E-Stability Does Not Imply Learnability

Abstract

Abstract. The concept of E-stability is widely used as a learnability criterion in studies of macroeconomic dynamics with adaptive learning. In this paper, it is demonstrated, via a counterexample, that E-stability does not in general imply learnability of rational expectations equilibria. The result indicates that E-stability may not a robust device for equilibrium selection.JEL Classification: D83, C62 Keywords: Adaptive learning, E-stability, learnability of REE Introduction and BackgroundIn recent years, there has been an explosion in research that studies macroeconomic dynamics with adaptive learning. 1 A key question in this literature is whether it is possible for agents that update their expectations using econometric algorithms (i.e. learn adaptively) to learn the rational expectations equilibrium (REE) as the sample size of their data set increases. This is also known as learnability of an REE. Since conditions for learnability are typically hard to pin down in a direct way, researchers have been looking for (and have often successfully developed) indirect ways for identifying them. One such popular approach is the E-stability criterion.In this paper, I revisit the concept of E-stability and demonstrate, via a counterexample, that E-stability does not always imply learnability of an REE. The example used is a generic reduced form model with expectations dated at time t and a lag of the endogenous variable. In particular, it is shown that, for certain parameter regions for which E-stability holds for a minimal state variable (MSV) solution, there is a learning algorithm (namely stochastic gradient) that does not converge to the solution, i.e. the REE is not learnable. Furthermore, I discuss some examples of economic models that can be expressed in this reduced form.The fact that E-stability may not always be an appropriate learnability criterion is not entirely new to the literature. This possibility has been pointed out by Barucci and Landi (1997) and further explored by Heinemann (2000). Nevertheless, in the former article, there is no example confirming the assertion. The latter article provides an example for which numerical simulations indicate that stability or instability under stochastic gradient learning is independent of the Estability conditions; in particular, it is shown that the stochastic gradient algorithm converges to an E-unstable solution, i.e. that E-stability is not a necessary condition for learnability. In the present paper, it is further shown that there may exist E-stable equilibria that are not learnable by a stochastic gradient algorithm, or in other words, that E-stability is not a sufficient condition for learnability.

Generalized Stochastic Gradient Learning

“…In models with dependence on expectations of future values of the endogenous variables, the correspondence between E-stability and convergence of SG learning no longer holds (see Barucci and Landi, 1997;Heinemann, 2000). Giannitsarou (2005) provides an economic example with lagged endogenous variables in which E-stability of the fundamental REE does not imply convergence of SG learning. 4 It would be possible to extend the analysis to models with lagged endogenous variables.…”

A note on the representative adaptive learning algorithm

“…Importantly, theoretical analyses of learning convergence have shown that these learning algorithms may lead to different learnability conditions of RE equilibria (Heinemann, 2000;Giannitsarou, 2005). The LS dominance also has been challenged in previous applied studies (see Bullard and Eusepi, 2005;Carceles-Poveda and Giannitsarou, 2007).…”

Adaptive learning in practice

“…For a derivation of the matrix see Giannitsarou (2005). REE), there is a positive probability that the estimates f t might not converge tof.…”