International Review of Economics and Finance 9 2000 123–137
Initial beliefs and the global stability of least squares learning
M. C. Chang
a
, C. Y. Cyrus Chu
b,c
, Kenneth S. Lin
b,
a
Graduate Institute of Industrial Economics, National Central University, Chung-li, Taiwan
b
Department of Economics, National Taiwan University, Taipei, Taiwan
c
Institute of Economics, Academia Sinica, Taipei, Taiwan Received 20 January 1998; accepted 31 March 1999
Abstract
This article provides an operational framework of global stability analysis when Ljung’s 1977 ordinary differential equation ODE approach is applied to the recursive stochastic
system. We first establish the notion of stable set under which sufficient conditions for the equivalence between them can be translated to restrictions on initial beliefs in agents’ forecasts.
The maximum ODE-stable set is simply the largest range of initial beliefs. We then demonstrate how to implement this operational framework for the analysis of stability beyond the local
sense in some well-known models in the learning literature.
2000 Elsevier Science Inc. All
rights reserved.
JEL classification: D83
Keywords: Least squares learning; Global stability; Initial belief
1. Introduction
Many recent researchers applied Ljung’s 1977 ordinary differential equation ODE approach to study the convergence of the least squares learning mechanism.
1
When a model satisfies Ljung’s regularity conditions, the local stability of the recursive stochastic system generated by the learning mechanism would be governed by a simpler
ordinary differential equation system. However, one drawback of this approach is that the stability of least squares learning mechanism beyond the local sense is difficult
to analyze. As Marcet and Sargent 1989a, p. 360 pointed out, sufficient conditions
Corresponding author. Tel.: 102-2341-8190. E-mail address
: kslinccms.ntu.edu.tw K.S. Lin 1059-056000 – see front matter
2000 Elsevier Science Inc. All rights reserved.
PII: S1059-05609900047-7
124 M.C. Chang et al. International Review of Economics and Finance 9 2000 123–137
for global convergence are “much more difficult to analyze in cases where agents include endogenous variables among their regressors.”
In this article, we propose an operational framework of global stability analysis when Ljung’s ODE approach is adopted. We first establish the notion of stable set
under which sufficient conditions for the equivalence between the recursive stochastic system and corresponding ordinary differential equations can be translated to restric-
tions on initial beliefs in the perceived law of motion for agents’ forecasts. The maximum ODE-stable set
is simply the largest range of initial beliefs so that one can use Ljung’s ODE approach to obtain the stability of the least squares learning
mechanism beyond the local sense. The explicit characterization of the maximum stable set is important and useful since it helps us understand the empirical significance
of the least squares learning mechanism, and recognize the justifiable range of compar- ative statics.
We will demonstrate that the maximum ODE-stable set is characterized by the combination of the following three conditions: 1 the autoregressive coefficient matrix
in the state variable transition rule must have eigenvalues all less than one in modulus; 2 the state variables must be bounded infinitely often with probability one; and 3
the differential equations which govern the original stochastic difference equations must be stable in the relevant range. Each of these three conditions places restrictions
on initial beliefs in the perceived law of motion for agents’ forecasts. If the maximum stable set turns out to be a universal set, then the learning mechanism under study
is globally stable.
To establish the equivalence between the stability of difference equation system and that of differential equation system, we need regularity conditions governing the
evolution of parameters in agent’s forecasts. For instance, as argued in Ljung 1987, p. 57, any approach such as the ODE approach and the martingale approach that
describes the behavior of a recursive stochastic system with the “small step” techniques requires the boundedness of state variables in the system as described in the second
condition. Otherwise, the equivalence between the solution of this system and the trajectory of corresponding ordinary differential equations can be easily destroyed by
an immediate jump of parameters in agents’ forecasts. Further, because any stochastic mechanism using small step techniques does not have its own mechanism to keep
relevant parameters on a desirable region, we also need an auxiliary facility that projects diverging parameters in agents’ forecasts back to the region. In this article,
we demonstrate that even if the differential equation system is globally stable, it is alway possible to find the projection facility that interferes with the trajectory of
differential equation system. Thus, without appropriate regulations on the projection facility, the traditional definition of global stability cannot be operational.
The remainder of this paper is organized as follows. In section 2, we point out the problem of the traditional definition of global stability, present the notion of ODE
stable set, and provide several propositions regarding the strong convergence of the learning mechanism. In section 3, we study a stationary version of Cagan’s 1956
model to show how we analyze its global stability. We also use Frydman’s 1982 model of market processes and Bray’s 1982 model of asset pricing to illustrate
different types of ODE stability. Section 4 provides concluding remarks.
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
