Learning About Rational Expectations Solutions

This paper by Bennett McCallum extends the work of a colleague, George Evans, on the least-squares learnability of rational expectations solutions. Rational expectations models require agents to understand how to calculate the solution to the model, but that is a very complicated mathematical problem so it is unclear how agents accomplish this task. The question in this work is whether agents can use simple linear learning rules (linear regressions) to learn about the complicated rational expectations solutions.

The paper shows that previous results of Evans and Honkapohja (and others) on the types of models that are learnable pertain to a broad class of models, broader than many might have suspected from the original work. This is important because, as McCallum argues, "learnability (and thus E-stability) should be regarded as a necessary condition for the relevance of a RE equilibrium" and this broadens the class of such models.

Since I don't expect many of you will be interested in wading through the paper which is necessarily technical, or even the introduction, let me highlight the part of the discussion on the relevance of RE equilibria.

Remember that, in RE models, agents are assumed to be able to calculate the solution. Suppose we give agents an ideal learning environment, i.e. they use the correct model, correct estimator, the structure is invariant, and so on. In such an idealized world, if agents cannot learn the RE equilibrium, then it is very unlikely they would be able to learn about it in a more complicated set-up. This helps us determine which models are useful representations of the economy. Models with RE solutions that cannot be learned in an ideal world are generally uninteresting and can be set aside:

The position that learnability (and thus E-stability) should be regarded as a necessary condition for the relevance of a RE equilibrium begins with the presumption that individual agents must somehow learn the magnitudes of parameters describing the economy’s law of motion from observations generated by the economy; they cannot be endowed with such knowledge by magic. Of course any particular learning scheme might be incorrect in its depiction of actual learning behavior.

But in this regard it is important to note that the LS learning process in question assumes that (i) agents are collecting an ever-increasing number of observations on all relevant variables while (ii) the structure is remaining unchanged. Furthermore, (iii) the agents are estimating the relevant unknown parameters (iv) with an appropriate estimator (v) in a properly specified model. Thus if a proposed RE solution is not learnable by the process in question—the one to which the E&H results pertain—then it would seem highly implausible that it could prevail in practice...

While I'm discussing colleagues, I also want to welcome the newest member of our Department, Jeremy Piger, mentioned today by Jim Hamilton in his discussion of the debate over business cycle dating that came in response to a question from Greg Mankiw, among others. I'm pretty happy to have Jeremy as a colleague. Here's the introduction to McCallum's paper:

E-Stability vis-a-vis Determinacy Results for a Broad Class of Linear Rational Expectations Models, by Bennett T. McCallum, NBER Working Paper No. 12441 (alt. link): 1. Introduction Much recent research in economics, especially in monetary economics, has focused on issues involving analytical indeterminacy—multiplicity of stable rational expectations solutions—often in dynamic general equilibrium models based on optimizing behavior by individual agents.1

In this context, the recent appearance of major publications by Evans and Honkapohja (1999, 2001) has stimulated new interest in the concept of E-stability, developed by DeCanio (1979), Evans (1985, 1986, 1989), and Evans and Honkapohja (1992).2

The reason is that E-stability is very closely linked with least-squares learnability, and the latter is arguably a necessary property for a rational expectations solution to be plausible as an equilibrium for the model at hand.3

In their book, Evans and Honkapohja (henceforth, E&H) provide conditions for E-stability of a class of linear multivariate models, but the class in question might appear to be rather restricted in scope. It is shown below, however, that the E&H specification is in fact quite broad, in the sense that essentially any model of the class analyzed by King and Watson (1998) or Klein (2000) can be represented in the implied form. It follows that analytical results shown to hold for the E&H class are actually of quite broad applicability.

In the present paper, consequently, I draw upon results of E&H (1999, 2001) and McCallum (1998) to develop simple proofs of two useful propositions pertaining to this broad class of linear rational expectations (RE) models. The first, Proposition P1, is that if a RE solution is determinate (unique dynamically stable), then it has the property of E-stability (and therefore least squares learnability). The second proposition, P2, is that there exist various cases with a multiplicity of stable4 solutions in which the one based on the decreasing-modulus ordering of the system’s eigenvalues is E-stable. Furthermore, it is a simple matter to determine whether the requisite criteria for E-stability are satisfied.

It should be stated clearly at the outset that all results presented here are based on the assumption that current values of endogenous variables are included in individuals’ information sets; if instead only lagged endogenous variables can be observed in the learning process then different E-stability and learnability results would be relevant.

Analysis of a few particular problems in monetary economics involving the latter specification has been conducted in a well-known paper by Bullard and Mitra (2002)5 while recent papers by Adam (2003) and Adam, Evans, and Honkapohja (2006) have emphasized that differing assumptions about information sets relevant for learning can lead to different conclusions.

The position that learnability (and thus E-stability) should be regarded as a necessary condition for the relevance of a RE equilibrium begins with the presumption that individual agents must somehow learn the magnitudes of parameters describing the economy’s law of motion from observations generated by the economy; they cannot be endowed with such knowledge by magic. Of course any particular learning scheme might be incorrect in its depiction of actual learning behavior.

But in this regard it is important to note that the LS learning process in question assumes that (i) agents are collecting an ever-increasing number of observations on all relevant variables while (ii) the structure is remaining unchanged. Furthermore, (iii) the agents are estimating the relevant unknown parameters (iv) with an appropriate estimator (v) in a properly specified model. Thus if a proposed RE solution is not learnable by the process in question—the one to which the E&H results pertain—then it would seem highly implausible that it could prevail in practice...

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1 In monetary economics such issues include indeterminacy under inflation forecast targeting  (Woodford, 1994; Bernanke and Woodford, 1997; King, 2000), deflationary traps (Benhabib, Schmitt-Grohe, and Uribe, 2001), the fiscal theory of the price level (Sims, 1994; Woodford, 1995; Cochrane, 1998; Kotcherlakota and Phelan, 1999; McCallum, 2001), and the validity of the “Taylor Principle” (Woodford, 2003). For a useful overview of several related points, see Bullard and Mitra (2002).

2 Evans and Honkapohja (1999) is an extensive survey article in the Taylor-Woodford Handbook of Macroeconomics, whereas their (2001) is a major treatise published by Princeton University Press.

3 This position is developed on pp. 2-3, while Appendix A briefly reviews relevant concepts.

4 Throughout, the unmodified word “stable” will refer to the presence or absence of dynamic stability of the rational expectations solution in question, not the learning process or the meta-time concept of E-stability.

5 Bullard and Mitra (2002) obtain a result analogous to P1, under the current-information condition, for a particular three-variable model relating to monetary policy, whereas Evans and Guesnerie (2003) present the same result for cases in which A11 is nonsingular and Evans and Honkapohja (2003) for cases in which C = 0. In none of these papers is broad applicability claimed.