Abstract
This chapter is devoted to the mathematical analysis of learning to become rational in the simplest purely dynamic case. More precisely, we consider a model in which the univariate endogenous variable depends only on its one period lagged value, the predictions of agents, and a disturbance term. Since this model exhibits a very simple structure (the estimates of agents as well as the endogenous variables are real valued) it is a suitable platform for introducing our mathematical approach. Due to this simple structure the mathematical techniques employed in this chapter are easy to understand and do not obscure the fundamental properties and problems.
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References
In the sequel we regard the stochastic process {θt} as the outcome of an adaptive on-line estimation algorithm, a concept which is well-known in engineering and control sciences (see, e.g., Ljujng/Söderström (1983) and Benveniste ET AL. (1990)) and for which the weighting coefficients γt are called `gains’.
We use the terms `learning procedure’ and `algorithm’ synonymously.
See, ROTTER (1954) for an early experimental study on the formation of ex¬pectations and the empirical evidence of the choice γt = l/t.
The conditions γr [0,1], γr=00 are the so-called Dvoretzky—Conditions which are a kind of natural conditions for sequences of gains of stochastic approximation procedures (see, e.g., Dvoretzky (1956)).
Notice that all information about the history of the model available at time t is contained in the σ—algebra Ft Notice furthermore that, by construction, yr is always Ft-measurable.
If k = 0 then a≠1 ensures the existence of a unique fixed point. If k≥1 more complicated conditions on a and 0 are necessary (see, e.g., Zenner (1992a,b)).
It is possible to admit also sequences of gains with an evolution not linked to the evolution of the estimation process. This case of `fluctuating gains’ is covered by the analysis in Zenner. (1994). We decided to disregard this generalization here since it gives only little additional gain but requires additional notational effort.
The reason why we exclude the case |θ|=1 is that we did not succeed in verifying (2.22) in that case without imposing additional assumptions. We are, however, convinced that (2.22) generally holds true with |θ|=1
Our concept of stability of rational expectations is closely related to the one of Evans (1989). But since his concept does not incorporate a real time learning procedure the two concepts do not fully coincide. The stability results obtained by Evans coincide with our stability results for learning procedures given by a sequence of gains with limit point one.
Notice that the conditional Lyapunov inequality jointly with condition (2.6) implies sup
Kottmann/Kuliherda (1990) made this mistake in the proof of their Theo-rem 2.4.
This result was already stated by Kottmann/Kuliberda (1990) for the spe-cial case of the OLS procedure but, as already mentioned, their proof was not correct.
Cyert/Degroot (1974) considered the linear model with additive forecasts one period ahead jointly with the OLS procedure and observed that, for the same deterministic initial values Bo and yo, some simulation runs showed convergence of the estimation process while other runs ended in divergence. They observed this phenomenon in the case of, for a stable as well as for an unstable REE. This finding indicates that if a fixed point is stable w.r.t. a learning procedure, then the respective learning procedure does not necessarily converges a.s. towards the fixed point if the evolution of the gains depends on the evolution of the endogenous variable.
Kottmann/Kuliberda (1990) remarked that for linear models with addi-tive forecasts more than one period ahead “almost everything can happen”. They considered the OLS procedure with a slightly different formation scheme for multi-period ahead predictions which results in a slightly different feedback function.
Although Kottmann/Kuliberda observed the same qualitative features of the model as we did they conjectured “that for models with k 003E 0 there is a bounded domain of attraction depending on p5 and a [chr(133)] containing all starting configure-tions [chr(133)] which give almost sure convergence” (p. 11). In fact, this conjecture is false in the stochastic case as shown in our own Monte—Carlo study of their model (see Zenner (1992b)).
By explosion we mean divergence of the estimation process and the data pro-cess caused by the feedback mechanism. For example, if the function f is shaped such that overestimation of the rational expectations parameter,leads to underestimation of the actual law of motion, θt < f (θt), such a feedback to cause explosion is likely to occur
Nevertheless, this does not necessarily imply that does not converge. Only if we impose additional assumptions on the disturbance terms, more precisely, if we assume {wt} to be normed regular in the sense of Marcinkiewicz and Zygmund, then implies that does not converge (see, e.g., Stout (1974), pp. 80 ).
Notice that, e.g., every sequence {wt} of independently distributed random vari¬ables such that xxxxx, and wt is uniformly integrable is a normed regular MDS in the sense of Marcinkiewicz and Zygmund. Other examples are MDS with xxxxx which are uniformly bounded by a constant.
See, e.g., Cyert/Degroot (1974), Bray/Savin (1986), Fourgeaud ET AL. (1986), Marcet/Sargent (1989a), Kottmann (1990) and Mohr (1990)).
In the next chapter we consider the same model without this simplification.
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Zenner, M. (1996). Univariate AR(1) Models. 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_2
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DOI: https://doi.org/10.1007/978-3-642-51876-8_2
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