Rational expectations models with partial information

Rational expectations models with partial information

Rational expectations models with partial information Joseph Pearlman, David Currie and Paul Levine This paper provides a general solution to the pro...

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Rational expectations models with partial information Joseph Pearlman, David Currie and Paul Levine

This paper provides a general solution to the problem of partial information in linear discrete time stochastic rational expectations models. The full information case isfirst reviewed and the solution of Blanchard and Kahn [4] extended. Then we consider the problem of part ial informat ion for the special case where only the current values of some variables are unobserved. The solution can be treated as a straightforward extension to the full information case. In the general problem where in addition to some current variables being unobserved, certain variables are unobserved for all lags, we provide a solution which requires the use ofKalman$lters. The paper concludes by examining the covariance properties ofthe rational expectations system under different informational assumptions. Keywords:Rational expectations models: Information variables: Covariance analysis

In models of rational expectations, it is frequently of interest to limit the information set on the basis of which agents form their expectations. This is because agents may observe certain variables of interest only after a delay, so that only lagged values enter the information set. In other cases, variables may not be observed directly at all: this may occur, for example, when a variable is observed with measurement error, or (as in our example below based on the model of Sargent and Wallace [20]) when the variable in question is the permanent component of an observed variable made up of both a transitory and a permanent component. Partial information poses an interesting problem of inference for agents in rational expectations models. This arises because, with limited current information,

Joseph Pearlman is at the Polytechnic of the South Bank, London; David Currie is at the Department of Economics, Queen Mary College, University of London. Mile End Road. London El 4NS; and Paul Levine is at the London Business School, Sussex Place, Regent’s Park, London NW1 4SA, UK. The financial support of the ESRC is gratefully acknowledged by Dawd Currie and that ofthe Nuffield Foundation and the ESRC by Paul Levine. Final manuscript

90

received I8 September

I985

variables contain (partial) current observed information about current disturbances impinging on the system; while inferences made from this information in turn influence the current state of the system, and hence the observations themselves. When certain variables are never observed, this inference problem must be combined with that of inferring past values of unobserved variables. The circularity inherent in this inference problem has been considered for special cases: thus Townsend [22,23] analyses it for the case where forward expectations are absent; Futia [9] analyses the case where predetermined variables are independent of the non-predetermined variables; while Minford and Peel [I81 examine it for a very particular model where all lagged variables are observed. More often, however, the circularity is neglected (see, for example, Barro [2], Saidi [19], Harris and Purvis [IO], Bhandari [3], Ford [8], King [15], Siegel [21], Weiss [24], Kimbrough [12,13,14], Duck [7] and Lawrence [16]). One reason for this is the inherent difficulty of solving the general problem. The purpose of this paper, therefore, is to provide a general solution to the problem of partial information in linear rational expectations models. To do this, it provides an explicit solution to the prediction problem for individual agents. This solution may then be used to derive simulations and forecasts from rational expectations models. Throughout the analysis, we

0264-9993/86/[email protected]

$03.OO’i.l 1986 Butterworth & Co (Publishers)

Ltd

Rational expectations models with partial iqjbrmarion: J. Pearlman et al

assume that agents share the same limited information set. We therefore do not address the problem of diverse information in rational expectations models, which will be the subject of a future paper. The partial information solution is of particular interest when considered in conjunction with the design of control rules in rational expectations models (see, for example, Currie and Levine [6] and Levine and Currie [17]). Changes in the control rule modify the dynamics of the system, and thereby alter the relationship between the free variables and current disturbances to the system. The informational content of the observed free variables is therefore dependent on the choice of control rule. But because the behaviour of agents depends on the information inferred from observed variables, this provides an additional route through which the control rule influences volatility in the system. Thus, in choosing the optimal rule under partial information, two considerations are relevant. First, the control rule operates to modify directly the dynamic response of the system: this is the route common to all control problems. Second, the rule operates to modify the information available to agents, and thereby indirectly the response of the system to disturbances: this route is particular to the rational expectations partial information solution.

Equation (1) represents aggregate supply as a function of the ‘surprise’ in the price level, (2) makes real demand a function of the expected real interest rate; (3) is a standard money demand function; while (4) represents an autoregressive money supply rule. Equation (4) defines the exogenous money supply, made up of two components: an autoregressive z component defined by (5) and a white noise component ad; Equations (6) and (7) represent equilibrium in the goods market and money market respectively. All behavioural relationships incorporate stochastic disturbances. Suppose that the interest rate is the only variable directly observable in the current period. Then we may write the system as: =t+1 [ Pr+ 1,t +

I[ =

P -c;’

0

(1 +c;‘j+u(K1

=t,* I[ 1

0

[ 0

-a(&’

pt,,

+c;‘c,)

expectations

model

with partial

A simple stochastic macromodel of the type presented by Sargent and Wallace [20] is: ~:[email protected]~-

P~,~I+~

(1)

.vP= -h(r,-pt+,,r+pt)+i:zr

(2)

d

m, =p,+c,~‘~-c~r,+~~,

(3)

m~=:,+~4,

(4)

I, + , = pzr + Es,

(5)

_l.d f = y: = _I’,

(6)

HI: = rnf = m 1

(7)

where Y represents the logarithm of real output, p the logarithm of prices, r the rate of interest (measured as a proportion), uz the money supply, z the autoregressive component of the money supply, tzi represents white noise disturbance, the ‘s’ and ‘d’ superscripts denote supply and demand respectively, and pt., denotes the expectation of p, formed at time s.

ECONOMIC

MODELLING

April 1986

--r

pt

(81

+u*

0

0 ofa rational

I[I

where

-b-l

An example information

+c;Ic,)

-(.;I

0 _c;l

1

1 0

Now consider the information sets on the basis of which agents form their rational expectations. We assume throughout that the right hand side matrices (denoted by G and H respectively) are known to agents. If agents were able to observe current values oft he state vector [~,p,]~, then I),,,=P, and the information set is given by:

This corresponds to the full information case analysed in the next section. It is easy to see that this information set implies full knowledge of those disturbances influencing the current state of the systems, ie full knowledge of u:-~- ,, ~f-~(i30). Suppose instead that agents can observe only the current rate of interest, r,, and that j: nz and p are observed only with a one period lag. Suppose also that c,=O, so that m follows a simple autoregressive process. Then m = 2, so that 2 is also known with a one period delay. It is easy to see that knowledge of 2, _ i and P,_~ (iz 1) implies knowledge of u/_,- 1 and u:- i (iz 1). However, agents cannot know the exact value of

91

current disturbances to the system (u:_, and zrf) since they observe only one variable. T,. However, if they also know the covariance matrix. U. of the disturbance vector. 11,. they can form optimal estimates of these disturbances conditional upon the observations, I’,. on the current interest rate. This inference problem is analysed later. For this case the information set is If=

Rewriting

Equation

(8) as

64t

:I -r

=

P,

jr,,r,,p,.G,H,~i(s
-r+

1

Pr+

1.r

0

where I’, is given by:

000

1

0

000

0

1

000

I

0

0

+(i.;‘(.II:,r+f,;‘i:3,)

(9)

This is the form of the problem analysed later. where the observation vector. \v,. reduces here to r,. Now suppose c,#O. Knowledge of 111no longer implies knowledge of -_. Agents have the problem of estimating the decomposition of ~1 into its random component, :; component, iz4. and its autoregressive and the true decomposition does not become known to them even with the lapse of time. We assume that agents use the Kalman filter to solve this inference problem. However, this inference problem has to be combined with the one discussed above, whereby current observations yield partial information about current disturbances. The general solution to this tricky inference problem is presented later in the paper. For this example the information set becomes:

vector \l‘, is given by:

where the observation

I1

-

I’,

\(‘, =

111, _, Pt-

I

I

I

0

0

i 0

0

01

0

zz

+

92

(‘2

-1

000

0

0

-c;r

1

0

0

0

0

0

0

0

0

0

0 0

+oooo I 0

0

0

0

0

Equations (10) and (11) are in the form of the problem analysed later. This example drawn from the simple SargentWallace model illustrates the main features of the problem analysed in this paper. However. it should be emphasized that the methods are perfectly general. and may be applied to models of much greater dynamic complexity.

Full-information We consider

!

1 +L.,‘-Cll(h-‘+C;‘c,) 0

solution

the following

equation

system:

(12) 0

0

0

0

0

0

0

0

0

0

0

0

-

of variables where Z, is an (11--111)x 1 vector predetermined at time t: x, is an 111x 1 vector of variables non-predetermined at time t: II, is an II x 1 vector of exogenous disturbances: A is an )I x II matrix and s, + , ., denotes the rational expectation of s,, 1 held at time t. In terms of the disturbances L, of the original model II, = Bc,. All variables are measured as deviations about the long-run equilibrium.

ECONOMIC

MODELLING

April I986

Aoki [l] shows that quite general dynamic systems may be put into state space representations, and therefore into the form of Equation (12). Blanchard and Kahn [4] note that Equation (12) incorporates a wide class of rational expectations models. and this is illustrated by our earlier examples. They also note that Equation (12) cannot encompass all such models. particularly those involving past expectations of current or future variables. We assume that Equation (12) is a dynamic representation of minimal dimension. so that A is nonsingular. We further assume that the disturbances 11 follow an ARMA (4.~) process. Provided qz~, following Aoki [ 11. we may expand the dimension of A to represent the system in a form where ~1is a white noise vector. (We do not assume that the elements of II are independently distributed.) In what follows. we therefore assume that 11is a white noise vector. Blanchard and Kahn [4] show that the model given by Equation (12) must have the saddle-point property if a unique non-explosive rational expectations solution is to be found. This requires that there should be (n - rn) eigenvalues of A within the unit circle: and ~11 eigenvalues of A outside the unit circle. We thereby exclude the possibility that one or more eigenvalues lie exactly on the unit circle. We assume throughout this paper that the models under consideration have the saddle-point property. In this section. we consider the full information solution to Equation (12). By full information. we mean that agents have knowledge of all variables (not including disturbances) up to and including the current period. The full information set is therefore

Partitioning

A. M and A conformably.

we have that

(14 In what follows, we assume that .klzr is ncn-singular. We discuss the significance of this assumption. and the consequences of singularity or near-singularity of Rl,? at the end of the section. Using E(.) to denote the mathematical expectations operator, we have from Equation (12)

-r+,,r=E(-_l+lIlf)=AI,-,+A,z.~,+~r:,, (1% Now uf is known at time t. so II:,, = l’,2L,‘;i~lf

Taking expectations at time t of Equation changed to r + I; gives

(12) with t

so that

(17)

Let

Let 11,=

11:

I_1 11;

.II: is the (rl- r~) x 1 vector ofdisturbances

in the equations for zl+ ]. I(: is the 111x 1 vector disturbances in the equations for x,+ ,,t with

of

The last 1)1elements written as

of Equation

where li.il > 1. We are interested It is clear from Equation (12) that If implies knowledge of II,‘_i for i > 0: and of [d,‘_I for i 3 0. Let M and A be square matrices such that MA = AM with A=

where

MODELLING

be

only in non-explosive

=t solutions for and thus for J‘~.Hence we require that [1s, taking /i = 1 we have !‘i.r + k.t = 0. In particular,

A, is an (r~- 111)x (n - !)I)

matrix with eigenvalues within the unit circle and AZ is an ITIx 111matrix with eigenvalues outside the unit circle.

ECONOMIC

(17) may then

April 1986

and inserting

this into Equation

( 16) gives

93

Rational expectations

Solving

models

with purtiul information: J. Prarlman et al

for .yr we have

x,= -M;2’M21z,-

M,‘K’(MdJ,z&‘+M,,b: (20)

From

Equations

(12) and (I 8) we have

Z,+, =C:,+Du, where C=A,, K1(Mz,U,2U;;

(21) - A,,M;,‘M,,

and D=[I,

-A,,M,

+MM,,)].

Equation (21) provides an (tt -tn) order dynamic system in terms of the predetermined variables and exogenous disturbances alone. It may be solved by standard solution procedures. Equation (20) may then be used to obtain the implied solution for nonpredetermined variables. To establish that Equation (21) is a stable system, it can easily be shown using Equation (14) that N;,‘C=A,N-’ where N;,‘=M,,-M,zM;;MZ,. Thus C= N, :A, N;,’ has just those eigenvalues of A that lie within the unit circle. Using a wellknown matrix identity (see for example Johnston [I 11) N, 1 is the top left hand submatrix of N = M - I ; furthermore we note from Equation (14) that

Thus Equation

(21) can be re-written

The case where Det M 22 = 0 is exceedingly unlikely to arise. More pertinent is the case where M,, approaches singularity, so that Det M,, is close to zero. In this case, a small change in parameter values can cause Det M,, to change sign. causing very large changes in the M;iM,, matrix which figures in Equations (20) and (21). In this region. the dynamic behaviour of the model will exhibit marked structural instability; and conclusions drawn for particular parameter values from dynamic analysis of the system will not be robust. These points concerning the singularity or nearsingularity of Mz2 apply equally to the partial information cases discussed in the remaining two sections of this paper.

Partial information solution (special case) In this section, we modify the information assumptions underlying the solution of the previous section by reducing the information available to agents about the current state of the system. An immediate consequence is that it is no longer necessarily the case that zt , = zr, x,,, = x,. Accordingly we must generalize Equation (12) to

as (23)

=,+,=N,,A,N;,‘~,+[!.-(h’1,AIM12M~~A;1+N,2) (M,,C’,,L:,’

+ Mz2)]u,

(22)

which apart from the presence of C.i,2 is the recursive solution presented by Blanchard and Kahn [4]. Noting the need to invert either Mzz or N, ,. it is clear that for the deterministic case (~,=0) solving Equation (22) will generally be computationally simpler if ttl
94

The partial rf=

information

set is assumed

(H.~,~~..u,~.G.H.K.L.L~.~~.~~~~

to be (34)

where

(2.5) Thus IV,represents a p dimensional vector of currently observable variables, while Z, and _yrare in general. observable only in subsequent periods. Note, however, that Equation (25) includes as a special case the possibility that some of the -I or .Y[ arc themselves directly observable (as illustrated by the example earlier). K and Lare of dimension (p x tt) and L’,is a p x 1 vector of white noise disturbances. As before we may assume without loss of generality that it is white noise. We also assume, in this section. that 11,‘.~2 and I‘, are uncorrelated white noise disturbances with L’ =COV(U,) and V=COV(U,). Although this is restrictive for the special partial information case considered in this section, as we shall see. it is not so for the general case in the next section.

ECONOMIC

MODELLING

April 1986

Rational

Equations (23H25) include previous analyses of partial information as special cases. Thus Futia analyses a similar system in which the predetermined on the nonvariables, -_,, have no influence predetermined variables, x, (ie G is restricted such that G, I = 0). Townsend [22,23] analyses a system in which forward expectations are absent, so that m = 0 (though he also considers the more general problem of diverse information which is neglected here). It is clear from Equations (23), (24) and (25) that 1: implies knowledge of uf for ss r - 2 and U: for ss r - 1, and that agents can have no knowledge whatever at time r of u:. However. the inclusion of M‘,in 1: means that they can make inferences about and u:, from which they may form expectations u:- ,,, and u:,~. Solution of this problem requires knowledge of the covariance structure ofthe disturbances, u and L’,which is why we assume that 1: contains U and I/: The possibility of such inferences gives rise to an interesting and rather complex problem. It is clear from that the nonsolution the full information predetermined variables at time t depend on current disturbances to the system (see Equation (20)). With partial information, it is plausible to conjecture that the non-predetermined variables will depend both on current disturbances and current inferences about current disturbances (and we confirm this conjecture later). But the inferences depend in turn on current nonof some of the observations predetermined variables. This gives rise to an obvious circularity, with inferences depending on observations and observations depending on inferences. In some models. this circularity can lead to self‘inferences’. with a ‘bootstraps’ confirming determination of the non-predetermined variables. But in an important class of models, this is not so. As we show, the assumption of rationality can be used to determine those inferences that will be consistent with subsequent developments. Agents must disentangle the influence of their (and other agents’) inferences on their observations, in order to obtain (partial) information about the underlying disturbances. Substituting for s, and z, from Equation (23) in Equation (2.5), we have

u:_,

expectations

MODELLING

April 1986

J. Pear/man

et al

+r,

(26)

-

K2G;;uf

+(L-K2G;;H2)

where

K =[K,K,],G=

[Zj=[Z::

:;:j

and H is similarly partitioned. At time r. everything is known in this expression apart from Du:_, - Fuf + r, and F=K,G;;. where D=(K, -K2G;;G2,) Inferences can now be made concerning u:- , , ~(2and r,. namely that

(DZ/,,DT+FU22FT+I/)-‘(Dt~:-,

-Fu;+t.,) (27)

where U =cov(u,) and I/= COV(L.,). Taking expectations of Equation (23) lagged one period and focusing on the upper block. we have z, - qr = u,1_ * - u,I * .,

WV

Taking expectations of Equation the lower block. we have

x~+~,,=(G~+H~) But from Equation

Subtracting x,-x,,,=

Equation

(23) and focusing

+

IIf.,

on

(29)

(23) lower block we have

(29) from (30) gives

-G,;‘G,,(-,--,,,)-G,‘(uf-Us,)

(31)

assuming that G,, is non-singular. We now assume that the saddle-point property holds for the matrix G+ H so that a unique nonexplosive rational expectations solution exists. Let M and A be square matrices such that M(G+ H)=AM with M and A partitioned as in Equation (14). Then in exactly the same way as for the full information case we have M

ECONOMIC

models with partial it&rmation:

21~,+~.r+M22~~,+~.r=O

(32)

95

Rarional expectations

models with partial ir$ormurion:

J. Pearlmutl

Taking expectations of Equation (23) at time t, premultiplying both sides by M and using Equation (32) we then have A&,=,,,

+ Mzzxr,r) + K&r

Thus using Equation

= 0

+(M,‘A;‘Mz2-

+u;-~

J-M;;A;‘M

22

uz1.1

+G,2(-x,-.QJ)+II:

=(G,,

+HiJ-_,

-4-,.J-(G,z+H,2)

+I~:_,.,)+M~~A~~M,,~~:,]

[&‘M~,(=~-II:-~

z[G,'G,,(zr:_ 1-u:_l,,)+ G,;‘(u; -u:,)]

using. Equations (29) (31) and Equation (27) we finally arrive at

(34). Then

using

-f + 1= c-, + (A - c)lI:_ , + I/: - Llu; -C)C,

lD’-(B-

Y(D!r;_ ] - FL/,’ + I.,)

7-)L&T] (35)

The partial information solution (general case)

r’p=(~(.,,G,H.K,L.L’.v~~~~:

+~,1)3,--,1(=,---,,,)+(G~z+~~2)~,,,

-[(A

(36)

In this section we reduce still further the information available to agents by writing the partial information set as

We can now combine Equations (23), (28) (29) (31) and (34) to obtain a dynamic relationship for z,. From the upper block of Equation (23) we have

- G,

Uz2Fr’l

Y( Du;_ 1 - Fu; + rJ

(33)

(34)

-ff,2(4-I

G;;)

(28) we obtain

X,,, = -M,LM21(-_r-~;_1

--,+i=(G,,

et al

+u;

(37)

where the model is still given by Equation (23) and the vector ofcurrently observable variables \t‘, by Equation (25). Note that Equation (25) still includes as a special case the possibility that some ofthe z, or X, are directly observable. Furthermore we show in the Appendix that the partial information model of the last section can be expressed as a special case of the model presented in this section. As for the full information case we may assume without loss of generality that II is again white noise. For notational convenience we also assume that II,‘, I$ and I‘, are uncorrelated white noise processes. Although this may appear restrictive it is, in fact. not so. Suppose that Equations (23) and (25) contain additional -random forms SC, and Tt>, respectively (where P, is independent of 11,and L.~)so that the noise terms are now correlated. The vector :, can be extended to include z: =L’, resulting in the system

where -(G,,+H,z)Ali,lMz,

C’=G,,

+H,,

A=G,,

-G,zG;;G2,

B=G,,G,; D=K,

-K2G&‘GZ,

F= K,Gy2 Y=(Db’,,D’+FL22F’+I’)m’ T=(G,,+H,,)~l,‘Ai’Mz,

Then combining Equations (27). (31) and (34) the nonpredetermined variables s, are given by

96

which is of the form of Equations

(23) and (24) with

c,+ I and I%,uncorrelated. L 1’1 I Similarly. if II/_, is correlated wlith 11,’or i’,. then z, can be extended to include u:_, and our assumption of independent noise processes holds once more. We again assume that the saddle-point property holds for G + H and we let M and A be square matrices such that M(G+H)=AM. Then as in the full information case Equation (32) holds.

ECONOMIC

MODELLING

April 1986

Rational expectations models with parrial infiwmation: J. Peurlmun et al

We further assume that an updated least squares estimator, as for Kalman filtering, is appropriate so that we may write

where C=G,,+H,,

-(G,2+Hlz)M;z’Mz,

W=(G’+H’)J(I+U)-’ and G and H are partitioned The second and third terms in the brackets together represent the best current estimator of \v,. The term --I.,

represents

information

to hand.

but the best

U -&,, -t is obtained using estimators of .Y; [1s, and z, based on information up to r - 1. The form of the J matrix is established in Theorem 1. Before stating our main results which describe the dynamics of zr and s,. we use the Kalman filtering assumption to obtain expressions for current estimators -_,.,and s,,, and the one-step ahead predictor :,+ l.,. We state the results in the form of the following two propositions: of K

estimator

Propositiorl

1

The updating equations and s, are given by

for the current

estimates

of :,

Proqf

Taking expectations at time r of zl+ 1 as given by the upper block of Equation (23) we have

zr+,,,=(GL+H’)

substituting

z____ -1--r

for

(43)

-1.1- 1 our

results

in the form

of the following

-

(39)

Theorem

I

(a) The stochastic E=(K,+L,)-(Kz+Lz)~~~~Mr,

and L= [L,.LJ

are partitioned

and as before.

difference equation

for z, is given by

~t+,=C-_,+(A-C)~,+[(C-A)PD7+(T-B)U22F7]

Proqf

Solving

(43)

21-1.1 -11ltJ(IiLJ)_'(~~,-E_,.,_,

-Mzz’M

K = [K,.K,]

1 "J 1

“” r %,I

from Equation (39) we r %., obtain the required result. Our main result is the presentation of a stochastic difference equation for the predetermined variables z, and a stochastic static equation for the nonpredetermined variables s,. These are obtained in terms of Zr defined by Then

We state theorem.

-r.r- 1

where

as before.

[email protected]:, - Fu; + r.,) - Bu; + 14,’ Equation

(38) for

(44)

where -,+,=A-,-Btrf+Ir:-(APD’+BC’22F’)

[email protected]:, - FL{; + l.,) (40) From Equation (32). replacing r by r - 1 we have s,.,_ 1= M;2’Mz,:r ,~ 1. Then substituting into Equation (40) and using the identities (I+JL)-’ = I-(I+JL)-‘JL and (I+JL)-‘J=J(I+LJ)-’ we obtain Equation (39).

A=(DPD’+FL&F and where P=cov(l,) equation P= APA’+

Proposit ioti 2

The filtering

+ V)-’ satisfies the steady-state

Riccati

BUz2B7 + U,, - (APD’ +BUz2F7)

4(DPA7 +FU,,B’)

equation

(45)

(46)

for zl+ l,, is given by Matrices

A, B, D, F and Tare

as defined for Equation

(35).

ECONOMIC

MODELLING

April 1986

97

Rational expectations

models with partial information:

(b) The non-predetermined

variables

J. Pearlman

are then given by

+[M,‘II,‘M,,U~~F~-G;;U~~F~

217

and we recall that D=K,

- My; M,, )PDT]A(DZ, - Fu,2 + u,) (47)

Proof of‘(u) We set out the proof of this central

where X=[G;;G,,-G,‘M;&M

x,= -Mz;lM2,(=1-;r)-G2;1GZIfr-G2;I~:

+ (G,;‘G,,

et al

I -G;2’M;2’A2M22]

-K2G;;

and F=K,G;i.

(iii) The d~mrni~.s qf$ Subtracting Equation (42) from the first row of Equation (23) and using Equation (50) we have

result in five steps.

(i) An espres.siorz,ftir s, - s,,,. From the lower block of Equation (23) we have (48)

.~~+~,~=~G~+~2,i,J+Gz~~~~~;~~+~~

Then combining

Equations

(32). (42) and (48) we have

Then using Equations

(39) and (51) we obtain

,+ I = A,-B~Z+I~:+(G,,XJ-G’J)(I-K~XJ)-’ (0, - Fuf + r,)

(52)

[M~,M~z~J(G+HI/;:I/+[~~][;T:;:~~~[~~~]=~

(49) which after applying [M2,M,,](G+H)=A2[M2,M22]

where we B=G,zG,‘.

recall

that

A=G,,-G,zG;;Gz,

(iz) The K~~Imar~,fi/ter.fijr j,. We rewrite Equation as

the relationship reduces to

: -,+* =z:,+y,

.Y,- .Y,.,= - G,’ G2, (-, - I,,~) - Gzz’uf -GzzlhrlzzlA2(MZ,_,,,+M2,.~,,) -

3

(ii) A rzen‘c~.up~e.s.sior~,/i~rH‘,.From Equation rewrite

and

(50)

(25) we can

(52)

(53)

where 4, i-s white noise and Z=A+(G21XJ-G’J)(I-K~XJ)-‘D cov(y,)=Q=

c’, , +([email protected])C27(B+OF)7

[email protected]’@

~=(G,2XJ-G’J)(I-KzXJ)~’ We rewrite

Equation

(51) as

\v, = N:, + E:, , , + I’,

(54)

where rr is white noise and

(M’,- E--,,,- ,I 1

N=(I+LJ)(I-K2X’J))‘D cov(r,)=R=(l+W)(I-

K2XJ)-‘(FL’,2F’

(I - K,XJ)using Equations (39) and (50). Then using Equation (39) again, after some manipulation we obtain ~~,-E~,,,_,=(!+LJ)(I-K2XJ)-‘(D~,-Fuf+~,) (51)

98

+ V)

‘(I +LJ)’

The system is then given by Equation (53) defining the dynamics of &, Equation (41) defining the dynamics of z,., _ , and Equation (54) giving the observable variables in terms of :t and -I.! _ ]. In Appendix 1 we show that the

ECONOMIC

MODELLING

April 1986

Rational

filtering

equations

=t + 1 .I =z:,,,_l

(I+U)-‘(M’t-EE=,,,_‘)

for I, are given by

+ (zPN7+S)(NPN7+R)-

Comparing

(IV,- N:,,,_ , - Ez,,,_ ‘)

(55)

; - = _I,r_’ +PN’(NPN’+R)-’ --,.r (M.,--Nf, t-’ -Ez,,,_

models withpartial [email protected]: J. Prurhnurl et al

expecrarions

Equations

(56)

(60) and (61) we obtain

J1(I-K2XJ)-1=~D7(DPD7+V+FL’,2F’)-’(62) Equations

,)

(61)

(59) and (62) are linear simultaneous J’ which can be solved in the usual equations in J = J’ way. Lengthy algebra gives the solution

[I

where J=

S=cov(r,,q,)=(G’,XJ-G’J)(I-K,XJ)-’ I/U - K,XJ)Y(I

+ LJ)7

M2;Ar’M22C.ZZF7--M~~M21PD7

r=(KlPD7-KzM;;~2,P~7

- K2XJ)-‘(1

+ LJ)’

and P satisfies the Ricatti

equation

I I-

(63)

where

+(B+(G’2XJ-G1J)(I-K,XJ)-‘F) C’,,F’(I

PDT

+ v

+KzM;;A;‘M22C’22F7)-1 In fact Equation (63) is not required to obtain Equation (44). Substituting for (I + LJ)- ‘(iv, - k,,, _ ‘) from Equation (51) into (41) gives

P=ZPZ’+Q-(ZPN’+S)(NPN’+R)-‘(NPZ’+S’) (57) We now recall that by definition f, + I = z, + I - z, + , .LSO that ?l+l.r=It,t-,=O. This means that from Equation (55) we have ZPN’+S=O

(58)

Substituting for Z, N and S into Equation after some algebraic manipulation (G12XJ-G’J)(I-KK2XJ)-‘=

(58) gives,

” _,+,,,=Cr,~,_,+(G1+H’)J(I-KzXJ)~’(D~,-F~r~,+~~,) (64) Then using Equations (59) and (62) directly we can verify that Equation (64) may be rewritten as ~t+l,r=C~,,,_‘+(CPD7+TTC’22F’) (DPD7+V+FL’22F7)~‘(Df,-F’~f+~,)

-(APD7+BU22F’)

(65)

(DPD’ + I/+ FUz2F7)(59) which, when substituted into Equation (52) gives Equation (45) as required. In the light of Equation (58). the Ricatti Equation (57) becomes P=ZPZ +Q. Hence from Equation (53) . /we have that I’=co\(i). r (I.) T/W rl~~r~trrnics q/‘z,. From have

our definition

=r,r= -r.r - -r.t - 1=J’(I+U)-‘(n,,-E=,,_‘)

of 5, we

(60)

using Equation (39). But from Equation (56) putting 1 _ ’ = 0 and substituting _,., for N and R we obtain :,,,=PD’(DPD’

ECONOMIC

+ I’+ FLI’,~F~)-‘(I-K~XJ)

MODELLlNG

April 1986

Finally adding Equation (44) as required.

(65) to (45) yields Equation

Pro~f‘of’ (h) From

the lower block of Equation

x,,,=s 1.1_l +J’(I

+ LJ)- ‘(w-

(39) we have

El.,_ ,)

(66)

and from the upper block, Lt.t= =r,r- I +J’(I+W)~‘(M.,-E_,,,_,)

(67)

Hence substitution of Equations (66) and (67) into Equation (50) and using (from Equation (33)) M 21=t.t-1

+M22~t.r-1

=O

99

Rarionul

expectations

models

bvith partial

[email protected]:

J. Pearlmurl

et al

Covariance analysis: a comparison of the partial and full information cases

we have

Then using Equations (51). (59) and (62), Equation (68) reduces to Equation (45) as required. We conclude this section by considering models studied by Sargent and Wallace [20] and Burmeister and Wall [5]. In these models expectations entering Equations (23) and (24) are based on information available at time t - 1. so that these equations are replaced by

(69)

and

(70)

(Note that Burmeister and Wall use the notation ~*(t,t) and p*(t+ 1~) to mean p,,,-, and p,+ ,.t-, in our notation.) If we take the appropriate estimator to have the form

For linear stationary dynamic systems without rational expectations, the assumption of full or partial information has no bearing on the long-term (asymptotic) covariance properties of the system. For systems with rational expectations. however. we shall show that not only does the form of information set affect the covariance properties. but the possession of more information may sometimes increase and sometimes decrease the value (in the positive definite sense) of the covariance matrix. To demonstrate this we first obtain expressions for the covariance under full and partial information (general and special cases) and then compare covariances for: (a) full information compared with partial information: (b) partial information (knowledge at time r of al) I, and z%for s < r. plus it‘,) as compared with knowledge at time t ofonly s, and :s for s < t. Throughout this section we assume that II:. 112and L‘, are uncorrelated white noise processes. For the case of full information the result is as follows: Thrown

.? (fir/l ir#nwurtiou)

Under full information. are given by Z, = CZ,C’

cov(:,) = Z, and cov(s,) = XF

+ L-, , + TCrlr T’

(75)

Xf= M,L(M,,Z,M:~A,‘hl,zl;zrM:zA~

‘)M,( (76)

where (replacing Equation (38)) then shown to take the form

our solution

can be

in the full information (35). Et+, =A~,+fr~-Bl/~-G’J(1I~,-Ei~,-~,))

case. C is as defined for Equation

(73) Pmf’ (74)

where I,‘,

=

J=

E:, + (D - E

)Z,- Flff +

I’,

The result follows immediately from Equations (20) and (21). For the case of partial information the results are as in the following theorems (where all covariances are asymptotic):

PD’

Theoret~l 3 (prrrt iul infi)rmlfiot1.

-G;;GZ,PD’+G;;LrzzF’

Under partial information (general case) cov(z,) = Z, and cov(.~,)=X~ are given by Z,= P+ &I where P satisfies the Ricatti Equation (46), k satisfies

(DPD’+ V+Fc’,2F’)-’ and P satisfies the Ricatti

Equation

(57).

qrtwrul

ctr.w)

iii=C&‘C7+(CPD7-+TCizzF1)A(DPC7+FU22T7) (77)

100

ECONOMIC

MODELLING

April 1986

Rariorwl expecrarions models with parrial ir!fivmufiorl: J. Peurlmu~~ et a1

Proof

where (we recall)

3 verifying that theorem Proceed as in E(:r+*U:J)= u22. We now turn to the comparison of covariances under the three information sets already given. Consider first Zp and Z,. Subtracting (77) from (75) we have

and X, satisfies X,=G,il(G2,PGS,+C’zz)G;zT+M221M2,~M:1M;: +M221(Ar1MZ2C’2.2FJ-M11PDJ) A(FU,,M;,A;J-DPM;,)M;~ -Gz;l(GZIPDT-

[CPC’ +TCz2TJ

ZF-ZP=C(ZF-ZP)Cr+ - (CPD

L&FT)A(DPG;,

-FL’,,)G;,T

+ TC’22Fr )A(DPCT + FL’?,T’))

- jAPA’+BL&B’

-(APDJ+BC~,2FJ)

(78) A(DPA’+ Matrices

C. D. 7: F, are as defined for Equation

Proof To calculate cov(:,) we need E(:,Z:) which we show to be equal to P. Write P=E(:,+ li:+I). Then from Equations (44) and (45) we can show that P=Ck4T-CiSDTA(FC’,,BJ+DPAJ)+(A-C)PTAT -(A -C)PDJA(FL’,2BT

F&B’))

(81)

(35).

+DPAJ)+BU,,BT

+ c:,, - BC’22FTA(FL22BJ+DPAJ) Then substituting P= P we obtain Equation (46) proving that P= P. After some algebra we can show from Equation (44) that Zp=CZpC7-CPC’+P+(CPDJ+TL~22FT)

The two terms in curly brackets can be shown to be positive semi-definite, so if the first is greater (in the positive definite sense) than the second. then Z, > Z,,. ie partial information reduces variances. Conversely if the second term is greater than the first then Z, > Z, . Both these cases are possible. Trivial sufficient conditionsforZ,>Z,are,4=B=O(ieG,,=G,,=O) C=T=O (ie and for Zf =CZ,CT-CC~22C7+.41?‘,,A’

A(DPC’ +FCz2TT)

+BL~22BJU,,

+U,, (89)

(79) Subtracting

Then substituting Zp= P+%l we arrive at Equation (77). The result Equation (78) is similarly obtained using Equation (47).

Equation

(80: from (79) gives

Z,>,-Z,> =C(Z,> -Z,>,)C’

+ (CC’ II D’ + TU

22

F’-)

A(DCT11C7+FL1Tf) Theorem

3 (ptrrtia/

ir!fi,rrwtior~,

For the special case. cov(:,)=Z,,.

speck/

cuse)

is given by Once again the sign of Z,,, -Z,, similar result holds for X,, -X,,

Zp=CZ,>.CJ-CC.22CT+AC’,,A7

is ambiguous

and a

.

+(CC,,DT+TL’22FJ)A(DC’,,CJ+FU22FJ)

References

-(AC,,D7+BC’22FT)A(DUI,AT+F&2B7-) + BC’,,B’ whilst Xp =X, and fi by Zp-

ECONOMIC

+ L‘, ,

in theorem U12.

MODELLING

(80) 3 with P replaced

1 M. Aoki. Opfind Conf rd ud .S~~srrn~ Theory it? D)vwnic Economic Anulysis. North-Holland, Amsterdam, 1978. 2 R. J. Barro, ‘A stochastic equilibrium model of an open

by I/,, 3

April 1986

economy under flexible exchange rates’, Quurterlp Journal qf Economics, Vol 92. 1978, pp 149-164. J. S. Bhandari, ‘Informational efficiency and the open

101

Rational

expectations

models with partial

infbrmation:

J. Pear/man

economy’, 4

Journal qf Money, Credit und Banking. Vol 14, 1982, pp 457478. 0. J. Blanchard and C. M. Kahn, ‘The solution of linear

difference

models under rational expectations’, Vol 48. 1980, pp 1305-1309. E.. Burmeister and K. D. Wall, ‘Kalman filtering estimation of unobserved rational expectations with an application to the German hyperinflation’, Journal of Econometrics. Vol 20. 1982. pp 255-284. D. A. Currie and P. L. Levine, ‘Simple macropolicy rules for the open economy’, The Economic Journal, Supplement, Vol 96, 1985, pp 6&70. N. W. Duck, ‘Prices. output and the balance of payments in an open economy with rational expectations’. Journal qfInrernurionu/ Ecortomics, Vol 16, 1984, pp 59-77. R. Ford, ‘Exchange rate and trade flow equilibrium in a stochastic macro model’, Canudiun Journul of Economics, Vol 15, 1982, pp 294307. C. A. Futia. ‘Rational expectations in stationary linear models’. Econometrica, Vol 49, 1981, pp 171-192. R. G. Harris and D. D. Purvis, ‘Diverse information and market efficiency in a monetary model of the exchange rate, Economic Journul, Vol 91. 1981, pp 829-847. J. Johnston, Economefric Merhods, McGraw-Hill, New York. 1963. K. P. Kimbrough. ‘The information content of the exchange rate and the stability of real output under alternative exchange rate regimes’. Journul of Infernuthu~ Morle~~ und Finunce. Vol 2. 1983, pp 27-38. K. P. Kimbrough, ‘Exchange rate policy and monetary information’. Journul o/’ Inrerrwrionul Monet und Fimwe. Vol 2. 1983. pp 333-346. K. P. Kimbrough. ‘Aggregate information and the role of monetary policy in an open economy’, Journal qf Pohictrl Econon~~, Vol 93, 1984. pp 268-285. Econometrica,

5

6

7

8

9

10

11 12

13

14

et al 15

R. G. King, ‘Monetary policy and the information content of prices’, Journal qf Political Economic. Vol 90. 1982. pp 247-279. 16 C. Lawrence. ‘The role of information and the international business cycle’, Journul qf Inrernufionul Economics. vol 17, 1984, pp 101-120. 17 P. L. Levine and D. A. Currie. ‘Optimal feedback rules in economy macromodel with an open rational expectations, Tile Europeun Economic Reriew. Vol 27. 1985, pp 141-163. 18 A. P. L. Minford and D. A. Peel, ‘Some implications of partial information sets in macroeconomic models embodying rational expectations’. T/IQ Munchesrer School. Vol 5 1. 1983. pp 235-249. 19 N. H. Saidi, ‘Fluctuating exchange rates and the international transmission of economic disturbances’. Journal o/‘ Morle~~, Credit and Banking, Vol 12. 1980, pp 575-591. 20 T. J. Sargent and N. Wallace. ‘Rational expectations. the optimal monetary instrument and the optimal money supply rule’, Journul ~~f‘Po/iricu~ Economic, Vol 83, 1975. pp 241-254. 21 J. J. Siegel. ‘Monetary stabilization and the information value of monetary aggregates’. Journul of‘ Poliricul Economic, Vol 90, 1982, pp 177-180. 22 R. M. Townsend. ‘Forecasting the Forecasts of others’. Journul of‘Polirictr/ Ecowmj: Vol 91. 1983, pp 546-588. 23 R. M. Townsend, ‘Equilibrium theory with learning and disparate expectations: some incomes and methods’, in R. Frydman and E. S. Phelps. eds. lndir~idutrl Forecusting cmd Aggregurr Ourromes. Cambridge University’ Press. Cambridge, 1983. 24 L. Weiss, ‘Information aggregation and policy’. Rerie\c q/‘Economic S!udies, Vol 49. 1982.

Appendix 1 The Kalman filter Equations (55) and (56) Our dynamic system consists of Equations which we recall to be

(41). (53) and (54)

where

Pr= ;-,.,-I :, + , ,, = C:,,, , + WI \I,,- E:,,,

,J

:, + , = zz, + qI

(41’) (53’)

[ /$j=

=r

c

I

0

with observations N=[E

\(‘,= .v:, + E:,,,

, + 1‘,

I

w/v z

I

N]

(54’) and

where C. Ml %. K. E. c)=cov((/,) and R=cov(r,) earlier. We may rewrite (41’). (53’) and (54’) as

are given

(Al) (A3

102

Then thelilteringequationsfor standard results

ECONOMIC

(Al)and(AZ)aregiven

MODELLING

April

by the

1986

Rationul expectations models with partial i~jbrmation: J. Pearlman et a)

p,+,.,=Mp,.,-,+(MPN+S)(NPN’+R)-‘(w,-Np, ,,-,1 I

(A3) and p,.r=pr,,-l

+phrT(NPNT+R)-‘(w,-Np,.,_,)

(~44)

where p satisfies the Ricatti equation recalling that S =cov(r,,q,). Substituting Equation (A5) it is easy to verify that

P=MPM~+Q-(MPN~+.T) (NPNT+R)-‘(NPM’fS’)

(As) p=

with

0 = cov(ij,), R = cov(r,) and .?=cov(j,,r,).

0

0

0

P

for M, N. L?and Q in

LI

where P is given by Equation (57). Then Equations (A4) give Equations (55) and (56) as required.

ie

(A3) and

Appendix 2 Derivation of the partial information special case solution from the general case, and the full information solution from the special case Special case from general case Equations (23) and (25) with partial information set as in Equation (24) can be set up in the form of the general case partial information set of Equation (37) by enlarging the z,+, vector to include z, and x,. This results in the system: The dotted lines indicate the break-up of the matrices into G,,. G12. G,,. G,,. fi, ,. fi,?. fi2,. Hz2. K,. K2. L,. L2. Note that

so that M(G + fi)= A.11 is satisfied

where

where IV and A are as in Equation (14). This means that fGf;;21M2, =[O 0 M,!Mz,]. Hence using the definitions of A. B. C. D. E and F in Equation (35) we obtain

and

ECONOMIC

by

MODELLING

-1

0

0

0

I

0

0

0

K,!

4pril

1986

10

lo I

K2

1

0

0

A= [ 0

0

0

0

0

0

-1 0 d

0

0

0

I

0

0

0

0

0

D

c= [

103

Ratior~al expectations

ii:

;

Pearlnml

models with purtial information: J

;]

F=

11

et al

+$Mj

It is now possible to show that the solution p of the Riccati Equation (46) with all variables in it having a bar above them is

+

0 , - c’, ,07 YDuT,)Sr+

P=S(c’,

5

(C,2-L’,,F7(li+FC’,,F’)-‘FC’22,[o

M,‘A12,-G,1G2, .A -c

+

G;:B’]

0

0

0

0

i! 0

0

G;z’G2,

0

c’,,

::,

C-A

-.M;2’Rf,,

1

0

0 0

I

0

0

+oo :

0

0

G;; H

00

0

0

I (I-L~,,D7YD)G,2

1

where I s=

-G;;G,, I

-G,ilL’,,F7(V+Fb’22FT)-‘D

A - BCz2FT( V+ FU22FT)-‘D

and Y=(FC’22F’+DL’,,DT+ V)-‘. It can then be shown that the following needed in the sequel. are given by PD’ (DPb’ +FL.22F’

L

1

F7Y]

two expressions.

I

+ ?)- ’ =

I

0

0

0

I

0

(I-I’,,D’YD)G,,

(T-L’,,D’YD)G,2

U,,D’Y

I (A6)

F.T((DPD+ FI:22F’

+ c’)- ’ =

[-F’l.DG,,Furthermore.

Using Equation

F’ YDG,z Fr Y]

(A7)

This follows from [G, ,G,,-I]B=Oand Equation (45): :,+, =ilf,-&;+ti;

,= -11; the

(AX) identities

, reduces to

-, +,=C:,-(C-A)(I-L’,,D’YD)tr;_,

[G,,G,2-I],?=0 and to the barredformof

+(T-B)C’,zFrY](-Fttz+~,)-Bt(z+t,: which is equivalent

to Equation

(35).

their application

-(APD’+BC’,,~7)A(D-,--~,:+t.,)

Using Equations (A6) and (A7) the barred form of Equation (44) can be written as

104

off,,

+(T-B)~‘,,F’YDI~:~,+[(C-A)~,,D’Y

we note that

[G,,G,2-I]:,.

(AX). the third component

Full information solution from special case When there is full information. variables can be written as

ECONOMIC

the set ofcurrently

MODELLING

observable

April

1986

Rarional erpecrarions

models with partial infirmatior~: J. Pearlnw~ et al

and

ie

Y-=(DC:,,DT+FU,,F7+v)-‘= K,=[J

&=[J

L,=[J

L,=[J

v=o

Furthermore, since z,,, =I, and s,.,= x,, it follows matrix H in Equation (23) satisfies H = 0. Thus B=G,2G2;L

A=G,,-G,,G,;G2,

L’;,’ +G;,C-‘G2, 22 G;,C,‘G2, L

-G,,M,;M2,

F=

ECONOMIC

D=[

- GLIGJ

Noting that UllDrY=[I Equation (35) reduces to

MODELLING

April 1986

GLWG,,

I

0] and

L’,,F7Y=[G,,G22].

~,+l=C~,+(A-C)~~_,+~~-B~~,Z O]+(B-TI[GI,

GJ)

(L_G~~G*~~‘~:~‘-[~l’]“i)=C~‘+u~-~’i This is equivalent Equation (21).

T=G,zM,1A;‘M22

22

that the

-((A-C)[I C=G,,

G;,C;;G

to the

full information

solution

of

105