Identification of rational expectations models

Identification of rational expectations models

Journal of Econometrics 16 (1981) 3755398. IDENTIFICATION North-Holland OF RATIONAL M.H. Company EXPECTATIONS MODELS PESARAN” Trinity Colleg...

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Journal

of Econometrics

16 (1981) 3755398.

IDENTIFICATION

North-Holland

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 ‘non-uniqueness’ 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 Aix-en-Provence. ‘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).

016557410/81/0000-0000/$02.75

Q 1981 North-Holland

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 non-linearities 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 ‘non-uniqueness’ 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 non-linearities, 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 non-RE 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+ l-ox, + 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 non-singular.

Ass~r~zptior~ 2.

The U, are non-autocorrelated.

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(B-t

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~,+{z-c(~+c)~’}~,,x,,=u,, and its reduced

(4)

form becomes

(5)

Yt=nox,,+n,x:,+n,x,,+v,,

where v, = B- ‘I+, and Z7,= -B-IT,,,,

(W

I7,=B-‘C(B+C)-‘r,,,

(6b)

ll,=

(6~)

-B-‘{I-C(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 non-trivial 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,,,)=2g-1 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 non-trivial 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 block-triangular 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 non-singular

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 {I-C(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-‘{z-C(B+C)-L)r02,

(17c)

17i+2= -B-‘{I--C(B+C)-‘i--C(B+C)-‘r,,~i},

(17d) i-l,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 non-policy 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 non-zero 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 non-zero 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)+I-i=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,,,)=2g-I, 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’iXr-i+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 non-uniqueness property of the solution of RE models with future expectations. But by

treating the free parameter as being non-stochastic, 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 non-stochastic difference equations. Shiller (1978. pp. 26~ 33) also provides a general solution of linear-RE 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 non-explosiae

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 non-singular, 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 non-singular, it follows that Rank (D) = h. If we now assume that all the h non-zero characteristic roots of D are distinct, fol some non-singular 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 non-zero characteristic root of D by Li and using the canonical form of D, (19a) becomes

Y,=A mt+lIQt)+% where jT,=p-lJ$

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

i-l,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 1-l

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 non-explosive. 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 non-singular.

We then prove:

Theorem 2. Under Assumptions one non-explosive 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 (B-C) to be non-singular. (iii) /PLil> 1. The relevant stationary solution for this case is the so-called ‘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 non-explosive solution for y, we need to assume B+ C and B-C are non-singular 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’ non-explosive 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 non-explosive 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 re-iterated by Revankar (1980, footnote 13), though sufficient to ensure a non-explosive 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. 29-33). Recalling that by Assumption 2, the U, are non-autocorrelated and noting that [see (19a)]

E(wt+jlQ2,)=-B-1

f

TiXt_i+Bm’U,

for

j=O,

r. txt+j-i*

for

jzl,

i=O

=-B-l

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 A-x , ,_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 non-linear 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 q-1 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’,

j-l,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 17ixt-i+vt,

where

lIi= Ai + f

D’d’,R,,,

i=O, 11 . , m,

W-9

i=m+l

(32b)

j=l

lITi= f

D’A,R,,,

,...,4-1,

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 second-order 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 non-linearities 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 Fisher--Rothenberg criteria for the local identification of non-linear 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. . . ..m-1.

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,

Dj-i-lB-lrj,

1,...,m-1.

j=i+l

If the above system of non-linear equations is to have a non-trivial solution, then clearly it would also be necessary for it to have a non-trivial 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 j2g-t 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 non-singular 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 (q-m - 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 non-RF. models. That is, if (q-ml)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 non-RE models and oic,e ve~su. This is unfortunate for macro-economic policy as RE models with Walrasian characteristics happen to imply strikingly different policy prescriptions as compared with non-RE macro-economic models. The latter. which hitherto have dominated post-war macro-economic 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 policy-makers 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 first-order

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

tj-1.

=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:

Xi-X,+,

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 = Y-Z,, 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 right-hand (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

t-1

y,=I_‘m,_,

+{w,-E(u’,p-l)}-

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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398

M.H. Pesuran,

ldentijiication

of

RE models

Pesando, J.E., 1975, A note on the rationality of the Livingston price expectations, Journal 01 Political Economy 83, 8499858. Poole, W., 1976, Rational expectations in the macro-model, Brookings Papers on Economic Activity, 463-505. Revankar, Nagesh S., 1980, Testing of the rational expectation hypothesis, Econometrica 48, 134771363. Rothenberg, T.J., 1971, Identification in parametric models, Econometrica 39, 577 591. Sargent, T.J., 1976, A classical macroeconomic model for the United States, Journal of Political Economy 84, 2077237. Shiller, R.J., 1978, Rational expectations and the dynamic structure of macro-economic models: A critical review, Journal of Monetary Economics 4. 1 ~44. Taylor. J., 1977, Conditions for unique solutions in stochastic macro-economic models with rational expectations, Econometrica 45. 1377 ~1386. Turnovsky, S.J., 1970, Empirical evidence on the formation of price expectations, Journal of the American Statistical Association 65, 1441 1454. Turnovsky, S.J. and M.L. Wachter, 1972, A test of the ‘expectations hypothesis’ using dtrectly observed wage and price expectations, Review of Economic and Statistics 54, 47 54. implication< of the ratronal expectations hypothw, Wallis, K.F., 1980. Econometric Econometrica 48. 49. 73.