M.C. Chang et al. International Review of Economics and Finance 9 2000 123–137 125
2. The stability beyond the local sense: A general framework
Let z
t
P R
k
be a k 3 1 the state vector of the economic system. The actual law of motion for z
t
is assumed to follow the linear stochastic difference equation: z
t
5 A
b
t
z
t
2
1
1 B
b
t
u
t
, 1
where u
t
is an m 3 1 vector white noise that is orthogonal to z
t2j
for j . 0, A b
t
and B
b
t
are matrices with the dimension k 3 k and k 3 m, respectively. Let z
1
t P R
k
1
and z
2t
P R
k
2
be subvectors of z
t
. Here z
1t
and z
2t
are, respectively, those variables whose future values agents care about and those used by agents to predict future
values of z
1t
. At time t, suppose agents believe that z
1,t
1
1
evolves according to z
1,t
1
1
5 z9
2t
b 1 v
t
1
1
, where v
t
1
1
is a vector white noise orthogonal to z
2t
. Let b
t
be the estimate of b at time t. The role of economic theory is to induce the mapping Ab
t
and Bb
t
, which determine the actual law of motion for the forecast of z
1,t
1
1
. For a given set of perceived values of b, it is possible to use the actual law of
motion [Eq. 1] to deduce the least squares projection of z
1,t
1
1
on z
2t
: E
ˆ [z
1,t
1
1
|z
2t
] 5 z9
2t
T b,
where Tb is determined by the least squares learning mechanism. A rational expecta- tions equilibrium is a stationary point of the mapping Tb → b, denoted b
f
. 2.1. The ODE approach
As agents use the least squares mechanism to learn about the actual law of motion, their belief b
t
evolves according to the following recursive formula: b
t
5 b
t
2
1
1 a
t
t R
2
1 t
2
1
z
2,t
2
1
[z9
1,t
2
1
2 z9
2,t
2
2
b
t
2
1
], 2a
R
t
5 R
t
2
1
1 a
t
t
3
z
2,t
2
1
z9
2t
2
1
2 R
t
2
1
a
t
4
, 2b
in which a
t
is a non-decreasing weight sequence of positive real numbers and R
t
is the estimate of the second moment of z
2
t at time t. When the value of b
t
, R
t
generated from the recursive formulas falls outside the set of reasonable values D
1
, agents project the observed vector b
t
, R
t
onto a most likely region D
2
. Formally, the projection facility can be described as
b
t
, R
t
5
5
b
t
,R
t
, if b
t
, R
t
P D
1
, some point in D
2
, otherwise.
2c To apply Ljung’s ODE approach, we need to assume that D
1
is an open bounded set containing the compact set D
2
Ljung, 1977, Theorem 4 throughout this article. The use of Ljung’s ODE approach in the study of the limiting behavior of b
t
and R
t
requires the mechanical transformation of the stochastic difference equations into the approximating ordinary differential equations:
3
b˙ t
R ˙ t
4
5
3
R t
2
1
M bt[Tbt 2 bt]
M bt2 Rt
4
, 3
126 M.C. Chang et al. International Review of Economics and Finance 9 2000 123–137
in which y˙t ; dytdt and Mb 5 E[z
2t
bz
2t
b9]. Here and henceforth, let yt denote the variable generated by an ordinary differential equation, and y
t
the one generated by the corresponding stochastic difference equation. Since Mb is positive
semi-definite for all t, Rt is positive definite so long as R0 is positive definite. Let R
f
5 M
b
f
and D
A
be a domain of attraction of b
f
, R
f
, a stationary point of Eq. 3. According to Ljung 1977, p. 567, if D
A
is not a singleton set, then there exists a Lyapunov function with the following properties:
b1. Vb, R is infinitely differentiable, b2. 1 . Vb, R 0 for any b, R P D
A
and Vb
f
, R
f
5 0, and b3. when evaluating along the solution of Eq. 3, dVbt, Rtdt 0 for any
bt, Rt P D
A
, and the equality holds only when bt, Rt 5 b
f
, R
f
. According to Hartman 1973, p. 548, if b
f
, R
f
is a unique stationary point and is locally stable, then D
A
is an open set so long as the domain of ordinary differential equation is open.
The central proposition of least squares learning convergence can be stated as follows, which is a slight variation of Theorem 4 in Ljung 1977:
Proposition 1. Let b
t
, R
t
be generated by the learning mechanism Eqs. 1 and 2. Suppose the five regularity conditions a1–a5 given in Appendix 1 hold. If
c1. z
t
is bounded infinitely often with probability one, and c2. b
f
, R
f
P D
2
, D
1
, D
A
, b
f
, R
f
is not on the boundary of D
2
, and there exists a Lyapunov function with properties b1–b3 such that Vb
a
, R
a
. V
b
b
, R
b
for any b
a
, R
a
Ó D
1
and b
b
, R
b
P D
2
, then b
t
, R
t
→ b
f
, R
f
almost surely as t → ∞
. Proposition 1 states that the limiting behavior of bt, Rt in 3 mimics the
limiting behavior of b
t
, R
t
in Eq. 2 under certain conditions. Notice that the sets D
1
and D
2
regulate the evolution of b
t
and R
t
in the stochastic difference equation, while the maximum domain of attraction D
A
determines the trajectory of bt and R
t in the ordinary differential equation. 2.2. The problem with the traditional definition of global stability
Three things have to be accomplished in the stability analysis of the least squares learning mechanism using Ljung’s ODE approach. First, we have to obtain a set of
restrictions on initial beliefs in which z
t
is bounded infinitely often with probability one. Second, we have to find the largest domain of attraction of the stationary point
of Eq. 3, b
f
, R
f
. Third and most importantly, we have to have a projection facility 2c so that the equivalence between the stochastic difference equations, Eqs. 2a–2c,
and the ordinary differential equations, Eq. 3, can be preserved. It is the projection facility that often blurred the notion of global stability. The
following example can illustrate our point. Even though D
1
and D
2
usually cannot be specified by economic theory, the definition of stability requires the specification of
these two sets. One natural candidate for the definition could be stated as follows.
M.C. Chang et al. International Review of Economics and Finance 9 2000 123–137 127
Fig. 1. The domain of attraction and projection facility.
The learning mechanism is said to be globally stable if b
t
, R
t
→ b
f
, R
f
for any bounded open set D
1
and any compact set D
2
such that D
2
, D
1
. Imagine that the solution of the ordinary differential equations in Eq. 3 is globally stable and Vb,
R is a Lyapunov function. As shown in Fig. 1, we can always find D
1
and D
2
such that D
2
, V
1
? 0 and V
2
, D
1
? 0, in which V
1
5 {b, R|Vb, R , c
1
} and V
2
5 {b, R|Vb, R , c
2
} with 0 , c
2
, c
1
. Consider a point in D
2
, say b. The trajectory of differential Eq. 3 may move point b to point a. This movement leads
to a value of Lyapunov function less than that in b, whereas point a following the learning mechanism [Eq. 2] will be projected back to D
2
. That is, even if the differen- tial equations system is globally stable, it is always possible to find a pair of D
1
and D
2
such that the projection facility interferes with the trajectory of ordinary differential equations. As a result, the trajectory of Eq. 3 no longer mimics that of the learning
mechanism characterized by Eq. 2.
2
Thus, without further restricting the ranges of D
1
and D
2
, the above definition of global stability is not operational. 2.3. An operational definition of global stability
To assure the legitimate correspondence between the ordinary differential equation system [Eq. 3] and the stochastic difference equation system [Eq. 2], we must
impose further restrictions on the sets D
1
and D
2
: Definition 1: Consider the system Eq. 1 and Eq. 2, and the ordinary differential
equations, Eq. 3 with the largest domain of attraction D
A
. Further, consider a connected set D which contains b
f
, R
f
and is a subset of D
A
. If for any neighborhood of b
f
, R
f
denoted 1b
f
, R
f
which is bounded and connected, and whose closure is contained in D, there exist D
1
. 1 b
f
, R
f
and D
2
. 1 b
f
, R
f
such that b
t
, R
t
converges to b
f
, R
f
almost surely,
3
then D is called an ODE-stable set for Eq. 1 and Eqs. 2a–2b.
This definition states that when D is an ODE-stable set, it is certain that there is a projection facility which guarantees the almost sure convergence of the estimate
and D
2
is “large” enough so that it can approximate D. How economic agents choose their projection sets D
1
and D
2
is beyond the scope of economic analysis. What we
128 M.C. Chang et al. International Review of Economics and Finance 9 2000 123–137
do is to specify the range in which D
1
and D
2
can be chosen appropriately to guarantee the stability of learning mechanism.
D must be restricted to be contained in D
A
, for otherwise the ordinary differential equations in Eq. 3 are unstable, which is clearly incompatible with the logic of ODE
stability. If the focus is on the local stability of Eq. 3, then the use of Eq. 3 is restricted to the stability analysis of Eqs. 2a–2b in the local sense. Only when Eq.
3 has a “more-than-local” domain of attraction can we use the ODE approach to discuss the stability of the learning mechanism in the large. The initial condition b
, R
does not appear in the above definitions because the projection facility Eq. 2c could project the outlying point b
, R back to D
2
. Definition 2: D , D
A
is called the maximum ODE-stable set if it is an ODE-stable set, and any other ODE-stable set is contained in D.
Definition 3: A learning mechanism is called ODE globally stable if D is the universal set.
The empirical relevance of the maximum ODE-stable set can be interpreted as follows. Suppose that an ODE-stable set D is very small, then the legitimate range
where D
1
and D
2
can be chosen is much restricted. Therefore, we do not expect that economic agents can perceive D
1
and D
2
accurately, and the ODE-stability of the learning mechanism cannot be assured.
The following proposition gives us the conditions which should be verified for a ODE-stable set.
Proposition 2: If regularity conditions a1–a4 in Appendix 1 are satisfied, and if the following conditions hold:
a5. the characteristic roots of Ab are all less than one in modulus for b, R
P D , D
A
, and c1. for any compact subset of D denoted D
b
, if b
t
, R
t
P D
b
for all t, z
t
generated by Eq. 1 is bounded infinitely often with probability one, then D is an ODE-stable set.
Proof: Because D
A
is a domain of attraction of b
f
, R
f
for Eq. 3, there is a Lyapunov function Vbt, Rt satisfying b1–b3. Consider a neighborhood of b
f
, R
f
denoted 1b
f
, R
f
, whose closure is entirely in D. We first select a connected compact set D
2
such that 1b
f
, R
f
, D
2
, D , and then pick D
1
5 {b, R|Vb, R ,
k } in which
max
D
2
V b, R , k , 1.
According to properties b2 and b3 of Vb, R, max
D
2
V b, R , 1, which guarantees
the existence of k. Under the above choice of D
1
and D
2
, condition c2 in Proposition 1 is satisfied, and hence Proposition 1 applies, QED.
It follows from the proof that if D is an open ODE-stable set then the closure of D
is also an ODE-stable set as long as the closure of D is also a subset of D
A
. Therefore, an ODE-stable set cannot be guaranteed to be either open or closed. As
M.C. Chang et al. International Review of Economics and Finance 9 2000 123–137 129
to the regularity conditions listed in the Appendix 1, most of them are not particularly important in our discussion. Specifically, a2 and a4 are not related to the parametric
specification of the model and a1 and a3 cannot be violated for a well-behaved economic system. Thus, what remain to be verified are c1 and a5. This is why in
Proposition 2 we explicitly single out a5 and c1 to emphasize that these two conditions are the ones need to be verified in applications.
4
2.4. A scalar parameter case If the focus is on the local stability of the learning mechanism, Marcet and Sargent
1989a show that the local stability of Eq. 3 is determined by the simpler differential equation:
b˙ t 5 Tbt 2 bt.
4 The advantage of studying Eq. 4 is that it has fewer dimensions and therefore is
easier to analyze than Eq. 3. However, the local connection between the characteristic roots of Eq. 3 and those of Eq. 4 does not necessarily establish the connection
between the domain of attraction for Eq. 3 and that for Eq. 4, which is necessary for the global stability analysis. The following proposition establishes that, when b
t
is a scalar, the domain of attraction for the simpler differential equations of Eq. 4 determines those of Eq. 3.
Proposition 3: Consider the systems Eq. 3 and Eq. 4 in which b t is a scalar. If there is a domain of attraction of b
f
denoted D
A
b
for the ordinary differential equation Eq. 4, then D
A
b
3 5
1
is the domain of attraction of b
f
, R
f
for the ordinary differential equations Eq. 3.
Proof: Since D
A
b
is the domain of attraction of b
f
for Eq. 4, there exists a Lyapunov function Vbt with properties similar to b1–b3. More specifically, when
evaluated along the solution path of Eq. 4, we have dV
btdt 5 V9bt[Tbt 2 bt] 0, ∀
b t P D
A
b
. The equality holds only when bt 5 b
f
. Now we want to search for a Lyapunov function Wbt, Rt for Eq. 3. It is natural to try
W bt, Rt
; Vbt. Then we check whether the W·, · function so defined is a Lyapunov function of Eq.
3. Suppose that bt, Rt is the trajectory of Eq. 3 and bt, Rt P D
A
b
3 5
1
. Then we see that dW
dt ; dVbtdt 5 V9btRt
2
1
M bt[Tbt 2 bt] 0.
The above inequality holds because Mb . 0 and Rt
2
1
. 0 by assumption, and the
equality holds only when bt 5 b
f
. This shows that the Lyapunov function Vbt for the differential equation Eq. 4 determines the convergence of bt, Rt to b
f
, R
f
for Rt . 0. According to Theorem 26.1 in Hahn 1967, any trajectory of Eq.
130 M.C. Chang et al. International Review of Economics and Finance 9 2000 123–137
3 starting from D
A
b
3 5
1
will eventually fall inside {b
f
, R:R . 0}. Once this happens, it is easy to see that bt, Rt → b
f
, R
f
as t → ∞
; QED.
3. Deriving the maximum stable set: Some examples
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