Abstract
A characteristic feature of dynamic economic models is that, if future states of the economy are uncertain, the expectations of agents matter. Producers have to decide today which amount of a good they will produce not knowing what demand will be tomorrow. Consumers have to decide what they spend for consumption today not knowing what prices will prevail tomorrow. Adopting the neo-classical point of view that economic agents are `rational’ in the sense that they behave in their own best interest given their expectations about future states of the ecomomy it is usually assumed that agents are Bayesian decision makers. But, as LUCAS points out, there remains an element of indeterminacy:
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References
For example,Nerlove(1958) suggests the use of the adaptive expectations formation scheme because it is able to explain why cobweb cycles are quite rare in agricultural markets although these markets are predetermined for these phenomena to occur since production is generally more elastic than demand.
From the mathematical point of view the definition of rational expectations given by Muth is in some sense ambiguous. On the one hand he speaks of expectations as probability distributions, on the other hand he states that these distributions coincide with the predictions of the relevant theory. But predictions are usually regarded as random variables or mathematical conditional expectations.
This ambiguity goes right through the RE literature. While Muth, in his mathematical analysis, treats rational expectations as (mathematical) conditional expectations, Lucas (1972) regards rational expectations as conditional probability distributions. Hence, there exist two versions of the REH; a strong version as suggested by Lucas, and a weak version as employed by Muth and the majority of studies on RE models, including this study.
In most microeconomic and macroeconomic models agents are assumed to be risk-neutral so that the mean value of agents’ subjective distribution of outcomes is the only relevant criterion for their decisions. Then the weak version of the REH is adequate. In other models, however, additional characteristics of the probability distribution may be relevant decision criteria. For instance, in financial market models the variance of the distribution may play an important role for agents’ decisions. In such models the strong version of the REH is appropriate.
Under a suitable probabilistic specification rational expectations are optimal in the sense of yielding minimum mean squared prediction errors. This optimality property, however, should not be confused with other optimality concepts. For example, rational expectations do not necessarily lead to utility maximization. See, e.g., Benassy (1992) for a recent study in which the RE scheme is not the utility maximizing expectations formation scheme.
If agents are very competent mathematical programmers, why should they not be also very competent statisticians?
In fact, as noticed by several authors (see, e.g., Townsend (1978)), in some models the REE may be regarded as some kind of Nash equilibrium. But this does not necessarily hold true in every model. For a counterexample see, e.g., Benassy (1992).
As Pesaran ( 1987, p. 32) points out “there is no doubt that individuals do learn from their own experience as well as from the experience of others. Generally speaking, learning takes place through two separate but closely connected mechanisms, namely repetition and understanding”. Bayesian learning as well as learning by statistical procedures incorporate both mechanisms assuming that agents have some prior information about the structure of the model and repeatedly update their estimates whenever additional information becomes available.
Under the stochastic specification of normally distributed random variables the Bayesian procedure and the OLS procedure are closely related, i.e., the OLS estimates may be regarded as Bayesian estimates based on a diffuse prior.
For example, if agents know the structure of the model (1.2) and, in addition, know the value of the parameter a, then the problem of forming rational expectations reduces to the problem of forming conditional expectations (in the Bayesian sense) of Ø’zt.
The use of an auxiliary moel which does not take into account the forecast feedback is usually defended by argueing that agents believe to act in a competitive world in which a single agents’ expectation does not matter, or that agents believe the market expectation already to be rational, or that it is impossible or too costly to gather information about other agents’ expectations.
But one can also adopt a more naive point of view. Suppose that agents have carried out an explorative data analysis which has revealed that certain time series influence the time series to be predicted, or that agents know, at least roughly, about the relationships between the economic variables of interest. Then they simply carry out a linear regression, i.e., they estimate some unknown, possibly only hypothetical parameters using the OLS procedure. To assume that agents use this procedure is reasonable since the OLS procedure is, probably, the only estimation procedure known also by non-statisticians.
Frydman (1982) shows that this estimability problem cannot be solved by introducing an intuition in the market which collects individual expectations and reports the aggregate expectation since agents would alter their predictions after receiving this additional information.
See, e.g., Basar(1989) andKottmann (1990, Part II).
See, e.g., Decanio (1979) and Evans (1983).
Bray ( 1982, Proposition 4)) gives a real time formulation of this kind of learning. She assumes that agents do not change their predictions during the learning period. This implies that the parameters agents are learning about do not change and agents eventually learn these parameters in infinitly many periods following the OLS procedure. After they have learned the exact parameter values agents simultaneously change their predictions and learn about the altered parameter values in another infinity of periods, and so on.
See, e.g., Evans (1989) and Marcet/Sargent(1988).
For the sake of expositional simplicity we do not present these approaches in their full scope.
We like to emphasize that at first hand the auxiliary model is purely fictitious and does not describe any real economic activity. It describes only what we assume agents believe in and serves us as a justification for what we assume agents actually do. If agents believe in the model (1.6) and, in addition, assume the (fictitious) disturbance term et+1 to be independent of the explanatory variables zs, s≤t, then it is rational for them to carry out a linear regression in order to form best predictions according to the mean squared prediction error criterion.
Notice that we do not assume the parameter 0 to be non-zero in each component. The vector zt may thus contain some variables which affect the endogenous variable only via agents’ predictions. These variables are usually calledsun-spotvariables since in a REE they do not affect the endogenous variable.
Obviously, the result of Robbins/Siegmund(1971) was not known to Bray/Savin (1986) since using this result simplifies the original proof considerably.
Notice that the stochastic approximation approach applies to considerably more complex models than model (1.2), in particular to (multivariate) models incorporating several prediction terms based on several different auxiliary models. For details see Kottmann (1990) and Mohr (1990).
The ODE approach does not require that that the auxiliary model includes all relevant time series. Furthermore it applies also to the case of multivariate endogenous variables as long as the state space representation (1.23) holds true.
Notice that the trajectories of the estimation process are denoted by θt and the trajectories of the ODE by (θt). Notice furthermore that the time index is used differently in (1.25) and (1.26).
The concept of expectational stability developed by Evans (1989) and applied
in a series of subsequent studies (see, e.g., Evans/Honkapohja (1994a) for references) is based on the study of the `smaller’ ODE (1.27) but incorporates no explicit real time learning procedure.
The set D1 has to be open and D2 has to be closed. Furthermore D1 has to be a subset of the domain of attraction DA of the ODE (1.26) and, in its first n components, also a subset of the stability region Ds. Moreover, the vector field of the ODE (1.26) has to point inside D1 everywhere on the boundary of D1.
The problem of a proper choice of the set D1 and D2 is overlooked in many studies applying the ODE approach. See, e.g., Moore (1993) and Evans/Honkapohja (1994c).
See, for example, the application of the ODE approach to specific economic models in the studies of Marcet/Sargent (1989a) and Chang ET AL. (1991a).
Some special cases are treated by Marcet/Sargent (1989a) and Chang ET AL. (1991b).
In engineering applications, for which the ODE approach was originally developed, this means no restriction since unstable processes do not play a prominent role in this field. In economic applications, however, unstable processes, like unit root processes or polynomial trends, are quite common.
In some studies considering overlapping generations models and general equilibrium models the term `learning’ is used in a somewhat euphemistic manner (see, e.g., Homimes (1991), Grandmont (1992), and Balasko/Royer (1994)). In these studies agents’ predictions are given by a fixed adaptive expectations formation scheme, for example, a moving average of lagged endogenous variables. If the predictions generated by these schemes converge towards a steady state equilibrium,then it is said that agents have learned this equilibrium value. But since agents do not adjust their expectations formation scheme over time this modelling lacks an important characteristic of learning.
The recently grown interest in this kind of adaptive models is somewhat surprising since, basically, it means a step back into pre-REH times. We believe that the reason for this interest can be found in the fact that, on the one hand, these models are easy to analyze mathematically since the endogenous variable is governed by a time-invariant law of motion and, on the other hand, they are able to explain not only convergence towards a steady state equilibrium but also business cycles and other kinds of interesting dynamical behaviour (e.g., chaotic movements) without leaving the neo-classical framework.
In some studies, however, different concepts are confused. Balasko/Royer (1994), for example, argue that the OLS procedure can be regarded as a special case of such adaptive schemes overlooking the fact that the estimates generated by the OLS procedure exhibit an adaptive structure, but not the predictions based on these estimates. Moreover, they compare their adaptive schemes for a first order autoregressive model to the OLS estimates generated by a regression on a constant as the only explanatory variable.
Some interesting studies not satisfying these conditions are given by Kirman (1975, 1983), Bray (1982), and Woodford (1990). In these studies agents learn about the relationships between endogenous variables (Kirman), between exogenous variables and a sun-spot variable (Woodford), or about the law of motion of an exogenous variable by a regression on the endogenous variable (Bray).
The results obtained by these studies are mixed. While BRAY shows a.s. convergence of the OLS procedure towards rational expectations under suitable parameter conditions Kirman and Woodford show that agents can learn to believe in a sun-spot or misinformed expectational equilibrium where “they are ignorant, incompetent but happy” (Kirman (1975, p. 152)).
The most restrictive assumption in the study of Fourgeaud ET AL. (1986) requires that the maximum and minimum eigenvalues of the matrix of moments diverge at the same rate. This holds true if, e.g., the involved time series are (covariance) stationary and ergodic. It is also satisfied if all variables explode at the same rate. But it fails to hold if the variables grow at different rates, for example, if some variables are stationary and some follow a polynomial trend. In Chapter 4 we shall show that we can relax the assumption on the eigenvalues and still obtain convergence if a<1/2.
Clearly, such a result is discouraging for an economist. Possibly for that reason Cyert/Degroot (1974) came to the conclusion “that the construction of inconsistent models is not the most fruitful way to progress in this field’’ (p. 524).
The global stability of least squares learning is analyzed properly by Chang ET AL. (1991a).
Chang ET AL. (1991b) point out that the formulation by Marcet/Sargent does not allow for an intercept term among the exogenous variables and provide a reformulation which overcomes this shortcoming.
Again, Chang ET AL. (1991b) provide a proper reformulation and give formal proofs of the propositions claimed by Marcet/Sargent (1989b).
See, e.g., Sargent (1991), Moore (1993), and Evans/Honkapohja (1994a).
See, e.g., Bullard (1992), Honkapohja (1993), and Evans/Honkapohja (1994b,c).
In Section 2.6 we shall restate, as a special case, the convergence result of Kottmann/Kuliberda (1990). This result applies as well to the stationary as to the non-stationary case.
The existence of endogenously generated cycles is shown in many recent studies, most of them based on the pioneering work of Grandmont (1985). These cycles are usually rational in the sense that they occur in spite of perfect foresight (the deterministic analogon to rational expectations) while the cycles observed by Zenner (1992a,b) should be regarded as a pathological outcome.
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Zenner, M. (1996). Introduction. In: Learning to Become Rational. Lecture Notes in Economics and Mathematical Systems, vol 439. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-51876-8_1
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