## THEORY OF EXPECTATIONS (Encyclopedia)

In economics, the word ‘expectations’ refers to the forecasts or views of agents about the future trend of relevant variables based on information or intuitions. In macroeconomic contexts, the importance of expectations was emphasised firstly by Keynes1, who stressed the central role of expectations in the determination of agents' behaviour. However, he did not have an explicit model of how expectations are formed, suggesting that agents often rely on the so-called “animal spirits”.
Starting from the 1940s, the first theories of expectations in economics were:
i) the theory of extrapolative expectations. In the 1940s, Metzler2 suggested that the expected value of a variable at time t depends on its value at  t-1 and on a certain correction parameter , taking into account the dynamic of the variable between t-2 and t-1:
(1)
This theory would be able to explain a certain viscosity of the variables' trend as a reaction to economic changes.
ii) the theory of adaptive expectations. In the 1950s, Nerlove3 formulated the theory of adaptive expectations, based on the assumption that the expected value of a variable at t depends on its expected values at t-1 and on the correction parameter , taking into account the difference between the “real” value and the expected value of the variable at  t-1.
A simple model of this theory is represented by the equation (2) where  is the expected value of the variable at t; is the expected value at t-1 and  is the effective value of the variable at t-1.
(2)
With being the correction parameter that ranges between 0 and 1. The theory of adaptive expectations can be extended to  periods.
See the equation (3):
(3)
where   is the actualised value of the variable forperiods in the past. Thus, the current expectations are a weighted average of the past values of the variable with decreasing weights going back to the past.
Both the extrapolative expectation and the adaptive expectation theories are based on distributed lag (DL) models. These two theories have three main shortcomings: i) they use ad hoc hypotheses; ii) they do not make an optimal use of the available information set; iii) they assume that people make systematic errors when predicting the future.
In 1961, in order to override these limitations, Muth4 introduced a new theory of expectations based on the assumption that the agents, being rational, make an optimal use of all the available information: the theory of Rational Expectations (RE). In his theory, Muth states that: i) the expectations formulated by rational informed agents have to be derived from the economic theory; ii) the economic agents make an optimal use of the scarce and costly information; iii) the expectations’ model is endogenous with respect to the economic system.
In particular, the subjective forecast of any agent is the mean of the expected value of the variable conditioned to the available information set. On average, agents can correctly predict the future trend of economic variables.
In fact:
(4)
where E is the conditional mean and  the information set available at t-1. Thus the theory assumes that agents do not make systematic errors when predicting the future, and deviations from the perfect foresight are only random. The forecasting error  is a stochastic variable with mean equal to zero and no serial autocorrelation. That is:
(5)

If  were not a stochastic variable the assumption of optimal use of the available information set would not be correct. A systematic error would imply the suboptimal use of the information set to formulate the expectations.
The RE hypothesis has been used to support some radical conclusions about economic policymaking by Thomas Sargent, Neil Wallace5 and R. Lucas6, the major “neoclassical” economists. They assume that in efficient markets with perfect or near perfect information the agents will anticipate the Government's policies, and will adjust their response accordingly. If the Government employed monetary expansion in order to increase output, agents would foresee the effects, and wage and price expectations would be revised upwards accordingly (“self fulfilling expectations”). Only stochastic shocks to the economy can cause deviations in real variables from their natural level, making the economic policy ineffective.
The hypothesis of RE is often criticised as an unrealistic model of how expectations are formed. Firstly, truly rational expectations would take into account the fact that information about the future is costly. The “optimal forecast” may be the best not because it is accurate, but because it is too expensive to get closer to accuracy. Furthermore, the fundamental uncertainty about future implies that the future cannot be predicted, so that no expectations can be truly “rational”. Lastly, the empirical literature on RE does not reach unique results.
Moreover, the announced policies with no impact on the real side of the economy have consequences on the monetary side. A discretional economic policy can determine an increase in the price level. Under the RE hypothesis, in the short term, an announced policy has the same impact on the economy as the one described by monetarists in the long term. Two others main objections to the RE hypothesis (REH) come from the bounded rationality literature. Firstly, it may be a very strong assumption that agents know the true stochastic process of the variables they need to forecast. Alternatively, one could allow agents to form expectations from less sophisticated schemes as in Evans and Honkapohja (2001), and Hommes and Sorger (1998). Another objection to the REH is that in an environment with heterogeneous expectations, economic outcomes depend upon the expectations of all the participants. Heterogeneous expectations may alter the stochastic process of aggregate variables. Thus, if agents with rational expectations know the form of this stochastic process, then they must be able to observe the expectations of all agents in the economy. The learning literature has also discussed expectation formation schemes with heterogeneity7. Furthermore, Adam (2005) and Guse (2005) have presented models of heterogeneous expectations where each agent uses one of the several available forecasting models to form expectations of a stochastic process. In this literature, one limitation is the restrictive assumption that the proportion of agents using each forecasting model is determined exogenously. A possible solution is to include a predictor choice in these types of learning models that would remove the possibility of agents continually using inefficient forecasting models. The predictor choice literature has focused mainly on deterministic models and not stochastic models of learning. A reason for this hole in the literature is that prior to Adam (2005) and Guse (2005) there were no learning models studied with multiple available forecasting models. In conclusion, it is worth to underline that several recent papers in macroeconomics and finance have used information theoretic ideas8 to develop models with expectations. While these papers develop some valuable insights concerning the expectations of individuals, it is worth noting that they have made assumptions, to allow tractable modelling, that are hard to defend and can lead to anomalous results9.
_______________________
1Keynes, John Maynard (1936)
2Metzler, L. A. (1941)
3M. Nerlove  (1958)
4Muth, J. (1961).
5Sargent, Thomas J. and Neil Wallace (1981).
6Lucas R. (1972, 1976)
7Evans, Honkapohja, and Marimon (2001), Honkapohja and Mitra (2006).
8See, for example, Mackowiak and Wiederholt, 2005; Mondria, 2005.
9For a discussion on this topic, see Sims (2005).

Bibliography

Evans, G. W. and S. Honkapohja (2001), Learning and Expectations in Macroeconomics Princeton University Press.
Friedman, M., (1968) The Role of Monetary Policy American Economic Review 58, 117.
Guse, E., (2005). "Learning with Heterogeneous Expectations in an Evolutionary World," Cambridge Working Papers in Economics 0547, Faculty of Economics, University of Cambridge.
Hommes, C. and Sorger, G. (1998). "Consistent Expectations Equilibria," Macroeconomic Dynamics, Cambridge University Press, vol. 2(03), pages 287-321, September
Keynes, J. M. (2007) [1936]. The General Theory of Employment, Interest and Money. Basingstoke, Hampshire: Palgrave Macmillan.
Klaus A., (2005). "Learning To Forecast And Cyclical Behavior Of Output And Inflation," Macroeconomic Dynamics, Cambridge University Press, vol. 9(01), pages 1-27, February.
Lucas, R. E., Jr. (1972). “Expectations and the Neutrality of Money.” Journal of Economic Theory, 4: 103-l 24
Lucas, R. E., Jr. (1976). “Econometric Policy Evaluation: A Critique.” Carnegie- Rochester Conference Series, 1: 19-46.
Metzler, L. A. (1941). The Nature and Stability of Inventory Cycles. Reviewof Economics and Statistics 23 (3), 113–129.
Mondria, J. (2006). "Financial Contagion and Attention Allocation," Working Papers tecipa-254, University of Toronto, Department of Economics
Muth, J. (1961), “Rational expectations and the theory of price movements”, Econometria.
Nerlove, M. (1958), 'Adaptive Expectations and Cobweb Phenomena', Quarterly Journal of Economics, vol. lxxii, 227-40
Sargent, T. J. and N. Wallace (1981). "Some Unpleasant Monetarist Arithmetic". Federal Reserve Bank of Minneapolis Quarterly Review 5 (3): 1–17
Honkapohja, S. and K. Mitra, (2006). "Learning Stability in Economies with Heterogeneous Agents," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 9(2), pages 284-309,
Wiederholt, M. and B. Mackowiak, (2005). "Optimal Sticky Prices under Rational Inattention," 2005 Meeting Papers 369, Society for Economic Dynamics.
Sims, C. A., (2005). "Rational inattention: a research agenda," Discussion Paper Series 1: Economic Studies 2005,34, Deutsche Bundesbank, Research Centre.

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