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 nonpredetermined 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
02649993/86/
[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
bl
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.
Fullinformation 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 (11111)x 1 vector predetermined at time t: x, is an 111x 1 vector of variables nonpredetermined 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 longrun 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 saddlepoint property if a unique nonexplosive 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 saddlepoint 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 ncnsingular. We discuss the significance of this assumption. and the consequences of singularity or nearsingularity 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 nonexplosive
=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 rewritten
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