# Brays’ Rational Expectations

Margaret Bray *Learning, Estimation, and the Stability of **Rational Expectations * JOURNAL OF ECONOMIC THEORY 26, 318-339 (1982).

## Abstract

The stability of the rational expectations equilibrium of a simple asset market model is studied in a situation where a group of traders learn about the relationship between the price and return on the asset using ordinary least squares estimation, and then use their estimates in predicting the return from the price. The model which they estimate is a well-specified model of the rational expectations equilibrium, but a misspecified model of the situation in which the traders are learning. It is shown that for appropriate values of a stability parameter the situation converges almost surely to the rational expectations equilibrium.

## 1. Introduction

The difficulty is that in many models with rational expectations equilibria … the objective distribution of variables depend upon agents’ subjective beliefs about the distribution.

Fully rational agents should estimate a correctly specified model, which takes into account the feedback from forecasts to outcomes. This is likely to entail a complicated learning strategy based on a considerable degree of understanding of the situation. Outside the rational expectations equilibrium, it is not usually rational to use estimation techniques which are based on a correct specification of the rational expectations equilibrium, such as [ordinary least squares] in the model considered here. Nevertheless the use of such techniques might be described as reasonable. The major propositions of this paper establish conditions, for the model presented here, under which ordinary least squares estimation ultimately generates rational expectations.

The model is basically an infinitely repeated version of the [a previous] model of an asset market with informed and uninformed traders. …

Two learning processes are studied. In the first, agents revise the estimates used in forecasting return at infrequent intervals. In the second, they revise the estimates each time a new data point is observed. …

## 2. The Model

Assumption 1

It is common knowledge that [the model parameters form] a sequence of independent, identically distributed, multivariate normal, random variables which are exogenously determined.Assumption 2. [Each informed trader has a specified demand function based on rational expectations].

Assumption 3.

It is common knowledge that… informed traders can learn nothing about [the return on assets] from [the supply] which they do not already know [from their information].Assumption 4.

Each uninformed trader[employ a specific demand function that depends on their forecast returns., based in a specific way on the history of prices and returns.]Assumption 5.

It is common knowledge that the market clears, and that the market clearing price … is determined by supply … information … and the past history of information, price and return … .

## 3. Forecasts and Expectations

DEFINITION. The market is in a

rational expectations equilibriumif it is in temporary equilibrium and if in addition the uninformed traders’ forecasts are the correct conditional expectation of [the returns] given [the history] and [the current price].Proposition 3.

A rational expectations equilibrium exists[ unless the expected return is uninformative in a specified sense].The equilibrium is unique[and satisfies a specified equality].

## 4. Ordinary Least Squares Learning

[The] uninformed agents who use [ordinary least squares learning] fail to take into account the fact that the relationship changes as they learn; their estimation procedure is based upon a misspecification of the situation.

I investigate two different regimes in which the uninformed agents use estimated regression coefficients in forecasting. In the first regime studied traders initially forecast [using fixed linear smoothing of price]. They continue to use this forecasting rule for a long period, during which they run a linear … regression of return on price. … At some date all the uninformed traders simultaneously drop the initial forecasting rule, and adopt [a similar rule but with updated coefficients. They then start to re-estimate the regression coefficients, [repeatedly] changing the forecasting rule … .PROPOSITION 4.

If the estimates are revised periodically, and the forecasting rule after the mth revision is[as above],expectations converge to rationality… .In the second OLS procedure studied, traders revise their estimates each time a new data point is observed.

Proposition 5.

Given[some stated assumptions, then:]In the limit expectations are rational.## 5. The Stability Parameter

[Proposition 3 depends on a condition that I (DJM) have interpreted as the expected return being informative. In the precise model of the paper this is represented by a parameter ‘k’.]

The parameter k is clearly crucial. … [Conditions are given under which the k parameter ensures stability. But:] learning tends to generate instability if either the ratio of uninformed to informed demand is large so the uninformed traders dominate the market, or the equilibrium value of the regression coefficient of price on return … is large.

## 6. Conclusions

The results of this paper suggest that a learning method can eventually yield rational expectations even if it is based upon a misspecication of the mode! in the situation when agents are learning. However-as one might expect-the stability properties of the system are different for different learning procedures, and instability seems to be a real possibility.

These results seem to lend weight to the rational expectations hypothesis. However, it must be borne in mind that expectations are not rational, and indeed are biased due to the misspecification of the model which is estimated, at all finite dates. Rational expectations are, if anything, a long run rather than a short run phenomenon.

## Comments

This paper has both a positive and a negative interpretation. If a market is sufficiently influenced by informed traders acting rationally, and if there is an equilibrium in that market returns are ‘statistically consistent’ with the expectations of the informed traders, then uninformed traders, using purely conventional statistical learning (or a crude approximation) will ‘in the long run’ tend to act as if they had approximately rational expectations. This seems reasonable, even much more generally than is claimed in the paper. It might even be regarded as ‘the normal case’. But there is more to say.

The negative interpretation is to point out the fragility of the assumptions. One has no reason to expect the uninformed to converge to uniquely ‘rational expectations’ (in my words) either:

- The demand in the market is mostly from those relying on a form of naïve induction, rather than any more informed understanding of the market.
- Asset prices are largely driven by returns (as against ‘fundamentals’).

Suppose, for example, that:

- Traders are themselves subject to naïve selective pressure, based on post performance, rather than any deep understanding of the theory.
- That both informed and uninformed traders need to perform similar statistical types of analysis, that is no costly for the informed than the uninformed. (I.e., the information doesn’t simply give the traders the required direction ‘on a plate’.)
- That it is costly to gain the required understanding and maintain the required information.

Then either uninformed or informed traders could have an advantage, depending on the cost/benefit balance from exploiting the information. If there is a sustained period for which the balance is against the informed traders, then they could lose influence, thus creating one of the conditions under which there may be no rational equilibrium, or possibly multiple pseudo-rational equilibria.

Alternatively, under severe naïve selective pressure demand for high-return assets might sky-rocket, thus creating the other condition for instability. Either way, the conditions for stability hardly seem sustainable, without some external mechanism to preserve them.

Stepping outside the study assumptions, if one supposes that genuine innovation is characteristic of vibrant real economies, then such innovation would seem to incompatible with rational expectations. But, turning this around, it does seem that the conclusions might apply to real economies in the medium term. That is, as long as traders believe in rational expectations, trading will tend to converge to an apparently rational expectation. It is just that form time to time one should ‘expect the unexpected’.

In addition to the logical aspects, the use of the terminology ‘rational expectations’ has some psychological aspects. Suppose, for example, that the UK chancellor thought that if he behaved as if he expected x% growth then the actual growth would be (x/2)%. Then it would be uniquely ‘rational’ for him to behave as if he expected 0% growth. But it hardly seems sensible. He would be usefully informed by an adviser telling him what the rational expectation was, but it might be unreasonable to act on it, naively. More generally, it seems reasonable to me for an adviser to identify relevant conditions under which there are no, unique or multiple pseudo-equilibria, and what they are. The existence of a ‘rational equilibrium’ seems a special case, albeit a fairly common one. More generally, the paper assumes that everyone has fixed characterisations (e.g., demand functions), which precludes any strategizing. In practice it might be reasonable to suppose, for example, that traders within large firms acting for institutional clients might act in a formulaic way, but I find it hard to suppose that informed individuals acting on their own behalf would not seek to think for themselves, and that those who do would not gain advantage. So the no-strategizing assumption seems unreliable (and probably false).

All in all, an important antidote to some common rational expectations dogmas.