this paper (based on a different data set) was presented at the 1992 North American Winter Meeting of the Econometric SocietyinNew Orleans, Louisiana.

We combine Nagel’s “step-k” model of boundedly rational players with a “law of effect” learning model. Players begin with a disposition to use one of the step-k rules of behavior, and over time the players learn how the available rules perform and switch to better performing rules. We offer an econometric specification of this dynamic process and fit it to Nagel’s experimental data. We find that the rule of learning model vastly outperforms other nested and nonnested learning models. We find strong evidence for diverse dispositions and reject the Bayesian rule-learning model.

this paper, we modify our previous framework to incorporate feedback. Section 2 gives several important economic examples to motivate our research. Section 3 studies a Robbins-Monro procedure with feedback (RMF) in a Banach space, and its almost-sure convergence. We prove results for almost-sure convergence in a certain topology under conditions similar to the weakest possible conditions for multivariate RMF (KC, 1978). Section 4 specializes to several RMFs in a Hilbert space, and obtains their almost-sure convergence in the weak topology. Sufficient conditions are provided in section 5 to allow the innovation process { Z n } to be a Hilbert space-valued near epoch dependent function of mixing processes. All the results in sections 3, 4 and 5 are new. In section 6, we apply our general theorems to establish convergence results for the economic examples in section 2. Section 7 is a brief summary. Some definitions and all proofs are collected into a mathematical appendix. 2. EXAMPLES Recently macroeconomists ( e.g., Marcet & Sargent, Woodford ) and game theorists ( e.g., Fudenberg & Kreps, Crawford ) have considered the consequences of dropping the mutual consistency assumption in the concept of rational expectations equilibrium (REE) or Nash equilibrium (NE), and instead search for plausible agent learning behaviors that lead to a "sensible" REE or NE. Many of these learning models are parametric stochastic approximation algorithms with feedback. However, parametric learning can lead to incorrect belief equilibria as shown by Kuan & White (KW) (1994). The following examples illustrate that it may be more natural to learn the true functional relationships nonparametrically. Example 2.1: (Generalization of Bray's Model) Bray's (1983) model is one of the first adaptive lea...

This paper characterizes equilibrium asset prices under adaptive, rational and Bayesian learning schemes in a model where dividends evolve on a binomial lattice. The properties of equilibrium stock and bond prices under learning are shown to differ significantly. Learning causes the discount factor and risk-neutral probability measure to become path-dependent and introduces serial correlation and volatility clustering in stock returns. We also derive conditions under which the expected value and volatility of stock prices will be higher under learning than under full information. Finally, we investigate restrictions on prior beliefs under which Bayesian and rational learning lead to identical prices and show how the results can be generalized to more complex settings where dividends follow either multi-state i.i.d. distributions or multi-state Markov chains.

We examine the prevalence of data, specification, and parameter uncertainty in the formation of simple rules which mimic monetary policy-making decisions. Our approach is to build realtime datasets and simulate a real-time policy-setting environment in which we are able to assess the actual performance of rules, had they been followed in real time. This approach allows us not only to track the performance of alternative rules over time (hence facillitating a type of model selection among competing rules), but also allows us more generally to assess the importance of the data revision process in the formation of macroeconomic time series models. From the perspective of real time data, our results suggest that the use of data which are erroneous, in the sense that they were not available at the time decisions based on forecasts from the rules were used, can lead to the selection of quantitatively di®erent models. From the perspective of policy rules, we find that: our version of "calibration" is better than naive estimation, although both are dominated by an approach to rule formation based on adaptive least squares learning using; rules based on seasonally unadjusted data are more reliable than those based on seasonally adjusted data; and rules based soly on preliminary data do not minimize mean square forecast error (MSE) risk. In particular, early releases of data can be noisy, and for this reason it is useful to also use data which have been revised when making decisions using policy rules.

In this paper the recurrent back-propagation and Newton algorithms for an important class of recurrent networks and their convergence properties are discussed. To ensure proper convergence behavior, recurrent connections must be suitably constrained during the learning process. Simulation results demonstrate that the algorithms with the suggested constraint have superior performance. 1. INTRODUCTION It is well known that feedforward neural networks may have difficulties in representing the sequential behavior of a target sequence and can perform only passive cognition. 4,13 This deficiency hampers the applications of feedforward networks in fields such as signal processing and dynamic control where temporal structure plays an important role. In contrast with feedforward networks, recurrent networks permit additional, internal feedback connections among units so that they are able to capture more dynamic characteristics and perform cognition even when inputs are static. This aspect o...

Le CIRANO est un organisme sans but lucratif constitué en vertu de la Loi des compagnies du Québec. Le financement de son infrastructure et de ses activités de recherche provient des cotisations de ses organisations-membres, d=une subvention d=infrastructure du ministère de l=Industrie, du Commerce, de la Science et de la Technologie, de même que des subventions et mandats obtenus par ses équipes de recherche. La Série Scientifique est la réalisation d=une des missions que s=est données le CIRANO, soit de développer l=analyse scientifique des organisations et des comportements stratégiques. CIRANO is a private non-profit organization incorporated under the Québec Companies Act. Its infrastructure and research activities are funded through fees paid by member organizations, an infrastructure grant from the Ministère de l=Industrie, du Commerce, de la Science et de la Technologie, and grants and research mandates obtained by its research teams. The Scientific Series fulfils one of the

This paper considers the optimal management of exchange rate target zones by regarding the operation of a target zone as a dynamic signalling game between the monetary authorities and the …nancial markets. A Sequential Open Loop (Feedback) policy of sterilised intervention is proposed that depends critically on the evolution of the policy maker’s credibility as opposed to the open loop precommitment strategy that has been implemented, for instance, in the Exchange Rate Mechanism of the EMS and in the Bretton Woods system. The width of the target zone and re-alignments are in turn determined optimally given the policy maker’s credibility. This ‡exible target This paper is a substantially revised version of an earlier paper with the same title. The revisions in this version have been stimulated by the consideration of the design of crawling pegs as in place in countries such as Hungary. Work on this revision was carried out while the third author was involved in an ACE project; P96-6149-R on Policy Making in a small open economy aimed at joining the European Union. We thank the partners on this project, in particular Axel Weber and Istvan Szekely, for a number of discussions on the issues raised by the formal design of Crawling Bands. A further revision of this paper which also considers the optimal rate of depreciation as well as the band width is currently in progress. 1 zone proposal is shown through simulation to stabilize the exchange rate to a substantial degree while retaining considerable ‡exibility and robustness in response to shocks.

To locate the root () o in H a Hilbert space of an unknown measurable mapping M: H-7 H, one can use the stochastic approximation (SA) method introduced by Robbins and Monro (1951). The Robbins-Monro (RM) procedure recursively approximates (Jo by