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A learning-to-forecast experiment on the foreign exchange market with a classifier system

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A learning-to-forecast experiment on the foreign exchange market with a classifier system
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  ELSEVIER Journal of Economic Dynamics and Control21 (1997) 1543-157s A learning-to-forecast experiment on the foreignexchange market with a classifier system Luca Beltramett?, Riccardo Fiorentinib, Luigi Marengo”,Roberto TamborinibT* a nstitute of Economics, University of Genoa, Ita1.vb Department of Economics, University of Padua, via de1 Santa 28, 35100 Paduva, Italy‘Department of Economics, University of Trento, Italy Abstract This paper reports on an experiment of learning and forecasting on the foreignexchange market by means of an Artificial Intelligence methodology (a ‘ClassifierSystem’) which simulates learning and adaptation in complex and changing environ-ments. The experiment has been run for two different exchange rates, the US dollar-Dmark rate and the US dollar-yen rate, representative of two possibly different marketenvironments. A fictitious “artificial agent” is first trained on a monthly data base from1973 to 1990, and then tested out-of-sample from 1990 to 1992. Its forecasting perfor-mance is then compared with the performance of decision rules which follow theprescription of various economic theories on exchange rate behaviour, and the perfor-mance of forecasts given by VAR estimations of the exchange-rate’s determinants. Keywords: Learning; Artificial Intelligence; Foreign exchange marketJEL classification: F31; C53 1. Introduction The search for a rational basis for forecasting economic variables is still open.As is well known, the advent of the rational-expectations hypothesis (REH)*Corresponding author. The authors would like to thank two anonymous Referees, Prof. Stavros A. Zenios, Prof. MassimoEgidi, Prof. Giovanni Dosi and Dott. Diego Lubian for their useful comments and suggestions onprevious version of the paper.0165-1889/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SO165-1889(97)00035-3  1544 L. Beltrametti et al. / Journal of Economic Dynamics and Control 21 (1997) 1543-1575 introduced the principle that rational forecasts should be based on the efficientuse of all available information, taking ‘the’ relevant model of the economyto represent the (‘true’) data-generating process. More recently, however,the research in the foundations of rational expectations equilibria has focusedon the agents’ learning process of the data-generating model. Now it isunderstood that rational expectations equilibria rest on the existence ofappropriate convergent learning paths. It seems to us that four main questionsemerge from this debate: (i) What is the nature of agents’ learning mechanism(s)?(ii) How can agents identify the phenomena that are relevant to the learningprocess? (iii) How can agents take into account other agents’ expectations whenforming their own ones? (iv) How can agents take into account the fact that theprocess to be learnt depends on the actions of the agents themselves who arelearning?These methodological issues are briefly discussed in Section 2. In this paperwe do not tackle the issues under (iii) and (iv), but we present a tentativeexperiment that focuses on the simpler and more preliminary questions under(i) and (ii). To address the issue of learning and forecasting in this context, wehave designed an experiment with artificial intelligence methods and real worlddata. The choice of artificial intelligence methods is based on two reasons. First,so far the results of research based on rational-expectations formal models oflearning have not been conclusive: there is no way of establishing what is the‘optimal’ learning technique a priori. As a methodological consequence, in thisfield the traditional criterion of optimality should be replaced instead with thecriterion of ‘goodness of fit’ (on this point see Arthur, 1992). Second, andrelatedly, the need for positive cognitive foundations of learning and forecastingactivities is now more strongly felt, and has drawn the economists’ attention toartificial intelligence methods (see e.g. Sargent, 1993). The artificial intelligencemodel we have employed in our experiment is particularly well suited to thesepurposes.We have implemented an artificial agent (AA) that observes a foreign ex-change market ‘from outside a window’ (that is, in a way that its actions have noimpact on the market outcomes), is instructed to observe a number of potentialexchange-rate determinants, bets on the exchange-rate going up or down, anduses gains and losses to learn to forecast. AA embodies one of the mostsuccessful learning mechanism that have been suggested by artificial intelligencescientists: the ‘classifier systems’ (CS) (Holland et al., 1986). In a number ofexperiments, such a learning mechanisms has performed well in reproducinghuman learning-based decision-making in repeated multi-choice problems(Arthur, 1990, 1991; Holland and Miller, 1991). Morever, the CS seems parti-cularly well suited to investigating learning processes because it is based on anexplicit logic of external signals classification, and of signal-action matching.Alternative devices such as neural networks, which are now increasingly used inforecasting activities, may not be as much informative on the ‘reasoning’ behind  L. Belwametti et al. / Journal of Economic Dynamics and Control 21 (1997) 1543-15751545 the agent’s actions. The basic principles and structure of CCs are illustrated inSection 3.Learning experiments with real-world data are not much developed, onereason being that artificial data-generating processes allow for greater control ofresults for theoretical purposes. On the other hand, we have found real datamore challenging and appropriate to the positive aim of our experiment.Financial markets, for perhaps obvious reasons, are the natural candidates forlearning and forecasting exercises and experiments. We have in particularchosen the foreign exchange market, namely the dollar-mark and the dollar-yenexchange rates, mainly because the theoretical debate over the determinants ofexchange rates is highly controversial (the most powerful econometric tech-niques have been implemented without success in order to test one modelagainst the others). Until recently, the profession was struck by the finding thata random-walk model of exchange-rate changes outperformed ‘fundamental’models (Meese and Rogoff, 1983). On the other hand, Granger (1992) reports ona renewed confidence in asset-prices forecastability springing from the applica-tion of new statistical techniques, and more fundamentally from the detection ofnon-linearities and regime switches. Quite in the same spirit, for instance,Goldberg and Frydman (1990) conclude thatno one set of fundamental variables is able to explain adequately the entireperiod of floating rates. Instead, different sets of fundamentals . . . are found toexplain the data reasonably well within the separate regimes of parameterconstancy (p.3).Their finding suggests that the data-generating process of exchange rates may beneither unique nor stable. This challenges any simple idea of learning mecha-nism based on recursive estimations of a given, invariant, structural model.In an earlier preliminary experiment, Beltrametti et al. (1995) only comparedthe CS forecasts of the dollar-mark exchange rate with those of a set oftheoretical fixed rule. Our present experiment has been run on two monthly datasets of the dollar-mark and the dollar-yen rates and their respective theoreticaldeterminants from 1973 to 1992. We have chosen two exchange rates asrepresentative of two possibly different market environments. The two data setshave been split into two sub-periods: the 1973-1990 sub-period has been used asthe training set, the 1990-1992 subperiod as the out-of-sample validation set.Both in-sample and out-of-sample AA’s forecasting performance is evaluatedagainst some selected theoretical fixed rules and a vector autoregression model(VAR) that we have used as a control device. We wish to stress that our AA hasbeen designed to simulate the basic ‘human’ learning techniques as they arecurrently understood in artificial intelligence; this means that the comparisonbetween AA and the VAR should not be taken as a test of whether AA is a betterpredictor than the VAR but only as a test of AA’s goodness of fit by means of  1546 L. Belrrametri et al. /Journal of Economic Dynamics and Control 21 (1997) 1543-1575 a formal statistical tool. Section 4 of the paper explains how the theoreticaldeterminants of the exchange rates have been selected, and Section 5 illustratesthe experiment design.Since we have used historical data instead of an in-built ‘true’ paper-model ofthe exchange rate (and of course we do not know the ‘true’ real-world model ofthe dollar exchange rate), our experiment can say nothing about the conver-gence of learning to RE. However, this experiment may help shed some light ona few important preliminary issues:(1) Can an individual in the conditions specified above learn to forecast theexchange rate?(2) What kind of learning takes place in the face of a seemingly random variable?(3) Do fundamental variables play any role in the learning process? And if theydo, which are they?The results of our experiment are presented in Section 6, and our conclusionsfollow in Section 7.2. Some preliminary methodological problems Learning in economics is the learning of an agent about the structuralrelationships existing among payoff-relevant variables. This is a very specialnotion of learning, although a strand of cognitive sciences support the view thathuman beings organize their knowledge by constructing ‘models of the world’ ina way that economists may find familiar with their idea of learning and evenwith their own practice of modelling.’The above conception of learning has long raised interest among economists(e.g. Knight, 1921, Chapter 7); however, it has become predominant over the lastdecade in the development of the REH, which is now viewed as appropriate tostationary states after the learning process of agents has been completed success-fully (Lucas, 1986; Marcet and Sargent, 1988; Sargent, 1993). In fact, the basictenet of the REH is that agents should form their expectations at time t on anyeconomic variable yt + it i 2 1, on the basis of all available information about therelevant determinants of y at times t + i. A general log-linear form for theexchange rate is (Frenkel and Mussa, 1985, pp. 726-727) e, = Kx, + ~E,(e,+i - e&(1)where i = 1, . . . , co, E,() denotes the expectation operator conditional on in-formation at t, a the sensitivity of e, to its own expected values, K the (row) ‘This is particularly true of ‘constructivism’ (see e.g. Tamborini, 1991).  L. Beltranwiti et al. /Journal of Economic Dwamics and Conrrol 21 II 997) 1543-15751541 vector of structural parameters, and x, the vector of determinants. The REHmakes it possible to solve (1) by iteration to obtainE,(e,+J = (1 + a)-‘C(a/l + a)ik’E,(~,+i),(2)that is to say, the conditional expectation of e,,; is the weighted sum of itsdeterminants into the future. Substituting (2) into (l), the current value ofe, under rational expectations results to be e, = (1 + a)-‘kk, + C(a/l + a)‘+‘k’E,(x,+J I (3)which proves that e, is fully determined by the current and discounted futurevalues of its determinants.’If the variables in x, evolve randomly, Eq. (3) also entails the strong resultsthat the time series (et-i, x,_~> will allow any observer to obtain an efficientestimate of a and k,3 while any future change e,+i - e, is not forecastable(efficient market hypothesis). However, it is apparent that we cannot constructEq. (2) from (1) - that is, we cannot use Eq. (3) straightforwardly - unless we haveproved that any agent is actually able to learn the structural relationship e,(x,)that appears in (1).Designing a learning machine makes it clear that taking Eq. (1) as the objectof learning of the individual agent raises two crucial problems that have in factbeen central to the debate over the REH. First, we should find what information,i.e. what x is to feed the learning process. Second, we should specifiy how thelearning procedure deals with the unobseroable market expectations thataffect e,.2.1. The problem of model heterogeneity Define (e,, z,> as an external state to an observer at time t, with XEZ. As is wellknown, the REH implies that (i) there exists one single set of determinants x,(ii) there exists one single set of structural parameters k, (iii) the above (i) and (ii)are free common knowledge to all agents (by implication, so is also the ‘true’model (3)). These conditions underlying the use of the REH have given rise toserious criticisms since they conflict with empirical observation of market life,with fundamental cognitive principles, and, for practical purposes, with theactual state of economic theory.4 At this stage, we concentrate on the last pointfirst. 2Note that for a -+ co the current value of x, disappears and e, = ~ik’E,(x,+i)).31n other words, RE equilibria, when they exist, are self-enforcing (see e.g. Grossman, 1981).40n this debate see Frydman and Phelps (1983, 1990), Frydman (1983), Pesaran (1987) andTamborini (1991).
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