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of Econometrics

21 (1983) 2455254, North-Holland

IDENTIFIABILITY CRITERIA EXPECTATIONS Leon L. WEGGE

Received

February

Company

FOR MUTH-RATIONAL MODELS

and Mark

University of California,

Publishing

Davis/Santa

1981, final version

FELDMAN* Barbara,

USA

received July 1982

The authors deal with complete static linear models that contain current Muth-rational expectations. Rank, order and variety conditions for the identifiability of the structural parameter are derived under general restrictions. We also correct statements that appeared in the literature. Our main finding is that, in general, the standard rank and order conditions are sufficient also for the identifiability of the Muth-rational expectations model parameter, whenever there are enough not fully anticipated exogenous variables. If the number of imperfectly forecasted exogenous variables falls short of the number of endogenous variables by g, then g extra restrictions are needed on every equation and the restrictions must meet easily verifiable variety conditions as well as an augmented rank criterion.

1. Introduction Consider the G equation model BY’ + B* Y*’ + TX’ = U’, where Y*’ are Muth-rational expectations of the endogenous variables Y’. Wallis (1980) derives necessary and sufficient rank conditions in terms of the reduced form parameters when the observable unrestricted reduced form is identified, i.e., when none of the exogenous variables are fully known at time t. In this paper we extend this analysis and obtain necessary and suflicient rank conditions on the structural parameters and its restrictions when all or some of the exogenous variables are fully anticipated at time t, such as lagged variables or fixed constants. This extension is needed in order to make the conditions of identifiability useful for applied work. Not surprisingly in light of Wallis’ order conditions, the order conditions obtained here put the number of required restrictions per equation equal to at least G+g where g is the positive difference, if any, between G and the number of exogenous *L.L.W. is at the University of California, Davis, while M.F. is at the University of California, Santa Barbara. We acknowledge our debt to Steven Sheffrin for valuable conversations, and particularly for inspiring our interest in this topic. We thank F.S. Mishkin for showing us how covariance restrictions help in his own related work. We also thank K. Wallis for his comments and the observation that our results imply the positive conclusion that with a sufftciently large number of not perfectly anticipated exogenous variables, i.e., with g=O, the static Muth-rational expectations model with current expectations does not need any extra restrictions for its identifiability. M.H. Pesaran made two useful suggestions and our final acknowledgement is to the referee for providing us with a much more concise proof and outline of our paper.

03044076/83/000&0000/$03.00

0

1983 North-Holland

246

L.L. Wegge and M. Feldman, Muth-rational expectations models

variables that are not fully anticipated. Totally new however is our statement of the rank condition in the structural parameter space and the finding that the restrictions meet certain qualifications not encountered in the standard econometric models. In particular, at least g must be on each row of (B, B*, C), not counting restrictions on B+ B* and counting two restrictions, one on bij and another on b$, as only one restriction. Furthermore at least g restrictions must be on each row of (B*, T). We explain these qualifications in terms of the transformations that leave invariant the mean and covariance of the reduced form. An example is used to illustrate the applied scope of our analysis and to show how the rank condition fails even when the order condition is satisfied. Since the rank can fail because of a failure to meet easily verifiable requirements, our results are important for the practicing econometrician. In the meantime and independently of our work, Pesaran (1981) in this journal has obtained single equation results for the static model with current expectations considered here. His paper is interesting for posing the problem of identification under a system of homogeneous linear restrictions on a single equation. Theorem 1 of Pesaran’s paper however is in error.’ When properly restated, our results imply his single equation results. The usefulness of expressing identification conditions for simultaneous equation systems in terms of the structural parameters has long been recognized. While rational expectations has received much attention recently in the econometric literature and in particular Wallis (1980) derives conditions for identification, here we formulate the identification requirements in terms of structural parameters. Our treatment includes nonlinear cross-equation restrictions, which is especially relevant in the context of rational expectations. Additionally, as in Pesaran (1981), we analyze the empirically important case where the vector of predetermined variables includes lagged components. In applied work this will be the usual situation, but it is treated awkwardly in the existing literature. Consider the G simultaneous equations holding over n periods, BY’ + B* Y*’ + TX’ = U’,

U’ = (u,),

t=l,...,n,

(1)

where B is G x G and non-singular, B* is G x G and B+ B* is non-singular, r is G x K, Y’, Y*‘, X’ and U’ have n columns with Y’=(yJ the matrix of endogenous variables, Y*’ = (E(y, 1sZ,_1)) is the matrix of expectations conditional on the information Q2,_1 at time t - 1, X’=(xJ is the K x n matrix ‘Pesaran (1981, p. 381) is incorrect in claiming that the rank of his block-triangular matrix TQ is given by the sum of the ranks of its diagonal matrices. Because of this error his paper misses the point that extra restrictions always can make up deticiences in rank.

L.L. Wegge and M. Feldman, Muth-rational expectations models

247

of exogenous variables, and U’ =(u,) is the matrix of structural residuals satisfying E(u,) = 0, E(u,, u;,) = 6,,,C, t, t’ = 1,. . ., n, with C positive definite and 6,,. the Kronecker delta. The expected values are Muth-rational [Muth (1961)], i.e., given the K x n matrix of expected values X*’ = (E(x, 1Q,_ 1)) for the exogenous variables X’, the matrix of expected endogenous variables is Y*‘= -(B+B*))‘TX*‘, which after substitution

(2)

into (1) implies

the observable

structural

BY’+T(X-x*)‘+B(B+B*)-‘Tx*‘=U’.

system (3)

The true structural parameter is the matrix (B,B*, T,C) and satisfies k restrictions $J(E)= (4i(c()) = 0, i = 1,. . ., k, where c(=(/I’, jJ*‘, y’, a’)’ with p, p*, y, CJ vectorizations of B, B*, r, C, respectively. We allow that some components of the exogenous variables x, are completely known at time t- 1. If (X-X*); are the K, rows of (X-X*)’ that are not null vectors, let S be the K x K, matrix consisting of K, columns of I, such that (X-X*)‘= S(X -X*);. In this notation, the observable reduced form equations corresponding to (3) are the equations Y’=fl,(X-x*);

+z7zX*‘+

I/‘,

I/‘=B-‘U’,

where the relations among the reduced form and together with the prior restrictions, are the system

(4) structural

parameters,

C-BQB'=O,

(5’4

BL’, +CS=O,

(5c)

4(B>P*, Y, 0) = 0.

(54

The special nature of the identifiability of the structural parameter of model (3) can be seen as follows. Suppose there exists a G x G matrix Tr such that T,TS=O. Then, for every G x G non-singular singular, it is readily verified from parameters,

(6) matrix T, such that T,+ T1 is non(5a), (5b) and (5~) that the structural

L.L. Wegge and M. Feldman, Muth-rational expectations models

248

reduced are observationally equivalent, i.e., all have the same observable form mean and covariance. Observationally equivalent parameters are obtainable by the usual transformations with T, of the system (1) of equations and also by adding to B* the parameters r,(B+B*) and by adding to r the parameters TIT. Therefore the problem of identifiability is a two-tier problem. The restrictions ~#1(a)=0 on the true structural parameter are needed to render inadmissible parameters (7) that are obtainable through the transformations T, and Tr.

2. Sufficient and for regular points necessary rank conditions2 Assuming Rothenberg

the rank (1971). Let

p((X-X*),,X*)=K,

+ K,

The Jacobian matrix of the system (5) with columns the vectors (y’, CJ’,/?*I, p’) is the matrix3

we

can

proceed

partitioned

as

according

in

to

J=

The structural parameter is locally identifiable if where cj,,A =24,.(I,@BQ). rank p(J)= GK + 3G2, which is the number of structural parameters. Since p(J)=p(J,)+p(J,-J,J;‘J,), where J, is the upper left corner identity matrix of order GK + G2 in J, and the other matrices are the submatrices in we have that the structural parameter the corresponding partition (;:$i) of J, is locally identifiable if the matrix J,-J,J,= has rank 2G2.

-I,@s’III; &W - &(I,

I, 0 (II; - sn;) @ n;)

4s’ - &‘(I, @ n;) + 24&G

8 sa) (8)

ZThe conditions were first obtained, following Bowden (1973) and Rothenberg (1971) by requiring that the information matrix bordered by the Jacobian matrix of the restrictions, is non-singular, assuming normality. The version of the proof that follows is due to the referee. We are grateful for this as well as for other suggestions to keep the paper clear and short. jIf z= vecZ is the transpose of the row of rows of a matrix Z, we have vecZ,Z,Z, = (Z, @ Z;) vecZ,, implying 8 vet Z,Z,Z,/8(vec Z,)’ =(Z, 0 Zs). It makes no difference for the analysis of identification if we consider all components of C or delete the symmetric elements below the diagonal as in Richard (1975).

I..!.. Wqye

trod M. Feldman, Muth-rational

We recall from Fisher (1966) that that the rank of the matrix J listed points in a neighborhood of a, and sufficient rank conditions stated below

expectations

models

249

a regular point is a parameter c( such above is a constant for all parameter also recall that at regular points the are necessary.

We state the main result of the paper: Proposition 1 (Sufficient Rank Condition). The Muth-rational expectations structural parameter (B, B*, r, C) of model (I) with B, and B + B* non-singular, C positive definite, with the K x n matrix of forecasts of the exogenous variables X*’ and the K, x n matrix of non-zero forecast errors (X-X*); having

rank

restrictions

X*)=

K, + K,

that satisfy p(@/&‘)

p((X -X*)

= k, if

is

locally

I,@S’T’ P

UT-Yug

0

u/g

identifiable

under

the

= 2G2,

k

(9)

where

YT - Y,* = (&

- &,)(ZG 0 B’) -24,,(ZG

0 C).

(11)

ProoJ:

In matrix (8) postmultiply the second column block by I, 0 B’ and add to it the first column block postmultiplied by I, @ B*‘. Then, postmultiply the first column block by I, 0 (B+ B*)‘. The result is the matrix in (9).4 Proposition 1 implies that the Muth-rational expectations parameter of the static model with current expectations is identifiable under the classical rank conditions p(Y,*) = G2 provided extra restrictions exist on the parameters of the ith equation, i= 1, . . ., G, when the rank of TS is less than G. If p(TS) = G, the classical conditions alone are sufficient. Corollary

Z

assumptions

(Order

and

of Proposition

4We verify that necessary condition

Variety

Conditions).

Let

p(TS)=

1 and if (B, B*, r, C) is a regular

the rank condition for the system

of Proposition

TI LS =O, d((vec T,B)‘, (vec((T,+

1 is sufficient

G,.

Under

the

point, it is locally

and for regular

points

a

T,)B* + T,B))‘, (vec(T, + TJT)‘, (vet TJTb)‘)=O,

to have the locally unique solution TO= I,, T, =O. With observational first and second moments, the identification results are distribution free.

equivalence

defined

by

250

L.L. Wqge and M. Feldman, Muth-rational expectations models

identifiable only if the total number of restrictions is at least G2 +G(G-G,). The parameters of the ith equation must be restricted by at least G+ G-G, restrictions. Of these, firstly at least G-G, must be restrictions involving elements of the ith row of (B*, I), and secondly at least G-G, must be restrictions involving the ith row of (B, B*,C), counting restrictions on both bij and b; as one restriction and not counting restrictions on B+ B*. This follows directly from the first and second form of the rank condition (9). To apply the rule stated in Corollary 1, the first task is to determine the rank of TS. In most practical situations this is either G, the number of endogenous variables, or the number of not fully anticipated exogenous variables. If p(TS)= G, then only the standard conditions must hold, as derived in Wegge (1965) or Rothenberg (1971). If the number of not fully anticipated exogenous variables K, is less than G, then one has to follow the counting procedure outlined in the Corollary. Pesaran’s corrected results are conditions for the identifiability of a single equation that can be derived directly from Proposition 1. In the matrices defined at (9), (10) and (1 l), put G = 1 and replace the matrices of derivatives of the restrictions with respect to all parameters by the submatrices of derivatives of the restrictions with respect to the first row of (B, B*, r) only. The first row is locally identifiable if the resulting matrix (9) has rank 2G, [2G- 1 when the normalization restriction is not counted as in Pesaran (1981)]. And if the restrictions are linear and independent of the covariances, then the rank condition for local identifiability is also the rank condition for global identifiability. This is Pesaran’s rank condition as stated in WeggeFeldman (this issue) and where he uses r,, for our TS. 3. Example We became aware of the problem of identification in Muth-rational expectations models through Wallis’ (1980) paper and through the problems encountered when we tried to estimate a simplified version of Sargent’s (1976) model. This version is a two equation aggregate demand and supply system ‘* (12a)

(12’4 where pt = log of GNP price deflator, y, = log of real GNP, m, = log of money supply, rt =long-term interest rate, Un,= unemployment rate, and t is calendar time. In matrix notation the structural parameter is

L.L. Wegge and M. Feldman, Muth-rational expectations models

251

which

multiplies in the variables (pt, J+, p:, y:, m,, rt _ I, (m -p),_ I, t, Un,_ 1, l), and satisfies the restriction bzl +bl, =O, in addition to all the other zero-order restrictions stated above.5 Six of the seven exogenous variables are predetermined and we made two alternative assumptions concerning the seventh exogenous variable, the money supply. First a forecast rnf for the money supply was constructed. In this case the matrix of forecast errors (X-X*)’ consists of one non-zero row, S is the first column of the identity matrix I,, TS =(A) and its rank G, = 1. From Corollary 1 we need three restrictions per equation with at least one restriction on each row of B or B* and at least one restriction on each row of (B*, r). But model (12) has eight iero-order restrictions (normalization and exclusion) on the first equation and five on the second equation, not To check the rank condition, as an counting the restriction b,, + b,,* -0. example, consider the subset of restrictions cl1 = 1, cl4 =O, cZ1 = 0, b,, = - 1, bT, = 0 and b:2 = 0. The criterion matrix (9) is Un,_,,

1

0

0

0

0

0

1

000

1

0

0

c24

0

0

0

0

0

0

1

0

0

0

0

1

000

0

0

0

0

0

0

-1

0

0

0

0

0

b,,

-10

0 c24

0

o-

0

0

0

0

0

0

10 b,z

-1

b”;,

0

0

0

0

0

which has rank 8 if bT1c24#0. We conclude that the model (12) with G, = 1, is identified under that condition, as well as under other alternative conditions. The time series forecast rn: for the money supply was so highly correlated with m, that the mom&t matrix between the two was nearly singular, For this reason a second alternative causing difficulties in estimation. assumption is that the money supply m, is known at time t - 1. Now we have that all seven exogenous variables are known, S is void and G, =O, implying that we will need a total of eight restrictions as a minimum, with at least four on each equation, at least two on each row of (B,B*) and at least two on each row of (B*, r). The total order condition as well as the partial order ‘With pt-l entering in the first equation, the model is dynamic. However since u, is white noise and maximum lags are speci!ied, the identification conditions of the static model are valid here also. as in Hannan (1971).

252

L.L. Wegge and M. Feldman, Muth-rational

expectations

models

conditions are all satisfied. However the rank condition of Proposition 1 fails. To see this write down the restrictions on the matrices (B,B*) in the order b,,+l=O, b,,+l=O, bT,=O, b:,=O, bl,=O, bzl+b$,=O, a total of six restrictions. However the matrix UT - Y,* in (11) has rank 3 only, one less than the required rank 4. There are indeed three restrictions on the second row of (B,B*,Z) but two are on bz2 and b&, which should count only as one, and the third specifies b,, + b,, * -0 which should count as none. Therefore the parameter of model (12) with G, =0 is not identifiable. If instead of the restriction b,, + bf, =O, we had a restriction specifying either the value of b,, or of bz,, then the parameter is identifiable. This is an example where the order conditions are satisfied, but not the further variety conditions specified in Corollary 1. For this reason the rank condition fails. The example illustrates the general rule that if all exogenous variables are known at time t, then we will need a restriction on either bij or on b$ for all i,j, assuming there are no covariance restrictions.‘j In the example b,, and b$, are not restricted and the restriction bzl +bfj, =0 is of no help in contributing to the rank of the criterion matrix. With covariance restrictions this requirement is relaxed and identifiability could be restored through Zrestrictions.

4. Conclusions We dealt with Muth-rational expectations models of the form BY’+B*Y*’ +TX’= U’ and we extended Wallis’ results in identifiability of the structure by removing the condition that none of the exogenous variables are known in advance. Instead we considered the case where some or all of the exogenous variables are perfectly anticipated, as well as allowing for a wider range of restrictions. Our principle finding, confirming Pesaran’s single equation result, is that the classical rank condition remains appropriate for Muth-rational expectations models whenever the rank of T(X -X*)’ equals G, the number always whenever the number of imperfectly of equations, i.e., almost anticipated exogenous variables equals or exceeds G and r has rank G. When p(T(X-X*)‘)= G,, then we need G-G, additional constraints on the parameters of each equation. These conditions are plausible given Wallis’ findings with zero-order restrictions. Our contribution is the statement of the rank condition for matrix (9) and 6Model (1) with all exogenous variables xl known at time t-l, has been studied by Muth (1981), a paper already written in 1960. He shows that without C-restrictions, the concentrated likelihood corresponding to (1) is exactly the same as the concentrated likelihood of the model assuming the structural residuals are normally distributed. Muth (B+B*)Y’+TX’=U*‘, concluded that we can estimate B+ B* by standard simultaneous equation estimators and that if expectations data are available, we can distinguish B from B*. The conclusion here is that prior information is needed to identify B and B*.

L.L. Wegge and M. Feldman, Muth-rational

expectations

models

253

its consequences. These are that the restrictions must satisfy the qualifications that there be at least 2G- G, on each structural equation and at least G- G1 on each row of (B, B*, C), not counting the restrictions on B+B* and counting restrictions on bij and b$ only once. Furthermore there should be at least G-G, on each row of (B*, r). Under Wallis’ assumptions the critical integer was the number of exogenous variables below G and here the critical integer is the number of not perfectly known exogenous variables below G. For each imperfectly known exogenous variable below G, an additional G restrictions, one per equation, are needed for identifiability of the structural parameter. While seemingly paradoxical that improved information can result in loss of identification, there is an intuitive rationale for this result. Each non-zero row of (X-X*)’ is an explanatory variable. The implication is that having perfect foresight of a series is equivalent to deleting a variable from the model. On the other hand every element of the asymptotic covariance matrix is a continuous function of the moment matrix T(X -X*)‘(X -X*)T’ when the conditions for identifiability are satisfied, and so is the variance of prediction n-‘( Y- Y*)‘(Y- Y*) a declining continuous function of the variance n- ‘(X - X*)‘(X - X*). Instead of viewing the problem in a discontinuous fashion, one should perceive that the interface between identifiability, estimation and prediction is a continuous relationship. Long before we reach the point of a discontinuous jump in the rank and its concomitant requirement of more prior information, we would be in a near singular moment matrix situation where the distinctions between some parameters become very confused, indicating that the parameter is close to being not identifiable. The rank condition for identifiability (9) captures this concept in a precise statement of the dependence of identification on the data matrix and the parameter restrictions.

References Bowden, R., 1973, The theory of parametric identification, Econometrica 41, no. 6, 106991074. Fisher, F.M., 1966, The identification problem in econometrics (McGraw-Hill, New York). Hannan, E.J., 1971, The identification problem for multiple equation systems with moving average errors, Econometrica 39, no. 5, 751-766. Muth, J.F., 1961, Rational expectations and the theory of price movements, Econometrica 29, July, 315-335. Muth, J.F., 1981, Estimation of economic relationships containing latent expectations variables, Ch. 17 in: R.E. Lucas, Jr. and T.J. Sargent, eds., Rational expectations and econometric practice (University of Minnesota Press, Minneapolis, MN). Pesaran, M.H., 1981, Identification of rational expectations models, Journal of Econometrics 16, no. 3, 375-398. Richard, J.F., 1975, A note on the information matrix of the multivariate normal distribution, Journal of Econometrics 3, no. 1, 57-60. Rothenberg, T., 1971, Identification in parametric models, Econometrica 39, no. 3, 577-592.

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models

Sargent, T.J., 1976, A classical macroeconometric model for the United States, Political Economy 84, April, 207-237. Wallis, K.F., 1980, Econometric implications of the rational expectations Econometrica 48, no. 1, 49-74. Wegge, L.L., 1965, Identifiability criteria for a system of equations as a whole, Journal of Statistics 7, no. 67-77. Wegge, L.L. and M. Feldman, 1983, Comment to the editor, Journal of Econometrics,

Journal of hypothesis, Australian this issue.

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