predicted exactly from those that can not be predicted exactly. We then derive necessary and sufficient conditions for the global identification of RE models with current expectations of the endogenous variables explicitly in terms of the structural parameters and the general homogeneous linear identifying restrictions. We extend our analysis to cover RE models with lagged exogenous variables and, at the same time, drop the unrealistic assumption of given expectations of the exogenous variables. We show that in the absence of LI priori restrictions on the processes generating the exogenous variables and the the RE and general distributed lag models will be disturbances, therefore the autoregressive and rational ‘observationally equivalent’; methods of expectation formation. in general, can not be distinguished from each other empirically.
M.H.
Pesuran,
Identification
of
RE models
377
In section 3 we consider the identification of RE models with future expectations of the endogenous variables and derive a ‘general solution’ that highlights the ‘nonuniqueness’ problem and from which all the other solutions given in the literature such as forward or backward solutions can be obtained. We then show that, in general, untestable and often quite arbitrary restrictions are needed if RE models with future expectations are to be identified. Assuming these latter restrictions are satisfied, it will be shown that order conditions similar to those obtained for the identification of RE models with current expectations will also be necessary for the identification of RE models with future expectations. Due to the presence of intractable nonlinearities, the derivation of usable necessary and sufficient conditions for the identification of future RE models does not seem to be possible; instead a necessary rank condition is obtained. Finally, we conclude that in the absence of a priori information concerning the order of lags in economic relations, RE and nonRE models can not be distinguished from each other on empirical grounds. This general underidentification of RE models clearly sheds serious doubts on the soundness of the recent attempts by Sargent (1976) Barro (1977, 1978) and others which favouring the proposition that purport to provide empirical evidence monetary and fiscal policies are incapable of influencing the path of real output and employment, even in the short run!
2. Rational expectation
models with current expectations
2.1. The model and its solution Consider
the following
linear multivariate
Bye+ lox, + CL’:’ =
UC,
RE model:
(1)
where y, and y: are g x 1 vectors of the obsermble actual and the unobservable expected values of the g endogenous variables at time t respectively; x, is a k x 1 vector of the current observable values of the exogenous variables, and u, is a g x 1 vector of disturbances. The structural parameters of interest are given by the g x g elements of B, the g x k elements of TO, and the g x g elements of C. In the event that expectations of one or more of the endogenous variables are absent from the model, the corresponding columns of C will contain only zeros.’
‘When discussing the identification problem both Wallis (1980) and Revankar (1980) prefer to employ a truncated form of matrix C with all its zero columns removed. We found formulation (1) analytically more convenient to work with.
37x
M.N. fesurun,
Identificution
of RE nwdrls
In this section it will be assumed that current expectations (whether of endogenous or of exogenous variables) are perceived by the public with the information available’ at time t  1. Formation of expectations according to the Muthian notion of ‘rationality’ requires that $=E(J’, / CT?_1) and x: = E(x, 1Q,_ 1), where Q, denotes the set of available observations on all the endogenous and exogenous variables up to and including time t. More specifically, Q,=(L.,,?;_ ,,...; x,,x,_~ ,... )={Qf;Of)., We now make the following assumptions: Assumption
1.
The matrices
B and B + C are both nonsingular.
Ass~r~zptior~ 2.
The U, are nonautocorrelated.
Assumption
The U, are distributed independently of the stochastic process generating X, and have zero mean and a finite variancecovariance matrix.
3.
Given Assumptions 1 and 2, taking conditional mathematical expectations of both sides of (1) with respect to the information set Q,_ 1 and solving for Y: yields
I::= (B+c)‘r,x:,
(2)
which if used in (1) gives us the following structural ut~ohsetwh/e expectations of the exogenous variables:
By,+I.,.u,C(Bt
model
in terms
of the
C)‘T,x~=u,.
Therefore, what ‘rational’ formation of expectations has achieved for us is to pass the problem of expectation formations of the endogenous variables on to the formation of expectations for the exogenous variables. We are still left with the task of modelling expectations of the exogenous variables, and. unless they are directly observable, any discussion of the identification of the structural parameters of (1) will inevitably be conditional upon the process generating {‘cl}. observable, the Even in the case that XT are given or are directly identification of (1) depends on whether at time t 1 some or all of the exogenous variables can be predicted precisely. In the extreme case that s, can be predicted without any error, the RE model will be ‘observationally equivalent’ to ByI+ I‘o_x,=u, and the hypothesis of the rational formation of expectations will not be testable. But since in general only some of the exogenous variables such as time trends and seasonal dummies can be
M.H. Pesaran,
Identification
of RE mode/s
319
predicted exactly, the identification of RE models for given {x:} can not be ruled out. Let x, be partitioned into xIr with k, x 1 elements that can not be anticipated precisely and a k, x 1 vector xzt (k, + k,= k), whose elements consist of time trends, dummies and etc. which are perfectly predictable (i.e., x$ = xZt). Partitioning TO = (ToI, r02) conformable to xi = (xi,, x;,), relation (3) can now be written as By,+~,,x,,C(B+C)‘~“,x~,+{zc(~+c)~’}~,,x,,=u,, and its reduced
(4)
form becomes
(5)
Yt=nox,,+n,x:,+n,x,,+v,,
where v, = B ‘I+, and Z7,= BIT,,,,
(W
I7,=B‘C(B+C)‘r,,,
(6b)
ll,=
(6~)
B‘{IC(B+C)‘)I,,.
2.2, Case of known XT, Here we treat assumption: Assumption
4.
xTt as given or observable
and make the following
further
The vectors x, and xTr are such that it will not be possible to express any one of their components as a nontrivial linear function of the remaining components, for all t.
This assumption in effect rules out the possibility of exact multicollinearities among the explanatory variables of (5) and, together with Assumptions 2 and 3, enables us to estimate the reduced form parameters l7,, 17, and n, consistently. The identification of the structural parameters will then depend on whether it is possible to get estimates of (B, To, C) from those of the reduced form parameters. Without any loss of generality we concentrate on the identification of the parameters of the first equation in (1). No restrictions on the variancecovariance matrix of the disturbance vector, u,, will be considered. Let a’= (h’, y;, y;, c’) be the first row of matrix A = (B,r,,. Toz, C) and consider the following r homogeneous linear a priori restrictions on the elements of LL: a’@ = 0,
(7)
where @ is a (2g+ k) x r matrix Theorem
1.
structural identified
ppramrters up to u scalar
Given Assumptions
of
of known
constants.
Then:
1 to 4 und the identijying
the ,first equation constant ij; und only. it .
qf
the
RE
restrictions
(7), the
model
will
he
to
the
(1)
Rank(
[email protected])+Rank(7,,,)=2g1 Proof:
Partition @‘=(@b. @I,,, @I,2,@:.), so that partitioning of a and write (7) as
Noting
it is conformable
that
write (6) as
=o,
(6a’)
B77,+C(n,+n,)=o,
(6b’)
Bn,+f‘“,
Then the equations relevant for the estimation the first equation in (I ) become
of the structural
parameters
of
h’rz, + ;‘; = 0,
Pa)
h’n,+i’;+“‘I7,=0.
(94
Relations (8) and (9) together form a linear homogeneous system of equations in vector a, which admits a nontrivial unique solution up to a scalar constant, if and only if the rank of Q is 2g+ k  1, where
(10)
381
M.H. Pesaran, Identijication of RE models
and Ikl and Ik, are unit matrices multiplying Q by the matrix
of order
k, and k, respectively.
and using results given in (6’) it is then easily established
r
[email protected]
0
0
Now pre
that
O1
@7,
TQ=
QjY2 (B+W’, which is a blocktriangular matrix whose ranks of its diagonal matrices. That is
rank
is given
by the sum of the
Rank(TQ)=Rank(
[email protected])+(k,+k,)+Rank(r,,). Now under Assumption may also be written as
1, T is nonsingular
and the above
rank
condition
Rank(Q)=Rank(
[email protected])+Rank(T,,)+k,
and the identification
condition
Rank (Q) = 2g + k  I implies
Rank (
[email protected]) + Rank (r,,)
= 2g  1.
Q.E.D.
(11)
Theorem 1 is a generalization of the familiar rank condition for the identification of the simultaneous systems originally derived by Koopmans et al. (1950). The following order condition for the identification of (1) can also be proved: Proposition 1. When x: are known the necessury (though not sufficient) condition for the identification of a given equation in RE model (1) is that the total number of exogenous variables plus the number of exogenous variables that can not be exactly predicted should be greater than or equal to the total number of variables included in the equation minus one.
M.H. Pesaran,
382
Ident$cution
of RE models
Proof: Since matrix Q is (2g + k) x (1.+ k + k, ), it is necessary that r+ k, 22g1.” Denoting exogenous (whether perfectly predictable or not) variables that are included in the ith equation then r = (g  S) + (k  F) + (g  I?) and the order becomes4
k+k,z(g+h+k)1.
2.3. Case qf’unknown
Q.E.D.
x;* with lagged
We now drop the unrealistic time generalize (1) to include manner: By,+
for it to have rank 2g + k  1 the number of endogenous, and expectional endogenous by g, k and h respectively, condition for identification
e.xogenous
(12)
vuriablrs
assumption that XT are given and at the same lagged exogenous variables in the following
5 rix,_i+cy,*=u,.
(13)
i=O
As before, solving we get
for $
and writing
(13) in terms of the exogenous
Again partitioning x, into those elements that can not be predicted (i.e., .xlr) and those that are perfectly predictable (i.e., xZr), the above may be written as
+ f {IC(B+C)‘:riXt
variables
exactly relation
(14)
iEUtr
i=l
where To = (I’, , , Toz) is partitioned (x ;,,x;~). We also have, xTr=E(xIIIQ,
so _ r).
that
it
is conformable
to
“This method of proof was suggested to the author by an anonymous referee. The same can also be obtained by noting that the ranks of the matrices
[email protected] and rlr, in (11) will at be r and k, respectively. ‘Note that Wallis’s order condition ia only valid when the model under consideration not contain constant terms, or other exogenous variables such as seasonal dummies and trends which can be predicted exactly.
x;=
result most does time
M.H.
Pesaran,
Identifiicution
383
of RE models
Faced with the problem of generating expectations of xIr relative to the information set 52,_r, the exogeneity of xIt is often taken to mean that yt ‘does not cause’ xIr in the Granger sense. Adding to this, the assumption that x1, is generated by means of a linear process, the optimal forecast of xIt conditional upon L?_ 1 is then obtained.5 Accordingly, ~T1=E(xltl52,,)=E(x,,152:~,;n:,)=E(x,,ln~~,), and by the linearity
xTt= f
assumption
we have
Rix,_i,
(15)
i=l
q is the order of x,process and Ri are parameter matrices of order Note that when none of the exogenous variables are exactly to assuming that x, follows a predictable, then x It=~t and (15) is equivalent multivariate qth order autoregressive (AR) process. By letting q tend to infinity and placing suitable restrictions on Ri, more general stationary and linear specifications of the x, process such as autoregressive moving average (ARMA) schemes can also be considered1 But for the sake of brevity we shall not deal explicitly with ARMA specifications of x, and confine our attention to (15). Replacing Ri in (15) by their consistent estimates say, ii obtainable from a regression of xIt on {x,_ r, x*_~, . . . , x,_,} enables us to obtain a consistent estimate of xT1 independently of the knowledge of the structural parameters of (13). The use of this estimate for the current expectation of xIt in (14) gives the following ‘empirically observable reduced form’ of (13), where
k, x k each.
(16) where Ho= BP’T,,,
(17a)
I7r =B‘C(B+C)lTor,
(17b)
17,= B‘{zC(B+C)L)r02,
(17c)
17i+2= B‘{IC(B+C)‘iC(B+C)‘r,,~i},
(17d) il,2
z,=~=$+,
I&i
and
,...’
m,
u,=B‘u,.
5Clearly, there are other equally, if not more, plausible notions of exogeneity that could bL adopted. It may be more appropriate to consider different stochastic specifications for the formation of expectations of policy and nonpolicy exogenous variables. A detailed discussion of these issues is beyond the scope of the present paper. However, see the remarks made at the end of this section.
From these results it is firstly clear that unless Z, #O, the RE model given by (13) and the simultaneous distributed lag model specified by By, +x:7’=. J.ix, i =u, are ‘observationally equivalent’ and the structural parameters of the RE model can not be identified. For zt to be nonzero it is necessary that ~1>fn. Therefore, we have :
Although it is possible to consider values of ~1such that q>m, this will not enable us to obtain consistent estimates of Z7, in (10) if the true order of the {u,) process is equal to or less than m. Moreover. there is no reason why 111 should be kept fixed when y is allowed to rise. In the absence of a priori information concerning the true values of m and rf it would be impossible to distinguish between the RE and the infinite distributed lag models. Clearly, q> m (or more strictly a nonzero z,) is a necessary but not a sufficient condition for the identification of (13), and other restrictions will be needed. Derivation of other necessary and sufficient conditions can be achieved by adopting exactly the same method used to prove Theorem 1. We first rewrite (17) as Brz”fP,,
=o,
(17a’)
Bn, +c(r7[I, fII,,)=O,
(17b’)
(B+C)17,+I‘,,=O,
(17c’)
CBfC)Cni+2 n,di)+Ii=O,
i=1,2
. . . . . m.
(17d’)
Concentrating on the identification of the first equation of (13) we again consider homogeneous linear restrictions on the elements of the first row of matrix A, which we now define as A = (B, P,,, I,,, I‘,. , T,,,. C). Then given q> 171,the necessary and sufficient condition for identification of the first equation of (13) is easily seen to be Rank(
[email protected])+Rank(r,,,)=2gI, where @ is the {2g + (m+ 1)k) x r matrix of the r homogeneous restrictions on the parameters of the first equation of (13).
linear
M.H. Pesuran, Identi&ution
The order condition
of RE models
for this case may also be written
385
as
of where k,, g and t? are defined as before, K stands for the number predetermined variables (current as well as lagged values of the exogenous variables) that are included in the equation under consideration, and K= (m+ 1)k. The addition of lagged values of the exogenous variables to (1) can help but also hinder identification. An increase in m can help fulfill the order condition (m+ 1)k + k, 2 (g+ 6+K)  1, but at the same time it may violate the condition ~7> m. Finally, since lagged values of the endogenous variables are known at time t, their inclusion either in (1) or (13) does not alter any of our conclusions. At time t, {yrmi; ill) are known exactly and can therefore be included as elements of xZt. This also means that formulation (15) is more general than it may at first look. The inclusion of the lagged values of y, as general triangular systems with elements of x2, enables us to consider feedbacks where xlf is determined not only by its own history, but also by the past values of the endogenous variables. Thus, general linear policy rules with feedbacks may be considered within the present framework without any need for further elaboration.
3. Rational expectation
models with future expectations
In this section we extend our investigation of the identification of RE models to cover the case when future expectations of the endogenous variables are specified to influence the current vector of endogenous variables. We consider general linear models of the form
By,+
2 I’iXri+C
i=O
E(Y,+,IQ,)=~,.
(18)
In this formulation we have assumed that future expectations of the endogenous variables are formed rationally using the current information set Q. Some investigators, however, have suggested using !2_ 1 when forming future expectations. We shall nevertheless adhere to (18), but in due course comment on the effect that replacing 52, by Q,_, may have for the identification of (18). The solution of models such as (18) has been discussed, among others, by Taylor (1977) and Blanchard (1978) using Muth’s method of undetermined coefficients. These authors derive an array of solutions for the scalar form of (18) in terms of a free parameter and therefore establish the nonuniqueness property of the solution of RE models with future expectations. But by
treating the free parameter as being nonstochastic, they overlook a class of stochastic solutions of (18) which differ from each other by a martingale process. These solutions are sometimes called ‘speculative bubbles’ in the literature. A more universal approach to the solution of future RE models is proposed by Gourieroux et al. (1979); it is similar to that used for the solution of linear nonstochastic difference equations. Shiller (1978. pp. 26~ 33) also provides a general solution of linearRE models when disturbances are serially independent and all the values of the exogenous variables (past, current and future) are assumed to be known at time t. Wallis (1980) and using the forward recursive Revankar (1980) obtain their solutions substitution method and implicitly concentrate only on OJF of the possible stationary solutions of (18).
3.1. Multiplicity Here, used to Since may be
vf nonexplosiae
solutiom
we propose another method for the solution of (18) which can be generate all the different solutions given in the literature. by Assumption 1 matrix B is nonsingular, the reduced form of (18) written as yr=D
E(y,+,lQO+w,.
where
D=B‘C
and
w,=K’
(1%)
In practice, C will rarely have a full rank. Denoting the rank of C by /I (2 g) and noting that B is nonsingular, it follows that Rank (D) = h. If we now assume that all the h nonzero characteristic roots of D are distinct, fol some nonsingular matrix P, the canonical form of D may be written as D =PAP‘, where n is a diagonal matrix with characteristic roots of D as its diagonal elements. Denoting the ith nonzero characteristic root of D by Li and using the canonical form of D, (19a) becomes
Y,=A mt+lIQt)+% where jT,=plJ$
and
W,=P‘w,.
387
M.H. Pesaran, Identification of RE models
The ith row of (20) may now be written yti=&
E(y,+,,ilSZ,)+w,i,
as
il,2
Yti = wti3
,...,
h,
(20a)
i=h+l,...,g.
Gob)
Thus, the solution of the multivariate RE model (18) entails solving the h univariate RE models given by (20a). As is proved in the appendix, a general solution for the ith equation in (20a) may be written as 1l
jti=li’m,i
C &4G_j,i,
(21)
j=l
where m,, represents the martingale process specific to the ith equation in (20a). That is, E(m,+,,~lQ,)=m,,. Due to the presence of m,, in (21), there will clearly be an infinite number of solutions to choose from, unless some u priori restrictions are placed on the y,process. In an attempt to avoid multiple solutions it may be plausible jti, to be nonexplosive. Although such a to require yt, or equivalently requirement considerably reduces the domain of possible solutions, as will be shown below, it nevertheless fails to provide us with a unique stable solution. We first introduce
the following
Assumption
5.
The processes
Assumption
6.
The matrix
further
generating
assumptions: x, and u, are stationary.
B  C is nonsingular.
We then prove:
Theorem 2. Under Assumptions one nonexplosive solution.
1, 5 and 6 RE model (18) will have at least
Proof. By Assumption 5 the process generating w, or W, will be stationary and whether (21) is stable or not depends on the value of lRil and the nature of the martingale processes m,,. Consider the following three possible cases: (i) 1Ai/ < 1. In this case, expectation conditional on Q, exists and we can define
of
the
infinite
sum
x:1
AjWji
(22)
It is now easily verified that ,hi is a martingale may also be written as’
jti=;l;If;i+
f
and the solution
1; E(q+j,i/a,),
given by (21)
(23)
j=O
which is often referred to as the ‘forward solution’. Now as t+co, i.i’fji will not explode, if and only if fti+O faster than &’ tends to infinity. But since jii is a martingale this will not be possible unless jji=O. Therefore the unique stationary solution for this case may be written as
ni=j~~~~a(~,+j,iln,),
i=1,2 )...) h.
(24)
(ii) (i.il = 1. U n d er t h’IS case it is evident from (21) that the variance of Jli will certainly explode and a stationary solution for y, does not exist. But since ii is a root of the determinantal equation IC+M =O, the solution l/1,/ = 1 is ruled out by Assumptions 1 and 6 that require the matrices (B+ C) and (BC) to be nonsingular. (iii) /PLil> 1. The relevant stationary solution for this case is the socalled ‘backward solution’ and can be obtained from (21) noting that when I&/ > 1 the infinite sum c,F= 0 j.imiG j, i converges. Define
then hfi is also a martingale.
Using this result in (21) yields (25)
Now as t+rj, I.;‘+0 and for any convergent choice of bri, the mean and variance of A;‘bti tends to zero as t +x. [see Doob (1953, p. 3 19)]. Therefore,
‘Note
that using (22)
ind smce nzti is a martingale
then
M.H. Pesaran,
when 1Ai1> 1, a stationary
solution
Yti=  f ri;jW,_j,i. j=l
IdentiJication
of RE models
389
for y, exists and is given by Q.E.D.
(26)
Thus to obtain a nonexplosive solution for y, we need to assume B+ C and BC are nonsingular and no other restrictions on the characteristic roots of D = B ‘C will be needed. But, to write down the correct nonexplosive solution of (18), a priori knowledge concerning those characteristic roots of the matrix D that lie within and those that lie outside the unit circle will be required. This information will not, normally, become available, unless structural parameters of (18). particularly the expression for B ‘C, can be identified and estimated. On the other hand, in the absence of direct observations of future expectations of the endogenous variables, identification and estimation of (18) will not be possible unless the ‘correct’ nonexplosive solution of yt can be chosen. We are bound up in a vicious circle. Therefore, we conclude :
Proposition 3. RE models with future expectations of the endogenous variables are unidentifiable even if the nonexplosive condition is imposed on their solution. The requirement that Ai should all lie within a unit circle as suggested, for example, by Wallis (1980, p. 59) and reiterated by Revankar (1980, footnote 13), though sufficient to ensure a nonexplosive solution for yI, is by no means necessary. It is however, possible to impose either { /Ji/ < 1; but as the i=1,2 ,..., h} or {lAi]>l; i=1,2 ,..., 11) not as ‘stability’ conditions necessary identifying restrictions. The choice between them is quite arbitrary and neither set, of course, is testable. We impose the former set of restrictions as the necessary condition for the identification of (18) and proceed to determine what other necessary conditions are needed if (18) is to be identified.
under 1Ai I < 1
3.2. Identification
Under {lAil
jt=
f j=O
h}, the relevant stationary solutions are given these relations into a g x 1 vector, it is easily seen
AjE(G,+jl,Rt)=P‘y,=
f j=O
Aj P’
E(w~+~IQJ,
390
M.H. Pesaran,
and since Dj=
Identification
PAjP',it immediately
of RE models
follows that
This is the familiar stationary ‘forward solution’ and is also obtainable by the forward recursive substitution method used, among others, by Shiller (1978. pp. 2933). Recalling that by Assumption 2, the U, are nonautocorrelated and noting that [see (19a)]
E(wt+jlQ2,)=B1
f
TiXt_i+Bm’U,
for
j=O,
r. txt+ji*
for
jzl,
i=O
=Bl
f i=O
the chosen
stable solution
of y, becomes
y,=  f
D’ t
j=O
i=O
B'Tix,*,j_i+v,,
(27)
where v, is defined as before, but unlike in the previous section, x,*, j_i now stands for the mathematical expectations of xt+jPi relative to the information set Q2, and not Q,_ i. Furthermore, since x,*~+~=x~~+~ for i =O, 1,2,. ,m and for all js i, the above reduced form relation, after some algebraic manipulation, may be written as
y, = f Ax , ,_j+ j=O
f
DjAox,*,j+v,,
w3)
j=l
where
dj=  5 D'jB'T,, ,j=Q,1,2,...,
m.
(29 I
izj
Important features of this result are the highly nonlinear way that the structural parameters enter the reduced form and the dependence of yr upon all the future expectations of the exogenous variables. Clearly if dummy variables and time trends are included among the exogenous variables, a partitioning of x, and x:+~ similar to that considered in the case of current RE models, should be made before identification of (18) by means of (28) can
M.H. Pesaran, Identqication of RE models
391
commence. But, in order to make the analysis manageable, we ignore this complication and proceed with the reduced form as it stands. Assuming all future expectations of the exogenous variables are known, using (28) it can be shown that RE models with future expectations are identified, if and only if for given future expectations of the endogenous variables [i.e., E(y,+ 1 1Qc)] t h en underlying structures are identified.’ We do not propose to go into the details of this extremely unlikely case. The situation is different when x:+~ are not known. Then, as to be expected, the identification will critically depend upon the nature of the x, process. Assuming an AR(q) vector representation of {x,}, the best linear predictor set Q2,may be written as Of xt+j relative to the information q1 xF+j'igo
j=l,L...,
Rijx*i,
(30)
where {R,r; i=O, 1,2, . . ..q 1) denote the k x k parameter matrices of the AR(q) process assumed for {x,}, in terms of which all the other matrices {Rij, j> l} can be obtained and estimated consistently in a recursive manner. In the simplest case when {x,} follows an AR (1) scheme with parameter matrix R (i.e., R,, =R), we have Roj = R’,
jl,2
Rij = 0,
Vi,jll.
,“.>
In the general case, replacing x*t+j given observable reduced form’ of (18) becomes 4
Yt=
1 i=O
by (30) in (28), the ‘empirically
1 17ixti+vt,
where
lIi= Ai + f
D’d’,R,,,
i=O, 11 . , m,
W9
i=m+l
(32b)
j=l
lITi= f
D’A,R,,,
,...,41,
j=l
and Ai are given by (29) with D =  B‘C. ‘This result does not, however, hold in the case of current condition for identification is given by (11).
RE models where the relevant
rank
M.H. Pesurun,
392
Ident(fkwtion
of RE models
The first important conclusion that emerges from the above result is that unless q>m+ 1 the structural parameters can not be identified. Therefore, even if the rational expectation model specified by (18) does not include any lagged exogenous variables, its identification requires the specification of at least a secondorder autoregressive process for (x,). It is interesting to recall that in the case of RE models with current expectations of the endogenous variables. the necessary condition for the identification of the structural parameters was less restrictive (i.e., q> vn). This, however, stems solely from the different amount of information that is assumed to be available when expectations of current and future endogenous variables are formed. If, instead of Q,, expectations of ytL 1 were made conditional upon Q,_ ,, then a necessary condition for the identification of both models would have been q>m. From this it also follows that when the same information set is used to form the current and the future expectations of the endogenous variables and all the eigenvalues of K’C are assumed to fall within the unit circle. then models (13) and (18) are ‘observationally equivalent’. RE models with future expectations can not be distinguished from RE models with current expectations. This result is true even if q >m. When qznrn the observational equivalence of (13) and (18) trivially follows from the fact that, when Q, in (18) is replaced by R,_ 1, both models arc observationally equivalent to R!,,+C~=“=,I.i.~I~i=II,. Suppose now the model under consideration is given by (18) and the two necessary conditions discussed above, namely i /ki / < I, i = 1,2, , h j and q > nz + 1. are a priori fulfilled. What other conditions are necessary if (I 8) is to be identified? Due to the irremovable nonlinearities that are present in relations (32), the derivation of rank conditions for the global identification of the structural parameters is not possible. Furthermore, the use of the FisherRothenberg criteria for the local identification of nonlinear systems in search of further does not seem to be very fruitful here. * Nevertheless, necessary criteria for identification. we first rewrite the relations in (32) as i=o. 817, + fTi +
CH, = 0,
1. . . ..m1.
i=m,
(33b)
BHi + CH, = 0,
i=m+
Hi=
i=O, 1,2,
1,
where f
Dj‘AoR,,,
j=l
“See Fisher
(1961)
and
Rothenberg
(1971).
(33a)
(33c)
M.H.
Pesaran,
Ident$cation
of RE models
393
and
5
Gi=~,
i=o,
DjilBlrj,
1,...,m1.
j=i+l
If the above system of nonlinear equations is to have a nontrivial solution, then clearly it would also be necessary for it to have a nontrivial solution even if Hi and Gi were replaced by their corresponding values evaluated in the neighbourhood of the identified structure. Although this approach does not give the sufficient conditions for local identification, it enables us to write down some useful order conditions. Introducing u’@=O as the relevant linear homogeneous restrictions on the parameters of the first equation of (18) where, as before, u’ stands for the first row of the g x {2g+ k(m+ 1)} matrix, A = (B, P,,r,, . ., Tmr C) and using (33), local identification of the j2gt k(m+ 1)) x 1 vector u requires that Rank (Q)=Zg+ k(nz+ l) 1, where
q no n,
Cp,
. n,,
n,,,
. . . n,_,
0
. .
0
0
0
...
0
. .
0
I,
0
. ..
G;,_,
Hz
H;+1
..
G': .
G;
n,
Hi_,
0’ = (@A,@I,, . . , @I,, ‘P:), and Go and HP denote the values of Gi and Hi evaluated in the neighbourhood of the identified structure respectively. Premultiplying the matrix Q by the nonsingular matrix 7; B
To
o
I,
0
I‘,
C
...
0
0
0
I,
0
rl
0 .
we obtain a matrix composed immediately follows that
of two lower triangular
matrices
from which it
Rank(TQ)=Rank([email protected])+(m+l)k + Rank(Hi+,, Hence the necessary of (18) will be
condition
Hz,,
,...,
H,O_,).
for the local identification
of the first equation
A number of important order conditions may now be inferred from the above result. Firstly, if q
where all the symbols are defined as before. If (qm  I)k happens to be greater than or equal to the number of endogenous variables, the rank of (HZ + , , M: + 2, . , H:‘ 1) at most will be g and (34) reduces to Rank (,[email protected])= g 1. This is the familiar rank condition for the identification of nonRF. models. That is, if (qml)kzg, the necessary condition for the identification of (18) could be investigated simply by treating all the expectational variables of (18) as if they were observable and cxogenouslq given.
4. Concluding remarks Although in principle it is possible to identify RE models. it is very doubtful that in practice economists could ever be in possession of the type of LEpriori information that is required for the identification of these models. All the order conditions given in section 3 for the identification of RE models with future expectations of the endogenous variables are valid only il it is known that the eigenvalues of B‘C fall within the unit circle. R priori knowledge of true order of lags in economic relations can rarely be inferred from theoretical considerations. As a result I, personally, do not believe that RE models can be empirically distinguished from nonRE models and oic,e ve~su. This is unfortunate for macroeconomic policy as RE models with Walrasian characteristics happen to imply strikingly different policy prescriptions as compared with nonRE macroeconomic models. The latter. which hitherto have dominated postwar macroeconomic modellinp, are employed to justify active fiscal and monetary policies for the achievement of output and employment targets, while the former are used to deny the policymakers of any intended and systematic impact that they may wish to have upon output and employment. Yet, it is perhaps ironical that as a result
395
M.H. Pesaran, identification of RE models
of inadequate a priori information, empirically indistinguishable.
in
general
these
two
models
are
Appendix In this appendix univariate firstorder
derive a general equation :
we shall difference
solution
Yt=A ~(Y,.lp,)+%
for the
following
(A.1)
. .}, and 2 is a scalar parameter. where fi2,=(y,,y,_r ,... ;w,, w,_ The proposed method of solution involves transforming (A.l) into a martingale process by means of the auxiliary variables K, X, and Z,, defined as r,
.
X=j.‘b%I,
(A.21
x, = /VW,,
(A.3)
Z,=E
for
tzl,
=o
for
t=O,
for
tj1.
=E
( 
i
X,/L+
Using (A.2) and (A.3) in (A.l), it is easily established
X=E(I:+,X,+&W Turning
to Z,, for t 2 1, we have
=E 01
(A.4)
j=r+l
that
(A.3
396
M.H.
Pesarun,
Identification
of KE models
Also for t = 0,
and similarly,
for tz  1, we have
Z,=E
!
=E(Z,+,

i:
XiX,+,
IsZ,
i=1+2
X,+,
la,).
Thus. for all r. we can write”
Z,=W,+,X,+$4).
(A.61
If we now subtract (A.6) from (A.5) and set q = YZ,, we get E(m,, , /Q,) =m,, which means that m, is a martingale,‘” and using (A.2) the general solution of (A.l) can be obtained as
Now for tz I, using the definition (A.l) becomes
of Z, given above, the general
and since for is t, E(w,l.Q,) = \v~, the above solution
simplifies
solution
of
to
“Note that since only finite sums are involved, the expression given in the righthand (A.6) exists if expectations E(X,IO,) exist. “For an excellent early treatment of martingales, see Doob (1953, ch. VII).
side of
M.H. Pesaran,
Identification
ofRE
If the information set !2, in (A.l) is replaced for y, can be shown to be
397
models
by Q,_i,
the relevant
solution
t1
y,=I_‘m,_,
+{w,E(u’,pl)}
c rjwt_j.
j=l
In the first instance, these solutions appear to be very different from the other solutions obtained in the literature. This is due solely to the martingale m, that has entered the solution. As there are an infinite number of stochastic processes that have a martingale representation, the solution of (A.l) is not unique and, as a result, a large number of apparently different but basically equivalent solutions of y, can be encountered.
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ldentijiication
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RE models
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