Rational expectations and forecasts

The perfect-foresight-with-error expectations model

The most frequently used model of rational expectations is the perfect-foresight-with-error model. This model is based on the assumptions that:

• Economic agents are rational – goal oriented, make no systematic mistakes
• All information is available and transparent, meaning:
  • - The economic model is known – i.e. effects of shocks can be calculated
  • - The type of shock hitting the economy is known – i.e. demand or supply, temporary or permanent

In mathematical notation we have then

X^e_t+1 = E_t (X_t+1 | I_t)

X^e_t+1 = X_t+1 + e_t+1 E_t (e_t+1) = 0 (no bias), E_t (e_t+1 I_t) = 0 (no inefficiency of information use)

For theoretical economists who deal with a particular economic model, where the model is known and the shocks and variables are known, the correct rational expectations are easy to identify.

"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." (Muth, 1961)

Rational expectations in non-stationary stochastic economies

All in all, the classic perfect-forecast-with-error model of rational expectations is a worthwhile theoretical hypothesis for the evaluation of theoretical models, but it does not describe the real world very well. The real world has more and different uncertainty which requires economic agents to exhibit different (but still rational) behavior, which may include the gradual adjustment of expectations to new information (“learning”) and which leads to ex post persistent forecast errors which observers subsequently identify as "bias" and "inefficiency". In the real world economic agents do NOT know the true economic model that describes their real world (even the expert economists disagree about the appropriate models). Also, real world shocks to the system are NOT necessarily drawn from stable and known statistical distributions (temporary shocks versus permanent shocks, non-normal distributions, regime shifts). Finally, economic analysis and forecasting is complicated by the influence of economic policymakers, the behavior of whom is NOT easily captured by random shocks and time-invariant policy rules (policymakers have economic objectives that tend to change over time, exhibiting regime shifts, frequently unknown to outsiders).

Even among the originators of rational expectations, the real world limitations of the perfect-foresight-with-error model were well-known.

For example, Muth (1961, p.316-17 emphasis added): "The [rational expectations] 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) [...] 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 formulation, are linear."

In his 1960 paper "Optimal properties of exponentially weighted forecasts", Muth even rationalized the adaptive expectations model. The implication of his work being that for non-stationary economic variables, with an unobserved mix of temporary and permanent shocks, i.e. IMA(1,1) processes, the adaptive expectations or backward-looking model can provide the best ex ante rational forecasts. In this framework, the discussion about rational expectations is not about the presence or absence of a backward looking component, but about the precise value of the coefficient(s) of the forecasting equation. The real world implication of rational adaptive forecasts is that ex post observed forecast errors will not be random and will, especially in finite samples, present the illusion of bias and/or inefficiency.

Lucas and Sargent (1979, p.13, emphasis added) state: "Benjamin Friedman and others have criticized rational expectations models apparently on the grounds that much theoretical and almost all empirical work has assumed that agents have been operating for a long time in a stochastically stationary environment. Therefore, agents are typically assumed to have discovered the probability laws of the variables they want to forecast. […] But it has been only a matter of analytical convenience and not of necessity that equilibrium models have used the assumption of stochastically stationary shocks and the assumption that agents have already learned the probability distributions they face. Both of these assumptions can be abandoned, albeit at a cost in terms of the simplicity of the model.”

Cook and Hahn (1990, p.14) state: "there is a distinction between the specialized form of the rational expectations hypothesis used in the literature ... and the general principle of rational expectations, which is that market participants use available information efficiently in forming their expectations."

Thus, empirical tests of expectations and forecasts relying uncritically on the REH-perfect-foresight-with-error model are simply testing a caricature, or, at best, a very general first approximation of reality. Rational expectations is NOT perfect foresight. The alternative hypothesis to the perfect-foresight-with-error model is NOT automatically irrational or non-rational expectations! The real world is more complicated than assumed in simple economic models, not necessarily irrational. We (should) all know that by now.

Note: Time series models that function as alternatives to the stationary stochastic and perfect-foresight model, are the ARIMA model with temporary and permanent components, and the hidden-state Markov switching model.

Note: The concepts of "bounded rationality" (see Sargent, 1993) or "less-than-rational-expectations" (Friedman,1979; Taylor, 1975) frequently mentioned as alternative models are misleading in this context. The failure of the perfect-foresight-with-error model does not result from limited rationality but is inherent to the nature of expectations in a more complicated non-stationary economic environment.

Note: The problem and effect of non-stationary stochastic economies is distantly related to another discussion: risk vs. uncertainty (Knight, 1921 Ch.7; Keynes, 1936 Ch.12). Again, rational behavior and expectations need to be interpreted differently and the alternative hypothesis is not automatically irrational behavior.

Adaptive and extrapolative expectations

Many economists seem to get confused when empirical results suggest that the dynamics of expectations and forecasts do not follow the basic perfect-foresight-with-error model of rational expectations. The usual default conclusion is irrationality and behavioral bias. There is no need for such conclusions. As mentioned above and illustrated by Muth (1960), the model of adaptive expectations can be an entirely rational mechanism to produce expectations, depending on the time series properties of the variable being forecasted. In fact, the empirical literature on expectations and forecasts suggests that forecasters do not arbitrarily use one single type of forecasting model or mechanism. The forecasting / expectations mechanisms they employ variously exhibit adaptive, regressive, extrapolative and 'perfect foresight' behavior, depending on the variable and the time horizon. This selective behavior is entirely consistent with a rational approach to forecasting and learning in an environment that is stochastically non-stationary and subject to uncertainty and costly information.

EMPIRICAL TESTS OF RATIONAL EXPECTATIONS AND FORECASTS

Many studies have examined economic expectations or forecasts and have tested their rationality. The basic properties of unbiasedness and efficiency were mentioned above.

Some key considerations with respect to appropriate testing of rational expectations and forecasts are (see also Webb, 1987):

• using the appropriate real-time information set, taking into account lags in the availability of information
• using the appropriate real-time information set, taking into account subsequent revisions in the measurement of variables (revised definition and estimation)
• taking into account the non-stationary stochastic nature of economic variables and economic models
• the effect of unit root variables (spurious regressions)
• the complication of uncertain regime shifts (learning and the credibility of policy changes)
• taking into account the rational trade-offs between forecast objectives: a smallest unbiased forecast error is not necessarily the only objective
• taking into account cross-correlation in panel data due to common (macro) shocks (Keane and Runkle, 1998)
• taking into account the skewed distribution of many variables (particularly financial variables). The appropriate forecast performance measure with respect to bias would be the median, not the mean forecast error (Gu and Wu, 2003)

Unfortunately, many empirical studies do not investigate and do not allow for these real world complications. Even recent studies continue to hide behind theoretical abstractions, favoring the quick and easy approach to empirical research. To be fair, it is not easy to identify the true rational expectations/forecasts, because they depend on assumptions about the time series behavior of variables and shocks and the nature of the economic model. But, again, one could start by recognizing that the alternative hypothesis to the simple perfect-foresight-with-error model is not automatically irrational or non-rational expectations and forecasts.

For example: The bias in (financial analysts') corporate earnings forecasts

Empirical studies have reported an optimistic bias in companies' earnings forecasts by security analysts. Mean forecast errors are found to be consistently positive and analysts appear to underreact to negative news and overreact to positive news.

• Some bias in mean errors is not necessarily irrational/inefficient. Forecasters interested in forecast accuracy (Mean Absolute Forecast Error, MAFE) need to forecast the median of the forecast distribution, not the mean. Many variables, particularly in finance, are not symmetrically distributed. Therefore, evaluating forecast performance using mean forecast errors will simply result in the mean-median difference (Gu and Wu, 2003).
• Rationality of earnings forecasts is rejected as long as two complications are not taken into account: (1) correlation in a given period of analysts' forecast errors due to 'macro' shocks affecting all analysts' forecasts, (2) discretionary asset write-downs creating skewed earnings distributions and which are not forecasted by analysts (Keane and Runkle, 1998).
• Aggregate long-run earnings forecasts appear to exceed the long-run economic growth potential of the economy and exhibit long cyclical errors. However:
• - Long-run earnings forecasts (in fact, defined as average 3 to 5 year ahead earnings growth) may exceed long-run economic growth rates because the share of labor (wage income) and capital (corporate profits) is not constant over time. During the 1970s and 1980s firms were catching up with their long-run profit share of total income, with higher growth rates of profits for many years.
• - Evaluation of long-run growth forecasts suffers from short samples (limited available data) and overlapping observations (serially correlated errors).
• - Many other methodological issues and data issues exist in evaluating the rationality of forecasts, similar to the issues in business cycle, inflation, interest rate, and other forecasts. Examples are stale forecasts, revised historical accounting data, errors in adjusting for stock splits, identifying the precise earnings definition being forecasted, ... (For a review of many issues see Ramnath, Rock and Shane, 2008)

Selected general literature

* Lucas, Jr. R.E. and T.J. Sargent (1979), After Keynesian macroeconomics, FRB Minneapolis Quarterly Review, Spring: 1-19. http://www.minneapolisfed.org/research/QR/QR321.pdf

* Mussa, M. (1975), Adaptive and regressive expectations in a rational model of the inflationary process, Journal of Monetary Economics, vol.1 (4) October: 423-42.

Satchell, S. and A. Timmerman (1995), On the optimality of adaptive expectations: Muth revisited, International Journal of Forecasting, vol.11 (3) September: 407-16. http://ideas.repec.org/a/eee/intfor/v11y1995i3p407-416.html

* Meltzer, A.H. (1982), Rational expectations, risk, uncertainty, and market responses, in P. Wachtel (ed.) Crises in the Economic and Financial Structure, Lexington Books: 3-22.

* Cukierman, A. and A.H. Meltzer (1982), What do tests of market efficiency show? August 1982. http://www.tau.ac.il/~alexcuk/pdf/Cukierman_and_Meltzer_1982.pdf

* Webb, R.H. (1987), The irrelevance of tests for bias in series of macroeconomic forecasts, FRB Richmond Economic Review, vol.73 Nov/Dec: 3-9. http://www.richmondfed.org/publications/research/economic_review/1987/pdf/er730601.pdf

Patton, A.J. and Allan G. Timmerman (2003), Properties of optimal forecasts, CEPR Discussion paper #4037.

* Keane, M.P. and D.E. Runkle (1989), Are economic forecasts rational?, FRB Minneapolis Quarterly Review, vol.13 (2) Spring: 26-33 http://minneapolisfed.org/research/QR/QR1323.pdf

* Keane, M.P. and D.E. Runkle (1998), Are financial analysts' forecasts of corporate profits rational?, Journal of Political Economy, vol.106 (4) August: 768-805.

* Croushore, D. (2006), An evaluation of inflation forecasts from surveys using real-time data, FRB Philadelphia wp 06-19 http://papers.ssrn.com/sol3/papers.cfm?abstract_id=940418#PaperDownload

* Andolfatto, D., S. Hendry and K. Moran (2008), Are inflation expectations rational?, Journal of Monetary Economics, vol.55 (2) March: 406-22. http://ideas.repec.org/p/wpa/wuwpma/0501002.html

* Bullard, J.B. (1991), Learning, rational expectations, and monetary policy, FRB St.Louis Review, Jan/Feb: 50-60 http://research.stlouisfed.org/publications/review/91/01/Learning_Jan_Feb1991.pdf

Biases in empirical results when using the perfect-foresight-with-error model of expectations

Several empirical studies have pointed out the important and misleading effects of using the traditional "rational expectations" hypothesis to evaluate various economic theories. For example,

* Evans, M.D.D. and K.K. Lewis (1995), Do expected shifts in inflation affect estimates of the long-run Fisher relation?' Journal of Finance, vol.50 (10 March: 225-53. http://ideas.repec.org/a/bla/jfinan/v50y1995i1p225-53.html

* Rudebusch, G.D. (1995), Federal Reserve interest rate targeting, rational expectations and the term structure, Journal of Monetary Economics, vol.35 (2) April: 245-74. http://ideas.repec.org/a/eee/moneco/v35y1995i2p245-274.html

* Kozicki, S. and P.A. Tinsley (2005), What do you expect? Imperfect policy credibility and tests of the expectations hypothesis, Journal of Monetary Economics, vol.52 (2) March: 421-47. (FRB Kansas City Working paper, no.01-02, 2001) http://search.ssrn.com/sol3/papers.cfm?abstract_id=276109