Archive for the 'Robert Shiller' Category

Making Sense of Rational Expectations

Almost two months ago I wrote a provocatively titled post about rational expectations, in which I argued against the idea that it is useful to make the rational-expectations assumption in developing a theory of business cycles. The title of the post was probably what led to the start of a thread about my post on the econjobrumors blog, the tenor of which  can be divined from the contribution of one commenter: “Who on earth is Glasner?” But, aside from the attention I received on econjobrumors, I also elicited a response from Scott Sumner

David Glasner has a post criticizing the rational expectations modeling assumption in economics:

What this means is that expectations can be rational only when everyone has identical expectations. If people have divergent expectations, then the expectations of at least some people will necessarily be disappointed — the expectations of both people with differing expectations cannot be simultaneously realized — and those individuals whose expectations have been disappointed will have to revise their plans. But that means that the expectations of those people who were correct were also not rational, because the prices that they expected were not equilibrium prices. So unless all agents have the same expectations about the future, the expectations of no one are rational. Rational expectations are a fixed point, and that fixed point cannot be attained unless everyone shares those expectations.

Beyond that little problem, Mason raises the further problem that, in a rational-expectations equilibrium, it makes no sense to speak of a shock, because the only possible meaning of “shock” in the context of a full intertemporal (aka rational-expectations) equilibrium is a failure of expectations to be realized. But if expectations are not realized, expectations were not rational.

I see two mistakes here. Not everyone must have identical expectations in a world of rational expectations. Now it’s true that there are ratex models where people are simply assumed to have identical expectations, such as representative agent models, but that modeling assumption has nothing to do with rational expectations, per se.

In fact, the rational expectations hypothesis suggests that people form optimal forecasts based on all publicly available information. One of the most famous rational expectations models was Robert Lucas’s model of monetary misperceptions, where people observed local conditions before national data was available. In that model, each agent sees different local prices, and thus forms different expectations about aggregate demand at the national level.

It is true that not all expectations must be identical in a world of rational expectations. The question is whether those expectations are compatible with the equilibrium of the model in which those expectations are embedded. If any of those expectations are incompatible with the equilibrium of the model, then, if agents’ decision are based on their expectations, the model will not arrive at an equilibrium solution. Lucas’s monetary misperception model was a clever effort to tweak the rational-expectations assumption just enough to allow for a temporary disequilibrium. But the attempt was a failure, because Lucas could only generate a one-period deviation from equilibrium, which was too little for the model to pose as a plausible account of a business cycle. That provided Kydland and Prescott the idea to discard Lucas’s monetary misperceptions idea and write their paper on real business cycles without adulterating the rational expectations assumption.

Here’s what Muth said about the rational expectations assumption in the paper in which he introduced “rational expectations” as a modeling strategy.

In order to explain these phenomena, I should like to suggest that expectations, since they are informed predictions of future events, are essentially the same as the predictions of the relevant economic theory. At the risk of confusing this purely descriptive hypothesis with a pronouncement as to what firms ought to do, we call such expectations “rational.”

The hypothesis can be rephrased a little more precisely as follows: that expectations of firms (or, more generally, the subjective probability distribution of outcomes) tend to be distributed, for the same information set, about the prediction of the theory (or the “objective” probability distributions of outcomes).

The hypothesis asserts three things: (1) Information is scarce, and the economic system generally does not waste it. (2) The way expectations are formed depends specifically on the structure of the relevant system describing the economy. (3) A “public prediction,” in the sense of Grunberg and Modigliani, will have no substantial effect on the operation of the economic system (unless it is based on inside information).

It does not assert that the scratch work of entrepreneurs resembles the system of equations in any way; nor does it state that predictions of entrepreneurs are perfect or that their expectations are all the same. For purposes of analysis, we shall use a specialized form of the hypothesis. In particular, we assume: 1. The random disturbances are normally distributed. 2. Certainty equivalents exist for the variables to be predicted. 3. The equations of the system, including the expectations formulas, are linear. These assumptions are not quite so strong as may appear at first because any one of them virtually implies the other two.

It seems to me that Muth was confused about what the rational-expectations assumption entails. He asserts that the expectations of entrepreneurs — and presumably that applies to other economic agents as well insofar as their decisions are influenced by their expectations of the future – should be assumed to be exactly what the relevant economic model predicts the expected outcomes to be. If so, I don’t see how it can be maintained that expectations could diverge from each other. If what entrepreneurs produce next period depends on the price they expect next period, then how is it possible that the total supply produced next period is independent of the distribution of expectations as long as the errors are normally distributed and the mean of the distribution corresponds to the equilibrium of the model? This could only be true if the output produced by each entrepreneur was a linear function of the expected price and all entrepreneurs had identical marginal costs or if the distribution of marginal costs was uncorrelated with the distribution of expectations. The linearity assumption is hardly compelling unless you assume that the system is in equilibrium and all changes are small. But making that assumption is just another form of question begging.

It’s also wrong to say:

But if expectations are not realized, expectations were not rational.

Scott is right. What I said was wrong. What I ought to have said is: “But if expectations (being divergent) could not have been realized, those expectations were not rational.”

Suppose I am watching the game of roulette. I form the expectation that the ball will not land on one of the two green squares. Now suppose it does. Was my expectation rational? I’d say yes—there was only a 2/38 chance of the ball landing on a green square. It’s true that I lacked perfect foresight, but my expectation was rational, given what I knew at the time.

I don’t think that Scott’s response is compelling, because you can’t judge the rationality of an expectation in isolation, it has to be judged in a broader context. If you are forming your expectation about where the ball will fall in a game of roulette, the rationality of that expectation can only be evaluated in the context of how much you should be willing to bet that the ball will fall on one of the two green squares and that requires knowledge of what the payoff would be if the ball did fall on one of those two squares. And that would mean that someone else is involved in the game and would be taking an opposite position. The rationality of expectations could only be judged in the context of what everyone participating in the game was expecting and what the payoffs and penalties were for each participant.

In 2006, it might have been rational to forecast that housing prices would not crash. If you lived in many countries, your forecast would have been correct. If you happened to live in Ireland or the US, your forecast would have been incorrect. But it might well have been a rational forecast in all countries.

The rationality of a forecast can’t be assessed in isolation. A forecast is rational if it is consistent with other forecasts, so that it, along with the other forecasts, could potentially be realized. As a commenter on Scott’s blog observed, a rational expectation is an expectation that, at the time the forecast is made, is consistent with the relevant model. The forecast of housing prices may turn out to be incorrect, but the forecast might still have been rational when it was made if the forecast of prices was consistent with what the relevant model would have predicted. The failure of the forecast to be realized could mean either that forecast was not consistent with the model, or that between the time of the forecast and the time of its realization, new information,  not available at the time of the forecast, came to light and changed the the prediction of the relevant model.

The need for context in assessing the rationality of expectations was wonderfully described by Thomas Schelling in his classic analysis of cooperative games.

One may or may not agree with any particular hypothesis as to how a bargainer’s expectations are formed either in the bargaining process or before it and either by the bargaining itself or by other forces. But it does seem clear that the outcome of a bargaining process is to be described most immediately, most straightforwardly, and most empirically, in terms of some phenomenon of stable and convergent expectations. Whether one agrees explicitly to a bargain, or agrees tacitly, or accepts by default, he must if he has his wits about him, expect that he could do no better and recognize that the other party must reciprocate the feeling. Thus, the fact of an outcome, which is simply a coordinated choice, should be analytically characterized by the notion of convergent expectations.

The intuitive formulation, or even a careful formulation in psychological terms, of what it is that a rational player expects in relation to another rational player in the “pure” bargaining game, poses a problem in sheer scientific description. Both players, being rational, must recognize that the only kind of “rational” expectation they can have is a fully shared expectation of an outcome. It is not quite accurate – as a description of a psychological phenomenon – to say that one expects the second to concede something; the second’s readiness to concede or to accept is only an expression of what he expects the first to accept or to concede, which in turn is what he expects the first to expect the second to expect the first to expect, and so on. To avoid an “ad infinitum” in the description process, we have to say that both sense a shared expectation of an outcome; one’s expectation is a belief that both identify the outcome as being indicated by the situation, hence as virtually inevitable. Both players, in effect, accept a common authority – the power of the game to dictate its own solution through their intellectual capacity to perceive it – and what they “expect” is that they both perceive the same solution.

Viewed in this way, the intellectual process of arriving at “rational expectations” in the full-communication “pure” bargaining game is virtually identical with the intellectual process of arriving at a coordinated choice in the tacit game. The actual solutions might be different because the game contexts might be different, with different suggestive details; but the intellectual nature of the two solutions seems virtually identical since both depend on an agreement that is reached by tacit consent. This is true because the explicit agreement that is reached in the full communication game corresponds to the a prioir expectations that were reached (or in theory could have been reached) jointly but independently by the two players before the bargaining started. And it is a tacit “agreement” in the sense that both can hold confident rational expectation only if both are aware that both accept the indicated solution in advance as the outcome that they both know they both expect.

So I agree that rational expectations can simply mean that agents are forming expectations about the future incorporating as best as they can all the knowledge available to them. This is a weak common sense interpretation of rational expectations that I think is what Scott Sumner has in mind when he uses the term “rational expectations.” But in the context of formal modelling, rational expectations has a more restrictive meaning, which is that given all the information available, the expectations of all agents in the model must correspond to what the model itself predicts given that information. Even though Muth himself and others have tried to avoid the inference that all agents must have expectations that match the solution of the model, given the information underlying the model, the assumptions under which agents could hold divergent expectations are, in their own way, just as restrictive as the assumption that agents hold convergent expectations.

In a way, the disconnect between a common-sense understanding of what “rational expectations” means and what “rational expectations” means in the context of formal macroeconomic models is analogous to the disconnect between what “competition” means in normal discourse and what “competition” (and especially “perfect competition”) means in the context of formal microeconomic models. Much of the rivalrous behavior between competitors that we think of as being essential aspects of competition and the competitive process is simply ruled out by the formal assumption of perfect competition.

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Excess Volatility Strikes Again

Both David Henderson and Scott Sumner had some fun with this declaration of victory on behalf of Austrian Business Cycle Theory by Robert Murphy after the recent mini-stock-market crash.

As shocking as these developments [drops in stock prices and increased volatility] may be to some analysts, those versed in the writings of economist Ludwig von Mises have been warning for years that the Federal Reserve was setting us up for another crash.

While it’s always tempting to join in the fun of mocking ABCT, I am going to try to be virtuous and resist temptation, and instead comment on a different lesson that I would draw from the recent stock market fluctuations.

To do so, let me quote from Scott’s post:

Austrians aren’t the only ones who think they have something useful to say about future trends in asset prices. Keynesians and others also like to talk about “bubbles”, which I take as an implied prediction that the asset will do poorly over an extended period of time. If not, what exactly does “bubble” mean? I think this is all foolish; assume the Efficient Markets Hypothesis is roughly accurate, and look for what markets are telling us about policy.

I agree with Scott that it is nearly impossible to define “bubble” in an operational ex ante way. And I also agree that there is much truth in the Efficient Market Hypothesis and that it can be a useful tool in making inferences about the effects of policies as I tried to show a few years back in this paper. But I also think that there are some conceptual problems with EMH that Scott and others don’t take as seriously as they should. Scott believes that there is powerful empirical evidence that supports EMH. Responding to Murphy’s charge that EMH is no more falsifiable than ABCT, Scott replied:

The EMH is most certainly “falsifiable.”  It’s been tested in many ways.  Some people even claim that it has been falsified, although I’m not convinced.  In the tests that I think are the most relevant the EMH comes out ahead.  (Stocks respond immediately to news, stocks follow roughly a random walk, indexed funds outperformed managed funds, excess returns are not serially correlated, or not enough to profit from, etc., etc.)

A few comments come to mind.

First, Nobel laureate Robert Shiller was awarded the prize largely for work showing that stock prices exhibit excess volatility. The recent sharp fall in stock prices followed by a sharp rebound raise the possibility that stock prices have been fluctuating for reasons other than the flow of new publicly available information, which, according to EMH, is what determines stock prices. Shiller’s work is not necessarily definitive, so it’s possible to reconcile EMH with observed volatility, but I think that there are good reasons for skepticism.

Second, there are theories other than EMH that predict or are at least consistent with stock prices following a random walk. A good example is Keynes’s discussion of the stock exchange in chapter 12 of the General Theory in which Keynes actually formulated a version of EMH, but rejected it based on his intuition that investors focused on “fundamentals” would not have the capital resources to finance their positions when, for whatever reason, market sentiment turns against them. According to Keynes, picking stocks is like guessing who will win a beauty contest. You can guess either by forming an opinion about the most beautiful contestant or by guessing who the judges will think is the most beautiful. Forming an opinion about who is the most beautiful is like picking stocks based on fundamentals or EMH, guessing who the judges will think is most beautiful is like picking stocks based on predicting market sentiment (Keynesian theory). EMH and the Keynesian theory are totally contrary to each other, but it’s not clear to me that any of the tests mentioned by Scott (random fluctuations in stock prices, index funds outperforming managed funds, excess returns not serially correlated) is inconsistent with the Keynesian theory.

Third, EMH presumes that there is a direct line of causation running from “fundamentals” to “expectations,” and that expectations are rationally inferred from “fundamentals.” That neat conceptual dichotomy between objective fundamentals and rational expectations based on fundamentals presumes that fundamentals are independent of expectations. But that is clearly false. The state of expectations is itself fundamental. Expectations can be and often are self-fulfilling. That is a commonplace observation about social interactions. The nature and character of many social interactions depends on the expectations with which people enter into those interactions.

I may hold a very optimistic view about the state of the economy today. But suppose that I wake up tomorrow and hear that the Shanghai stock market crashes, going down by 30% in one day. Will my expectations be completely independent of my observation of falling asset prices in China? Maybe, but what if I hear that S&P futures are down by 10%? If other people start revising their expectations, will it not become rational for me to change my own expectations at some point? How can it not be rational for me to change my expectations if I see that everyone else is changing theirs? If people are becoming more pessimistic they will reduce their spending, and my income and my wealth, directly or indirectly, depend on how much other people are planning to spend. So my plans have to take into account the expectations of others.

An equilibrium requires consistent expectations among individuals. If you posit an exogenous change in the expectations of some people, unless there is only one set of expectations that is consistent with equilibrium, the exogenous change in the expectations of some may very well imply a movement toward another equilibrium with a set of expectations from the set characterizing the previous equilibrium. There may be cases in which the shock to expectations is ephemeral, expectations reverting to what they were previously. Perhaps that was what happened last week. But it is also possible that expectations are volatile, and will continue to fluctuate. If so, who knows where we will wind up? EMH provides no insight into that question.

I started out by saying that I was going to resist the temptation to mock ABCT, but I’m afraid that I must acknowledge that temptation has got the better of me. Here are two charts: the first shows the movement of gold prices from August 2005 to August 2015, the second shows the movement of the S&P 500 from August 2005 to August 2015. I leave it to readers to decide which chart is displaying the more bubble-like price behavior.gold_price_2005-15

S&P500_2005-2015


About Me

David Glasner
Washington, DC

I am an economist in the Washington DC area. My research and writing has been mostly on monetary economics and policy and the history of economics. In my book Free Banking and Monetary Reform, I argued for a non-Monetarist non-Keynesian approach to monetary policy, based on a theory of a competitive supply of money. Over the years, I have become increasingly impressed by the similarities between my approach and that of R. G. Hawtrey and hope to bring Hawtrey's unduly neglected contributions to the attention of a wider audience.

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