Abstract
Generally, the subject of ‘learning to become rational’ is attached to stationary economic environments; and nearly all studies on this subject are restricted to the case that at least the exogenous variables entering the model are stationary and ergodic processes. Apart from questions of mathematical convenience this point of view is based on the idea that learning, as a repeated trial-and-error mechanism, requires a stable, or stationary, environment in order to be successful. How should agents learn something in a non-stationary environment?
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If the law of motion is time-varying other learning procedures have to be considered. BULLARD (1992), for example, considers a learning procedure based on the Kalman filter. He assumes that agents believe in an auxiliary model with time-varying parameters and, consequently, adopt a learning procedure which takes into account the time-varying nature of the parameters. Of course, one cannot expect this learning procedure to converge towards a REE since it is not designed to give consistent parameter estimates. Nevertheless, BULLARD argues that “if agents correctly view parameters as time-varying, however, the local convergence properties of least squares are, in general, overturned” (p. 166 ). Notice that in his model the parameters are time-varying for the only reason that agents’ predictions are time-varying during the learning phase. Hence the agents in his model are not actually `more rational’, as BuLLARD claims. Although they recognize that parameters are time-varying they do not understand why parameters are time-varying. If they would really understand the time-varying nature of the parameters, they would not apply the Kalman filter learning procedure.
The RLS estimates are usually employed in Monte—Carlo studies to avoid the problems associated with singular matrices Z r In particular, the RLS estimates can be calculated recursively without any explicit matrix inversion. This considerably speeds up the calculation of the estimates.
The convergence result for the OLS procedure given by FOURGEAUD ET AL. (1986) relies heavily on that condition. Apart from this condition they assume that a 1/2, Arna=(Zt) = O(.mín(Zt)) a.s., and E[ztwt+1] = 0. It is interesting that also FOURGEAUD ET AL. need the parameter condition a 1/2 while the studies of BRAY/SAVIN (1986) and KOTTMANN (1990) only require a 1. (However, since the latter studies rely crucially on the assumption of stationarity and ergodicity the result of FOURGEAUD ET AL. is not covered in its full scope by the results of the other authors.) The coincidence that FOURGEAUD ET AL. and we both need a 1/2 possibly stems from the fact that their convergence analysis, as well as ours, relies only on algebraic properties of the involved time series.
We are grateful to THOMAS SIwuc for writing this program.
As mentioned by GRAUPE (1984, p. 73), “ß should be sufficiently below the square root of the maximum computer word. In pactice, the identification algorithms are not too sensitive to a choice of ß which may range from ß = 5 to 10,000. There is little point to go beyond ß = 100”.
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© 1996 Springer-Verlag Berlin Heidelberg
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Zenner, M. (1996). Univariate Non-Stationary 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_4
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DOI: https://doi.org/10.1007/978-3-642-51876-8_4
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