A learning-to-forecast experiment on the foreign...

ELSEVIER Journal of Economic Dynamics and Control

21 (1997) 1543-157s

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

Luca Beltramett?, Riccardo Fiorentinib, Luigi Marengo”, Roberto TamborinibT*

a Institute of Economics, University of Genoa, Ita1.v

b 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 foreign exchange market by means of an Artificial Intelligence methodology (a ‘Classifier System’) which simulates learning and adaptation in complex and changing environments. The experiment has been run for two different exchange rates, the US dollar-D mark rate and the US dollar-yen rate, representative of two possibly different market environments. A fictitious “artificial agent” is first trained on a monthly data base from 1973 to 1990, and then tested out-of-sample from 1990 to 1992. Its forecasting performance is then compared with the performance of decision rules which follow the prescription of various economic theories on exchange rate behaviour, and the performance of forecasts given by VAR estimations of the exchange-rate’s determinants.

Keywords: Learning; Artificial Intelligence; Foreign exchange market JEL 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. Massimo

Egidi, Prof. Giovanni Dosi and Dott. Diego Lubian for their useful comments and suggestions on

previous 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 efficient use of all available information, taking ‘the’ relevant model of the economy to represent the (‘true’) data-generating process. More recently, however, the research in the foundations of rational expectations equilibria has focused on the agents’ learning process of the data-generating model. Now it is understood that rational expectations equilibria rest on the existence of appropriate convergent learning paths. It seems to us that four main questions emerge 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 learning process? (iii) How can agents take into account other agents’ expectations when forming their own ones? (iv) How can agents take into account the fact that the process to be learnt depends on the actions of the agents themselves who are learning?

These methodological issues are briefly discussed in Section 2. In this paper we do not tackle the issues under (iii) and (iv), but we present a tentative experiment that focuses on the simpler and more preliminary questions under (i) and (ii). To address the issue of learning and forecasting in this context, we have designed an experiment with artificial intelligence methods and real world data. The choice of artificial intelligence methods is based on two reasons. First, so far the results of research based on rational-expectations formal models of learning have not been conclusive: there is no way of establishing what is the ‘optimal’ learning technique a priori. As a methodological consequence, in this field the traditional criterion of optimality should be replaced instead with the criterion of ‘goodness of fit’ (on this point see Arthur, 1992). Second, and relatedly, the need for positive cognitive foundations of learning and forecasting activities is now more strongly felt, and has drawn the economists’ attention to artificial intelligence methods (see e.g. Sargent, 1993). The artificial intelligence model we have employed in our experiment is particularly well suited to these purposes.

We have implemented an artificial agent (AA) that observes a foreign exchange market ‘from outside a window’ (that is, in a way that its actions have no impact on the market outcomes), is instructed to observe a number of potential exchange-rate determinants, bets on the exchange-rate going up or down, and uses gains and losses to learn to forecast. AA embodies one of the most successful learning mechanism that have been suggested by artificial intelligence scientists: the ‘classifier systems’ (CS) (Holland et al., 1986). In a number of experiments, such a learning mechanisms has performed well in reproducing human learning-based decision-making in repeated multi-choice problems (Arthur, 1990, 1991; Holland and Miller, 1991). Morever, the CS seems particularly well suited to investigating learning processes because it is based on an explicit logic of external signals classification, and of signal-action matching. Alternative devices such as neural networks, which are now increasingly used in forecasting activities, may not be as much informative on the ‘reasoning’ behind

L. Belwametti et al. / Journal of Economic Dynamics and Control 21 (1997) 1543-1575 1545

the agent’s actions. The basic principles and structure of CCs are illustrated in Section 3.

Learning experiments with real-world data are not much developed, one reason being that artificial data-generating processes allow for greater control of results for theoretical purposes. On the other hand, we have found real data more challenging and appropriate to the positive aim of our experiment. Financial markets, for perhaps obvious reasons, are the natural candidates for learning and forecasting exercises and experiments. We have in particular chosen the foreign exchange market, namely the dollar-mark and the dollar-yen exchange rates, mainly because the theoretical debate over the determinants of exchange rates is highly controversial (the most powerful econometric techniques have been implemented without success in order to test one model against the others). Until recently, the profession was struck by the finding that a random-walk model of exchange-rate changes outperformed ‘fundamental’ models (Meese and Rogoff, 1983). On the other hand, Granger (1992) reports on a renewed confidence in asset-prices forecastability springing from the applica- tion of new statistical techniques, and more fundamentally from the detection of non-linearities and regime switches. Quite in the same spirit, for instance, Goldberg and Frydman (1990) conclude that

no one set of fundamental variables is able to explain adequately the entire period of floating rates. Instead, different sets of fundamentals . . . are found to explain the data reasonably well within the separate regimes of parameter constancy (p.3).

Their finding suggests that the data-generating process of exchange rates may be neither unique nor stable. This challenges any simple idea of learning mechanism based on recursive estimations of a given, invariant, structural model.

In an earlier preliminary experiment, Beltrametti et al. (1995) only compared the CS forecasts of the dollar-mark exchange rate with those of a set of theoretical fixed rule. Our present experiment has been run on two monthly data sets of the dollar-mark and the dollar-yen rates and their respective theoretical determinants from 1973 to 1992. We have chosen two exchange rates as representative of two possibly different market environments. The two data sets have been split into two sub-periods: the 1973-1990 sub-period has been used as the 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 evaluated against 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 has been designed to simulate the basic ‘human’ learning techniques as they are currently understood in artificial intelligence; this means that the comparison between AA and the VAR should not be taken as a test of whether AA is a better predictor than the VAR but only as a test of AA’s goodness of fit by means of

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a formal statistical tool. Section 4 of the paper explains how the theoretical determinants of the exchange rates have been selected, and Section 5 illustrates the experiment design.

Since we have used historical data instead of an in-built ‘true’ paper-model of the exchange rate (and of course we do not know the ‘true’ real-world model of the dollar exchange rate), our experiment can say nothing about the convergence of learning to RE. However, this experiment may help shed some light on a few important preliminary issues: (1) Can an individual in the conditions specified above learn to forecast the exchange 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 they do, which are they?

The results of our experiment are presented in Section 6, and our conclusions follow in Section 7.

2. Some preliminary methodological problems

Learning in economics is the learning of an agent about the structural relationships existing among payoff-relevant variables. This is a very special notion of learning, although a strand of cognitive sciences support the view that human beings organize their knowledge by constructing ‘models of the world’ in a way that economists may find familiar with their idea of learning and even with 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 last decade in the development of the REH, which is now viewed as appropriate to stationary states after the learning process of agents has been completed success- fully (Lucas, 1986; Marcet and Sargent, 1988; Sargent, 1993). In fact, the basic tenet of the REH is that agents should form their expectations at time t on any economic variable yt + it i 2 1, on the basis of all available information about the relevant determinants of y at times t + i. A general log-linear form for the exchange 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 information 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) 1543-1575 1541

vector of structural parameters, and x, the vector of determinants. The REH makes it possible to solve (1) by iteration to obtain

E,(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 its determinants into the future. Substituting (2) into (l), the current value of e, 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 future values of its determinants.’

If the variables in x, evolve randomly, Eq. (3) also entails the strong results that the time series (et-i, x,_~> will allow any observer to obtain an efficient estimate 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 construct Eq. (2) from (1) - that is, we cannot use Eq. (3) straightforwardly - unless we have proved 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 object of learning of the individual agent raises two crucial problems that have in fact been 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 the learning procedure deals with the unobseroable market expectations that affect 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 well known, 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 to serious criticisms since they conflict with empirical observation of market life, with fundamental cognitive principles, and, for practical purposes, with the actual state of economic theory.4 At this stage, we concentrate on the last point first.

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) and Tamborini (1991).

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It is apparent that the prescription ‘use all available information’ is not operational. It has long been recognized that learning is ‘theory-laden’ (Lakatos, 1970), that is, it takes place within a pre-definedframework of relevant phenomena. Cognitive research has shown that human beings select ‘relevant phenomena’ through a variety of heuristics which may depend on the object of learning, their outer environment, social or even genetic conditioning. In a very simplified view of the economic agent, we may agree with the REH that the current economic theory provides the main selection device of relevant phenomena. However, the restrictions imposed on z by the current economic theory is still insufficient, since it actually consists of different competing models. As a consequence, there is not a unique set of determinants which the learner might look at, but there are heterogeneous sets.’

To deal with this basic problem, we have followed the practice of deploying a general ‘eclectic’ model, in our case of the exchange rate, which is capable of reproducing more specific models by means of restrictions on parameters (e.g. Frydman and Goldberg, 1990,1993,1996). This general model has allowed us to sort out a vector of ‘potentially relevant’ determinants of the exchange rate which we have used as the x vector for AA (see Section 4). In this way, we have endowed AA with an ‘open-minded’ attitude towards the selection of the exchange-rate determinants.6

2.2. The problem of market expectations in the learning process

The second problem we have met in designing our learning machine is the presence of market expectations in Eq. (1). This basic fact raises two major questions, at the individual and at the aggregate levels: (i) how the individual learner can take into account the market expectations in Eq. (1) - the so-called ‘beauty contest’ problem,’ (ii) what happens to the learning process while all agents are trying to learn to forecast e, by means of Eq. (l), and hence they keep on changing Eq. (1) through their own learning - the so-called ‘market self- referentiality’.’ The problems of the beauty contest and of self-referentiality are

‘See especially Frydman and Phelps (1983, 1990) and Frydman (1983).

6Nonetheless, we should warn that no model exists which is ‘eclectic’ enough as to encompass all exchange-rate theories, thus it may be that some variables which are deemed to be relevant by some theory are neglected a priori.

‘Keynes (1936, Chapter 12) named this the ‘beauty contest’ bias that affects forward-looking markets.

*Also in this field the literature is now quite extended. We simply recall the papers collected by Frydman and Phelps (1983), Bray (1982), Bray and Kreps (1986), Pesaran (1987, Chapter 3), Marcet and Sargent (1988) and Sargent (1993).

L. Beltrametti et al. J Journal of Economic Dynamics and Control 21 (1997) 1543-1575 1549

now being examined extensively in the literature on learning and expectations.’ However, in the experiment we present in this paper we have chosen not to tackle these problems and to focus on more basic issues of individual learning, such as signal detection, variable selection, creation of action rules, etc. Thus we have assumed that AA observes the exchange rate vis-a-vis the set of theoretical determinants x, and should choose whether to sell or buy, while its actions do not affect the observed exchange rate. However, a residual problem is whether or not to assume rational expectations in the exchange rate models generating the theoretical determinants used in the experiment. As recalled above, if we assume that all agents in the market have learnt the ‘true’ model of e, (1) then their rational expectations will reveal such ‘true’ model to any external observer. But since discovering what is rational to believe was the task of our experiment, we found it inconsistent for us to predetermine the REH in Eq. (1) from the outset. Operationally, we have instead assumed static expectations in all the theoretical exchange-rate models (as shown by (3), static expectations conditional on the observation of x are also rational if the variables in x follow a random walk).

3. On classifier systems

Classifier systems” are computer-based systems which emulate adaptive learning according to two main cognitive principles: (i) the representation of the ‘external state’ of the world into an ‘internal state’ of the system, and (ii) the action-response-based updating of the internal state in order to increase the system’s adaptation to a complex and uncertain environment.” It should perhaps be stressed that (i) ensures that ‘adaptive’ in this context does not mean

‘The results to date are mixed since they are highly sensitive to the specification of the learning problem, e.g. whether the agent has to learn the relevant determinants and/or their parameters, whether or not the agent uses a priori knowledge, what is the learning technique adopted by the agent, whether interaction between the learner(s) and the market outcome is allowed or not. See e.g. the different results that Bayesian updating, square error minimisation and other statistical techniques yield under different circumstances in the papers by Frydman (1983) and Bray (1982, 1983).

“‘Cf. Holland (1986) and Holland et al. (1986) for a thorough presentation; Arthur (1992) and Holland and Miller (1991) for a general discussion of their usefulness for economics; Marimon et al. (1990), Palmer et al. (1994), Marengo (1995) Marengo and Tordjman (1994) for some economic applications.

“Complexity and uncertainty can be given the following meaning in the present context. Consider the external state at time t {e,, z,} as defined previously. The complexity of this state is greater the greater the number or combinations (nonlinearities) of the elements in r,. The uncertainty of the given state is basically of two types: (i) stochastic, if z, contains random elements and (ii) structural, if t, contains spurious variables.


‘mechanica!ly backward-looking’ as e.g. in the simple pre-rational expectations models. On the contrary, the mapping from external to internal states replicates the general idea that rational actions should be based on forecasts of their consequences in a ‘model of the world’. Classifier systems try to catch the idea that the learning agent is not endowed with a model of the world which is the ‘true’ one, or at least isomorphic to reality, and that the learning process cannot be reduced to a (Bayesian) refinement of the parameter estimates. The learning agent is assumed to hold only an imprecise, partial, possibly wrong and even internally inconsistent model of the world: learning becomes therefore essential- ly a matter of structural modification of such a model. Classifier systems implement such a process of model building and modification in a typically adaptive fashion.

3.1. Basic principles and structure

A classifier system (CS) consists of a set of condition-action rules, which constitute the learning system’s representation of the environment, and a series of mechanisms which drive the process of learning and adaptation on such representation. Adaptation and learning take place in an evolutionary fashion, that is by the combined action of selection among the existing rules and by generation of new rules. The evolutionary process is led by a fitness (payoff) function which measures the ‘goodness’ of the actions performed by the CS.

The core of a CS is a condition-action rule, that is, a syntactic rule of the ‘if...then...’ type. Conditions provide the system’s mapping from external into internal states. An external state is a vector of signals, and conditions ‘classify’ the environment into classes of signals which are considered equivalent for the purpose of action (i.e. they all activate the same action). Thus, an action a is activated when the system receives signals which fall into the condition C associated with a.

In most applications external signals are strings of n symbols over the alphabet (0, 1), for example, they can be binary numbers if the relevant environmental properties are numeric values, or they can encode the presence or absence (on/off) of certain binary properties. The condition part is also a string of n symbols over the alphabet (0, 1, #}, where the # is a ‘do not care’ symbol. I2 A condition is met when

- either a # appears in the condition,

‘*It can be easily shown that, when signals represent numeric variables, the ‘#’ symbols in the condition amount to defining (unions of) intervals on the space of signals.

L. Beltramerri et al. /Journal of Economrc Dynamics and Control 21 (1997) 1543-1575 1551

_ or the condition has a 0 or 1, and the same symbol appears in the corresponding position of the last detected external state13 (the system could be given a longer ‘memory’ simply by introducing more conditions which depend on previous states of nature, but this is a complication which we do not consider in our present model).

More formally, a condition

Ci := Cir . . . Cij . . . Gin with cij~{O, 1, #}

is satisfied if

Si := sir . . . siz . . . sin with SijE (0, 1)

is the last detected external state, and

either cij = #

or Cij = Sij. The action part is instead a string of length p over the same alphabet which

codifies a chosen action within the set of possible ones. If we define an action

Ai = air . . . aij . . . aiP with UijE (0, l,#},

then we shall define Ri the ith rule ‘if Ci then Al:

Ri: cil . . . Cij . . . C, * ai1 . . . Uij . . . ai,,.

Therefore, the state of a CS at a given point in time t is identified by the finite set R of order q of its rules:14

R(q, t) = {RI,, ... Rit, ... Kg}.

13Hence, note that each condition encodes a collection of subsets of the external states. Conse- quently, a set of condition-action rules defines a representation of the environment which, in general, is nor a partition of the environment itself but a subset of its power set. This kind of non-partitional representation allows therefore for such phenomena as partial ignorance, surprise, systematic mistakes, conflicting hypothesis, and allows for a model of learning as a reduction of such cognitive limitations and biases rater than as a simple update of probability estimates.

r4The fact that the set of rules is fixed introduces a limited computational capacity, in the sense that the repertoire of classification rules and actions is finite and usually ‘small’ relative to the environmental complexity. Moreover, it is kept fixed throughout the entire simulation.

1552 L. Beltrametti et al. /Journal of Economic Dynamics and Control 21 (1997) 1543-1575

External state Internal state conditions actions

Cl Rl - s 0 1

ul 0 l- Al

__ # 1 R2_ c2 A2

# #R3 c3 A3

c4 0 #__!.?_. A4

r---,---7

I I __----__---_i

/_____;

I I I I

/ I /___l___i (_______-___I

Fig. 1.

An example with an external state consisting of a string of length 2 is given in Fig. 1. Suppose, for instance, that the ith state is Si = (0 l}. Then all conditions (0 I},

{# l>,{# #>,{O #> are satisfied and the relevant rules of action are activated. How one of the possible actions Ai . . . A4 is chosen will be explained below.

An obvious extension is to consider each condition string as the catenationi of several sub-strings (sub-conditions) which classify different parts of the environment. This is the case of the simulations presented below, where each condition is actually composed by some sub-conditions, each of them classifying a different variable. The condition is satisfied when all the sub-conditions are satisfied.

In the simulations presented below, the CS receives five different signals, each being the value (in its binary encoding) of a ‘fundamental’ variable which could determine the exchange rate (see below, Section 4). Thus, each condition is made up of five sub-conditions, each of them classifying one of such variables.

A CS is able to develop ways of discriminating between contradictory signals, like (in our example of exchange rate determinants) different variables moving in directions which suggest opposite actions. The CS in fact produces rules with thresholds and intervals in which one variable will be considered more influen- tial than another, possibly depending on the value(s) taken by other variables. Fundamentals (in their binary encoding) are the input signal given to the CS, which adaptively builds ways of classifying such signals. It can be easily shown (e.g. Holland et al., 1986) that the condition part of a single classifier rule defines a set of intervals on the space of the input variables: a classifier rule can be therefore read as: ‘if the value of fundamental A belongs to the union of some

ISGiven two strings of symbols aI . . . a2 . . . a, and b, . . . bz . . . b, the catenation is simply the new string of length n + m which is originated by joining the two strings: a, . a2 . . a, ._. b, . bz . . . b,.

L. Beltrametti et al. 1 Journal of Economic Dynamics and Control 21 (1997) 1543-1575 1553

intervals and the value of fundamental B belongs to the union of some other intervals and so on for all the fundamental, then do some prescribed action’. A fundamental part of the learning process in CS consists of adaptively modify- ing such intervals.

3.2. Evolution of rules and learning

At the beginning of the simulation (t = 0) the CS is set in a state of ‘absolute ignorance’: all its rules are generated randomly. The state of the CS is changed after each action following the procedures that will be explained below. In each state, each rule is assigned a strength and a specijcity measure. The strength measures the past usefulness of the rule, that is the payoffs cumulated every time the rule has been applied (minus other parameters which will be examined later); the spec$city measures the strictness of the condition. In our case Ri will have the highest specificity value if CL contains no #‘s and therefore is satisfied when one and only one particular external state occurs, whereas Ri will have the lowest specificity if Ci contains all #‘s and is therefore always satisfied by the occurrence of any external state. The reciprocal of this specificity coefficient is a measure of the generality of the rule: it reaches its maximum value when the condition part is entirely formed by #‘s and thus it is always satisfied by every environmental state whatsoever.

As already mentioned, this set of rules undergoes two different processes: selection and generation of new rules. Selection among the existing rules is performed by processing the set of existing rules through the following steps:

(1)

(2)

Condition matching: A message Si is received from the environment which informs the system about the current state. Si is compared with the conditions of all the rules: those rules whose conditions are matched by Si are said to be activated and enter the following step.16 Competition among activated rules: All the activated rules compete in order to designate the one which is allowed to execute its action. To enter this competition each activated rule makes a bid which is a function of its strength and its specificity. More precisely, the bid of each selected rule is proportional to its past usefulness (strength) and its relevance to the present situation (specificity) according to the following formula:

. Bid(Ri, t) = gl(g2 + g$pecificity(RJ)Strength(R,, t),

160f course, several (possibly all) conditions associated with different rules may be satisfied by the present state of the world. It is also possible that none of the existing rules is matched by the environmental signal, in this case (which is a state of ‘surprise’ for the CS) we introduce a new rule which is formed by ail ‘#‘s in its condition part and is therefore matched by all possible signals and has a randomly selected action. In order to keep the number of rules constant, whenever such a new rule must be introduced, the weakest existing rule is discarded.

1554 L. Beltramerii et al. /Journal o/Economic Dynamics and Control 21 (1997) 1543-1575

(3)

where gl, g2 and g3 are constant coefficients.” The winning rule is chosen randomly, with probabilities proportional to such bids.

It must be pointed out that making the bid depend on the specificity coefficient allows the endogenous creation in a CS of what Holland (1986) calls hierarchies of default, i.e. an internal organization of rules which contains both specific rules for well-understood and frequently experienced states of the world, which, given their higher specificity, tend to be preferred when such states occur, and more generic rules which apply to the other cases. Action rewarding and strength updating: the winning rule executes the action indicated by its action part. The action receives a payoff (positive or negative) according to the evaluation given by the exogenous payoff function. Then the relevant rule has its own strength reduced by the amount of the bid and increased by the payoff that the action receives. If the jth rule is the winner of the competition, we have:

Strength(Rj, t + 1) = Strength(Rj, t) + Payoff(t) - Bid(Rj, t).

So far we have examined the mechanism of adaptive selection among a set of existing rules, but the system must be able not only to select the most successful rules, but also to discover new ones and modify the very structure of the learning system’s representation of the world. This is ensured by applying genetic operators which, by recombining and mutating elements of the already existing and most successful rules, introduce new ones which could improve the performance of the system. In this way new rules are constantly injected into the system, and scope for new search is always made available.

Genetic operators generate new rules which explore other possibilities in the proximity of the presently most successful ones in order to discover the elements which determine their success and exploit them more thoroughly: the search is not completely random but is influenced by the system’s past history. New rules so generated substitute the weakest ones stored in the system, so that the total number of rules is kept constant.

Genetic operators are applied” to the rules which have the highest strength: a few rules are randomly selected with probabilities proportional to their current strength and they can generate new rules which are copies of themselves modified by the genetic operators. The rationale is that successful rules must

17These coefficients are chosen so as to ensure the dynamic stability of the system. We refer the reader interested in this technical detail to Holland et al. (1986).

‘*The frequency with which genetic operators are applied is usually lower than the one of the selection operators, because useful genetic material (i.e. useful building blocks for effective decision rules) must have the time to be selected before being used in the generation of new rules.

L. Belhvnelti er al. 1 Journal of Economic Dynamics and Conrrol 21 (1997) 1543-1575 155s

contain useful elements for the classification of the environment, and thus further exploration in their ‘neighbourhood’ could produce even better rules.

The number of rules present at any time is kept constant by discarding in the system a number of existing rules equal to the number of newly generated rules. Such rules which are eliminated are selected among the weakest ones, with probability inversely proportional to their strength.

Our CS has been endowed with two genetic operators for the condition and one for the action part of each rule. The latter is simply a mutation: a new action, among the possible ones, is randomly chosen.

The two operators used for the condition part deserve more attention because of their role in modelling the evolution of the state of knowledge embedded into the system. They operate in opposite directions.

(1) Specijication: A new condition is created which increases the specificity of the parent one. Wherever the parent condition presents a ‘#‘, this is mutated into a ‘0’ or a ‘1’ with a given (small) probability.

(2) Generalisation: the new condition decreases the specificity of the parent one. Wherever the latter presents a ‘0’ or a ‘l’, this is mutated into a ‘#’ with a given (small) probability.

Specification and generalisation are two possible cognitive attitudes which tend to drive the learning system towards, respectively, specific rules which apply to more specific external states and more robust rules which instead cover a wider set of states. Different degrees of specification have been simulated both by means of different combinations of these two genetic operators and by varying the coefficient g3 of specificity in the bid equation: the higher this coefficient, the more highly specific rules will be likely to prevail over general ones.

4. Theoretical determinants of exchange rates

As explained above, our first step has been to identify a number of theoretical determinants of the exchange rate which are eligible for feeding the learning process. Because of the problem of model heterogeneity, we have first devised a general-equilibrium macro-economic model of the markets for money, goods and currency where no a priori restrictions on parameters or variables appear (for a similar model see Frydman and Goldberg, 1993). Then we have proceeded to several different specifications.

4.1. The general model

The general model is in log-linear form and is built for the US as the domestic country, whereas Germany (Japan) is the foreign country with variables denoted

1556 L. Behametti et al. /Journal of Economic Dynamics and Control 21 (1997) 1543-1575

with (*). The same symmetric model holds for the foreign country (i.e. as usual it is assumed that the parameters are equal in the two countries).

(44

y, = b(i, - 71,) + b2yt + b3k + P: - pt) + by: + u,, PW

c,(e, + p: - pt) + c2(yy1 - y:) + c3(it - i: - E,(de)) = 0.

(4a) is the equation of money market equilibrium, where m, is money supply and a,~,, a2y,, a3i, are respectively the price, income and interest components of demand; (4b) is the equation of goods market equilibrium, where y, is the supply of goods, n, is the inflation rate, bl(i, - 71,) and b2y, the real interest rate and the income components of domestic demand, b3(et + p: - p,) and b4y: the real exchange rate and the foreign income components of foreign demand, U, repres- ents exogenous excess demand shocks (e.g. public deficits); (4~) is the equation of the balance-of-payments (currency market) equilibrium, where c,(e, + p: - pJ is the current account depending on the real exchange rate, and c3(it - i: - E,(de)) is the capital account depending on the expectations aug- mented interest rate differential. Moreover, E,(de = e,+{ - e,) is the expected rate of appreciation, < 0, or depreciation > 0, of the exchange rate. Obvious a priori restrictions are c1 = b3 and c2 = bq. We also assume c3 ---, oc), or perfect capital mobility, which implies i, = ir - E,(de), or uncovered interest parity.

Established exchange-rate theories differ on two grounds: (i) the identification of exogenous and endogenous variables, and (ii) the value of the parameters. We shall consider the two traditional alternative specifications given by the monetarist and the Keynesian theory.

4.2. The monetarist modellg

The monetarist model takes the money supply m, and the output supply y, as exogenously fixed by the central bank, and by technology and consumer tastes, respectively. Hence, the three equations of the structural model determine the three respective endogenous variables (p,, it, e,). The same holds for the foreign country. Hence, we have five linearly independent equations for five variables (IA, L P:, if, e,).

The monetarist model is also characterised by the following restrictions on parameters: a, = 1 since money demand is homogeneous of degree 1 in the general price level; a2 > 0 is the reciprocal of the velocity of circulation; a3 = 0

lgAmong the many general treatments of the monetarist model, see Frankel(1983), and Frenkel and Mussa (1985).

L. Beltrametti et al. /Journal of Economic Dynamics and Control 21 (1997) 1543-1575 1557

since there is no demand for money as store of value; br < 0 is the real interest rate elasticity of demand; b2 = b, = 0 capture the permanent income hypothesis; cr -+ co reflects the assumption of homogeneous goods and perfect arbitrage, which implies pt = e, + p:, . 1.e. purchasing power parity. With the addition of E,(de) = 0 we obtain the following reduced-form equation for the dollar:

e, = kr(m, - m:) + kZ(yt - y:) + k3(u, - vf), k, > 0, k2 < 0, k3 < 0. (5)

According to this model the dollar depreciates, e, > 0 (appreciates, e, < 0) if the US relative money supply (m, - m:) increases (decreases) or the US supply of goods (y, - y?) falls (grows) or the US aggregate demand asymmetric negative (positive) shock.

4.3. The Keynesian mode12’

relative has an

To obtain the Keynesian specification of the general model, the money supply m, is still taken as given by the central bank, whereas output supply is assumed to be infinitely elastic at the given prices. Consequently, the level of output is determined by aggregate demand yt while the general price level pt is fixed unless exogenous changes (mainly in costs) occurs. Therefore, now we have (m,, pt, m:,

p:) as exogenous variables and (y,, it, y,*, if, e,) as endogenous variables. Moreover, the Keynesian model has the following typical restrictions on

parameters that differ from the monetarist ones: a3 < 0 because of the demand for money as store of value, b2 > 0 and b4 > 0 are the marginal propensities con consume and import, cr > 0 reflects trade specialisation and the real exchange rate elasticity of trade balance. With these parameters and E,(de) = 0 we obtain

e, = k,(m, - m:) + k5(p, - p:) + kg(v, - II:), k4 > 0, k5 > 0, k6 > 0 (6)

Therefore, in the Keynesian world the dollar depreciates (appreciates) if the US relative money supply increases (decreases) or the US inflation differential has a positive (negative) shock or the US aggregate demand has an asymmetric positive (negative) shock.

4.4. The observational set of the artijcial agent

Both the monetarist and the Keynesian model impose severe restrictions upon the determinants of the exchange rate with the respect to the general balance-of-payments equation (4~) and to the variables to which operators seem to attach importance in their assessment of exchange-rate dynamics. The reason

20We refer to the Mundell-Fleming family of models (see e.g. Dornbusch, 1980).

1558 L. Behamefti et al. /Journal of Economic Dynamics and Control 21 (1997) 1543-1575

why some of the ‘popular’ determinants of exchange rates disappear from Eqs. (4c), (5) and (6) is that such variables are treated as endogenous variables themselves along with the exchange rate. However, as is well known in econo- metrics too, if endogeneity is not perfect, say because of some time lag in adjustment, the exclusion of these variables would entail a loss of information. This is in particular the case of the interest rate differential, which in the presence of slow adjustments of prices dominates the short-run dynamics of the exchange rate (Dornbusch, 1976). Similar considerations suggest the inclusion of the lagged exchange rate value among the determinants as well. For sheer technical reasons - limiting the extent of x, - we have excluded the demand shock variables; this choice may be justified with the widely held view that aggregate demand shocks are higly correlated across integrated countries, so that (u, - $) = 0.

Therefore, we have presented AA with the whole set of ‘intermediate’ and ‘final’ determinants that appear in the theoretical models of the exchange rate, and one lagged value of the exchange rate, that is, the following vector of theoretical determinants:

xt: {k-l, Ah - m:), A(y, - ~3, A@, -p:), (i, - if)>,

where the A operator stands for the first difference of the variable with respect to the previous period, and all data are expressed in logarithms except interest rates.

5. Experiment design

Our experiment’s aim is to assess whether an AA represented by a CS can learn to forecast the exchange-rate changes profitably, in comparison with alternative forecasting techniques, by observing the set of theoretical determinants x, defined in Section 3. As we have already explained, there is no interaction between AA’s actions and the market outcomes.

5. I. The artificial agent’s task

The experiment runs on two data sets to assess the learning and forecasting ability of AA in possibly different environments. The first data set contains the 1973:6-1992: 12 monthly dollar-mark rate, where X, refers to US and Ger- many’s monthly variables, the other the 1973: 6-1992: 12 monthly dollar-yen rate, where x, refers to US and Japan’s monthly variables. The data for each variable are described in the appendix.

The data sets are split in two sub-periods each. The 1973:6-1990: 12 subsets are the training sets for AA and the VAR. The 1991: 1-1992: 12 subsets are the

L. Beltramerti et al. / Journal ofEconomic Dynamics and Control 21 (1.997) 1543-1575 1559

validation sets for out-of-sample forecasting. It is perhaps unnecessary to explain the importance of out-of-sample forecasting tests (see e.g. Granger, 1992). In the case of CSs, and in general for similar recursive learning machines, the need for this kind of test is particularly felt because of the possibility of ‘overfitting’ inbuilt in the system’s recursive exposure to the same data set during the training runs.‘*

Following the general principles explained in Section 3, we have first run the CS on the training sets. AA is supposed to be completely ignorant at the outset: it is endowed with a fixed number of 50 rules generated by the computer randomly. Each monthly realisation of the training set {e,,x,} is an external state to AA. The ‘fundamentals’ in x, are given as binary numbers (as Gray numbers, to be precise, in order to avoid problems of discontinuity in the representation), the exchange rate is given only as a payoff (see below). For each external state there exist five conditions, one for each variable in x, that has to be classified. Conditions are associated with the 50 rules, and the rules trigger three possible actions conditional on x,:

Al do nothing,

A2 buy 1 dollar,

A3 buy 1 mark (yen).

Suppose, for instance, the state of the ‘fundamental’ variables in t is x,, then a typical internal rule of AA reads like the following:

if {de,_, < de,-1 < de,_l and A(m, - m:) < A(m, - rn:) < A(m, - m:) - and. . .}

then do action Ai, i = 1, 2, 3,

where the upper and lower bars indicate the upper and lower limits of the interval of each variable that characterizes the condition part of the rule. Different conditions, and hence rules, are identified by different intervals which codify the external state of the ‘fundamental’ variables. Note that AA does not identify any external state of the market exactly, but creates ‘patterns of configuration’ which are in fact collections of market states such that all states that belong to a certain pattern trigger the same action in spite of the fact that they differ in some respect. Such a fuzzy logic of action has proved to be a powerful instrument of intelligent behaviour in noisy environments.

“On the other hand, it should be considered that the CS starts from completely random rules and

no a priori information (a much worse condition than ignorance or incompetence). Moreover, the

CS does not have a true ‘memory’ of what happened in the previous training runs because its rules

are constantly reshuffled by genetic operations.

1560 L. Beltrameiti et al. /Journal of Economic Dynamics and Control 21 (1997) 1543-1575

Table 1 The payoff scheme of actions

Al A2 A3

de, > 0 de, < 0

-6 -6 P(f) < 0 P(t) > 0

P(C) > 0 P(t) < 0

de, = 0

5 P(t) = 0

P(t) = 0

Once a decision has been made, AA is given the true exchange rate e, and its decision is immediately rewarded or punished according to the variation of the exchange rate on the previous period de, = e, - e,_ 1. The decision is rewarded if AA has switched away from the depreciated currency, or if it has not acted when the exchange rate has remained constant (as if there existed a small transaction cost). AA has zero initial wealth but unlimited free ‘borrowing’ if wealth is negative (learning is not constrained by the initial endowment). Before each run, AA’s cumulated wealth is set back to zero.

The payoff scheme P(t) of each possible action is reported in Table 1. P(t) is proportional to the absolute value of de,, which has been normalised to make it possible to compare the payoffs on different markets, The cumulated payoff &P(t) has been used as a measure of the forecasting ability of AA.22 It is important to recall that during the training runs, the payoff is the only force which drives AA in the search for rules which allow for a better performance. P(t) enters monotonically the function that generates the strength of the rules as explained in Section 3. The learning procedure consists of the repeated ‘experi- ences’ of the training set, but only the rules which have survived in the last month of the previous run are carried over.

The set of rules giving the best forecasting performance on the training set, are then ‘frozen’, and AA is exposed to the validation data set. The mechanisms of action and reward are the same as before, except that now no generation of rules is allowed.

5.2. Alternative forecasting techniques

The performance of AA has been assessed against several alternative forecasting techniques: some theoretical fixed rules (TFRs) deduced from the exchange

2zWe have not deduced risk and transactions costs from the cumulated payoff because we simply assume an absolute measure of successful actions that will be compared with that of alternative forecasting techniques.

L. Beltrametti et al. / Journal of Economic Dynamics and Control 21 (1997) 1543-1575 1561

Table 2 Theoretical fixed rules

44 - ma 4Y, - Y3 4Pc - P3 d(i, - i:)

EM 0 + - + _

EK 0 + + + _

PM 0 + 0 0 0 PPP 0 0 0 + 0 AM 0 0 0 0 _

OK 0 0 + 0 0

RW + 0 0 0 0

EM = Eclectic Monetarist, EK = Eclectic Keynesian, PM = Pure Monetarist. PPP = Purchasing Power Parity, AM = Asset Market, OK = Old Keynesian, RW = Random Walk.

rate theory in Section 4, the random walk model of exchange rate (RW)23 and an estimation of the exchange-rate data generation process based on vector autoregression (VAR).

The TFRs we have drawn from the models treated in Section 3 are sum- marised in Table 2. A (+) in the table indicates that the sign of the expected change in the dollar exchange rate is the same as the sign of the corresponding variable; the opposite holds for a (-). Thus, the actions Al, A2 or A3 generated by these rules have been treated exactly in the same way as those generated by AA obtaining a comparable cumulated payoff path.

As already recalled in the introduction, the RW model is widely considered as a good description of exchange rate movementsz4 and is often used as a benchmark for assessing the empirical properties of exchange rate theories. In our case it can be written as

de, = Ae,.el + E,, Ed N (0, a2)

and it implies that this period observed exchange rate variation is the best forecast of the next period exchange rate variation, so that the best action an agent can take is to sell (buy) dollars when a current depreciation (appreciation) is observed.

z3The use of the random walk model of exchange rate as an alternative forecasting technique was suggested by Prof. Stavros A. Zenios.

%ee for example Meese and Singleton (1982), Meese and Rogoff (1983), Wasserfallen and Kyburz (1985j, Baillie and’McMahon (1987, 1989). and Fiorentini (1991).

1562 L. Behametti et al. /Journal of Economic Dynamics and Control 21 (1997) 1543-1575

The VAR methodology is based on the idea of interdependence among macroeconomic variables. According to this methodology, it is possible to analyse the relationships between economic variables without the use of any ‘a priori’ theoretical structure since it is assumed that each variable affects and is affected by all the others. Theoretical inference can be drawn after the econometric model is estimated. The VAR methodology, therefore, is a ‘atheoretical’ way to study the economic reality: we observe data first and then we test whether economic theories (if any) fit the observed macroeconomic series. VAR models can be used not only to test economic theories but they are also useful tools for forecasting purposes. 25 Because of its lack of theoretical foundations we think that the VAR methodology is closer to the CS than other econometric methods and this is the main reason why in our experiments we compare the results obtained by a CS with the results of a VAR model applied to the foreign exchange market.

From a technical point of view, it is assumed that the time series are generated by a multivariate autoregressive process, so that a typical VAR analysis is performed by estimating a simultaneous equation system in which lagged values of all the variables in the model appear on the right hand side of each equation. To take into account the presence of heteroskedasticity in the data, the model was specified as a five variable VAR(3) with ARCH(l) errors26 as follows:

where pr = (PI - p:), m, = (m, - m;“), it = (it - i:), yt = (y, - y:).

The VAR length was chosen on the basis of the Schwarz‘s Criterion test in the following way: we recursively calculated the Schwarz’s Criterion on a fixed size sample running from 1973:6 up to 1992: 12. The result was that the lag that minimises the criterion is n = 3 in both the German and Japanese case. In the above equation, therefore, L(xij) = XIijL + x2ijL2 + x3ijL3.

25A good introduction to VAR methodology can be found in Liitkepohl (1993).

“This specification was suggested by a referee and was supported by the results of ARCH LM test applied to both the Mark and Yen data sets; see Engle (1982).

L. Beltrametti et al. /Journal of Economic Dynamics and Control 21 (1997) 1543-1575 1563

6. Experimental results

In this section we shall present our main experimental results. These will be analysed for the two data sets, the training set and the test set, separately. For each set we shall compare the performance of AA with that of the alternative forecasting techniques. Only the most significant comparisons and results are presented. The full account of the experiment is of course available on request.

6. I. Learning and forecasting in the training sets

The dollar-mark rate and the dollar-yen rate paths in the training sets are plotted in Figs. 2 and 3. As to the dollar-mark rate, we can easily spot three different sub-periods: in the first (from 1976:6 to early 1980) and in the third (from 1985 to 1992: 12) we have an appreciation of the mark, whereas from early 1980 to 1985:3 the mark falls. The dollar-yen behaviour is more complicated: we can identify five sub-periods, in the first (1976:6-1976: 12) the yen rate is relatively stable, while in the subsequent periods appreciation and depreciation alternate.

Data with relatively long trends such as these are double-edge training tools. On the one hand, trends, as long as they last, make AA’s task easier for the reason that a poor internal state, where naive and conservative rules are predominant and the genetics of better rules is weak, may well produce good results. On the other hand, it should be considered that the exchange-rate trends are quite noisy, and, more importantly, that trends eventually have turning points and detecting them is the acid test of intelligence in the financial community.

0.6

0.2 74 76 78 80 82 84 86 88 90

Fig. 2. Dollar-mark exchange rate (73 : 6-90: 12)


7000 -

6000-

5ooo-

4ooo..

3000-

2000-

lOOO_

Fig. 3. Dollar-yen exchange rate (73:6-90: 12).

Fig. 4. Mark cumulated payoff (training data set).

As already explained, the CS typically displays ‘emergent behaviour’, that is, the tendency of its internal state to find a stable configuration of rules spontan- eously after a certain number of runs. AA showed this tendency around the fifth, sixth run on both the training sets. Figs. 4 and 5 plot AA’s cumulated payoffs over the two training sets at the eighth run (remember that AA’s initial wealth was set to zero at the beginning of each run).

The following points are worth noting:

L. Beltrametti et al. / Journal of Economic Dynamics and Control .?I (1997) 1543-1575 I565

(1)

(4

(3)

74 76 78 80 82 84 86 88 90

Fig. 5. Yen cumulated payoff (training data set).

AA’s payoff is consistently positive over the whole period for both the dollar-mark rate and the dollar-yen rate. Put differently, the frequency of right actions is greater than that of wrong actions and reaches about 70% in the two different markets. Checking the AA’s actions report given by the computer, we see that AA proved to be able to capture the fundamental trends in the two exchange rates with a satisfactory degree of approximation. It consistently bought marks until mid-1980, then switched to the dollar until mid-1985, when it went back to the mark. Likewise, it mostly bought yen from 1976 to 1979, dollars from 1980 to mid-1985 then again yen until end-1986. From 1987 onwards, the dollar-yen rate shows a sharp downward sloping trend: AA almost always bought dollars. AA’s performance around the turning points of trends is given by its marginal payoffs. To give a visual assessment of AA’s behaviour in these circumstances, we reproduced the turning points in the exchange-rate paths of Figs. 2 and 3 on the time axis of Figs. 4 and 5. For instance, around the first turning point in the dollar-mark rate, from 1979: 10 to 1980:9 AA went wrong 50% of the time, with losses concentrated in the first six months and amounting to 14% of its previous total wealth. The other episodes show a similar pattern of behaviour of AA, i.e. a speed of adjustment to the new trend within five or six trials (months).

Figs. 4 and 5 also compare AA’s total payoff with the VAR’s total payoff on the same market. As we have already explained, we have used the VAR estimates and forecasts as a benchmark of AA’s performance. In the case of the

I566 L. Beltrametti et al. / Journal of Economic Dynamics and Control 21 (1997) 1543-1.575

Table 3 AA and VAR payoff summary statistic (training set)

Dollar-mark

Var AA

Dollar-yen

Var AA

Mean 30.57 25.92 17.73 25.57 Median 97.67 100.86 97.12 100.54 Maximum 112.58 112.58 109.56 109.32 Minimum - 107.78 - 110.75 - 109.09 - 112.22 Std. Dev 95.61 99.63 98.85 99.59

Variance F-test 1.085 1.015 P-value 0.277 0.457

dollar-mark exchange rate the payoff path is strikingly resemblant to that of AA in the first and third subperiods, while it is dominated by the AA payoff in the second period in which the dollar appreciated. At the end of the training set the cumulated payoff of the VAR is slightly greater that the AA‘s one. On the contrary, the latter outperforms the VAR systematically on the yen market.

A possible explanation of the AA’s better performance in the yen market is that it can be the result of higher payoff paid for higher risk. Accordingly, using the payoff variance as a surrogate for risk, the AA’s payoff variance should be significantly greater than the VAR one. This is not indeed the case, as it can be easily seen looking at the payoff summary statistics reported in Table 3. In fact, at a first glance the VAR and AA’s payoff standard deviation seems not to be very different in both markets. Such a conclusion is supported by the result of a formal F-test on variance equality. The values of the two F tests reported in Table 3 lead to the conclusion that the hypothesis H,,: rr$ar = c&, versus H1 : ai* > asar cannot be rejected, so that the good performances of the AA are not simply due to riskier positions taken in the foreign exchange market. We interpret this result as meaning that the adaptive learning technique employed by AA has generated forecasts whose results are not inferior to those given by a formal ‘state of the art’ statistical technique.

Now let us compare AA’s performance with that obtainable by applying the TFRs collected in Table 2. For the sake of brevity, we shall illustrate the dollar-mark rate only (the dollar-yen case giving broadly the same results).

In the first place we have considered the two principal ‘eclectic’ sets of theoretical rules, the Keynesian (EK) and the monetarist (EM) one. Fig. 6a shows the cumulated payoff of these two rules along with that of RW. Fig. 6b repeats the same kind of experiment for the other TFRs listed in Table 2.

L. Belrrametti et al / Journal ojEconomrc Dynamics and Conrrol 21 (1997) 1543-1575 1567

3000 -

-1000 , , , I , , / , , , , / , , , , (at 74 76 78 80 82 84 86 88 90

Fig. 6(a). Mark cumulated payoff of TFRs (training data set).

PPP

-2000 -

-3ooo- , , , , , , , , , , , , , , , I (b) 74 76 78 80 82 84 88 88 90

Fig. 6(b). Mark cumulated payoff of TFRs (training data set).

We first observe that AA, without any a priori knowledge but by searching for new rules through genetic algorithms, fares substantially better than any of the theoretical rules. Among the TFRs, the Keynesian ‘eclectic’ rule attains the best result, followed by the PPP rule, while the AM rule is the worst with a payoff falling below zero. However, even the payoff of the best TFR (EK) is about half the payoff of the AA.

As to the RW rule, we may note that its performance is quite good, even if not good enough to reach the AA’s payoff level. However, Fig. 6b shows that our

1568 L. Beltramelti et al. /Journal of Economic Dynamics and Control 21 (1997) 1543-1575

results are consistent with the empirical findings that in the foreign exchange market the RW behaves better than any other theoretical model.

We interpret the results on the training sets as evidence that the adaptive- evolutionary learning mechanism of AA is successful as compared with the benchmark given by a standard statistical technique and some standard theoretical rules. Some tentative inferences about learning and forecasting from these results will follow in the concluding Section 7.

6.2. Forecasting on the validation set

As already explained, this part of the experiment is aimed at assessing AA’s out-of-sample forecasting ability. More precisely, the test was expected to give information about whether the successful rules generated in the training runs could be carried over to a newly observed data set, where no genetics of rules was allowed. Note that such a test rests on a joint null hypothesis, namely that the new data generating process is structurally the same as in the training set. Here we limit the comparison between the two more successful techniques, the AA and the VAR.

Let us begin again with a plot of the dollar-mark and dollar-yen rates in the validation sets spanning from 1991: 1 to 1992: 12 - see Figs. 7 and 8. As can be seen, the mark rate remained on a fairly stable path (after the steady increase on the last four years of the training set), while the yen rate was on a rising trend (after the downward sloping trend in the last two years of the training set).

AA’s cumulated payoff is shown in Figs. 9 and 10 for the dollar-mark rate and the dollar-yen rate, respectively, while payoff summary statistics are reported in

0.65-

0.60-

Fig. 7. Dollar-mark exchange rate (91: l-92: 12).

L. Beltrametti ei al. / Journal of Economic Dvnamics and Control 21 (1997) 1543-1575 I S69

0.85

0.80-

Fig. 8. Dollar-yen exchange rate (91: l-92: 12).

-3001 ,, ,, ,, ,, ,, ,, ,,,,I 91fOl 91104 91!07 91ho 92101 92104 92f07 92:lO

Fig. 9. Mark cumulated payoff (test data set)

Table 4. First of all, it is apparent that in both cases the cumulated payoff does not exceed zero significantly at the end of the tests. This is the results of 50% incorrect actions on both markets; however, an important point is that AA did not change what it was doing at the end of the training set (i.e. buying marks against dollars, buying dollars against yen). We interpret this as a consequence of the fact that the rules inherited by AA were ‘frozen’, whereas both exchange- rate paths changed moving from the training to the validation set.


600

-SOOl,,,,, ,, ,, ,, ,, ,, ,I 91 :Ol 91:04 91 f07 91 f10 92101 92104 92f07 92110

Fig. 10. Yen cumulated payoff (test data set).

Table 4 AA and VAR payoff summary statistic (validation set)

Dollar-mark

Var AA

Dollar-yen

Var AA

Mean 3.06 - 0.35 5.88 - 0.29 Median 4.14 -0.13 104.08 106.82 Maximum 107.63 107.63 104.08 106.82 Minimum - 100.27 - 112.94 - 102.44 - 104.08 Std. Dev 101.97 105.65 100.21 104.22

Variance F-test 1.073 1.081 P-value 0.433 0.426

Support for this interpretation may come from the VARs parallel performance in Figs. 9 and 10. On both markets, the VAR’s cumulated-payoff path is generally above AA’s. However, while in the dollar-mark case it is very close to AA’s from 1991: 8, in the dollar-yen case the difference is greater. The VAR chose the wrong action 50% of the time on the former market (making exactly the same sequence of decisions as AA from 1991: 8), but 33% of the time on the latter (making the opposite sequence of decisions of AA). Overall, also the VAR’s out-of-sample performance is poorer than the in-sample performance, but it helps point out that AA’s failure is more critical on the dollar-yen market, i.e.

L. Beltramettr et al. /Journal of Economic Dynamics and Control 21 (1997) 1543-1575 1571

where the change in the exchange-rate path was sharper. We defer further comments on this finding to the next section.

7. Concluding remarks

The experiment we have presented in this paper was addressed to explore a few preliminary issues emerged in the recent research on learning and forecasting, namely: (i) whether an individual can ever learn to forecast a seemingly random price such as the exchange rate by observing some selected potential determinants and using gains and losses as signals in his/her search for better trading rules; (ii) what kind of learning takes place, if any, and (iii) whether the ‘fundamentals’ play any role in the learning process. We have created an AA using a CS as a learning device that simulates the basic features of human learning, and we have compared its forecasting performance against the VAR econometric technique and some theoretical rules.

Our results are mixed, but, overall, we find them informative on the extent of human learning in market environments, on the relationship between learning and forecasting in general and, more in particular, on the way the CS works.

First, within the limits of this experiment, we may say that our AA ‘dis- covered’ and exploited some empirical regular patterns in the co-evolution of the exchange rate and its selected theoretical determinants in a noisy environment. AA’s performance was at least as good as the one of the VAR. This is not to say that AA found out a structural relationship between e, and X, in an econometric fashion; rather, it was able to classify the external states that evolved through time into patterns that when were recognised triggered appropriate actions.” This is the typical behaviour of CSs. It is produced by a simple hierarchical system of rules, where very general rules are employed along with more specific ones which deal with special cases, and which in particular signal the swings of the trends.

As to the ‘weights’ of the variables in x,, after examining the internal state of the CS we can only say that the strongest rules at the end of the training runs ignored e,_ 1, whereas all the other variables were present. This is too scant a basis to make inferences about exchange-rate forecastability. However, our result may suggest that the belief that no information whatsoever can be extracted from (some selected) ‘fundamentals’ may be questioned. In a word, we may rather subscribe to the idea that ‘theoretically informed predictions’

*‘Cognitive research has found evidence that people do make use of ‘linear mental models’ in the

presence of multivariate problems of inference or forecasting (see e.g. Arkes and Hammond, 1986).

However, the variables entering these mental models are assembled to form patterns associated with

rules of action, such as ‘if money supply is high and interest rates are low, then.. .‘, etc.


may do reasonably well (see e.g. Frydman and Goldberg, 1990, 1993, 1996). Further support to this intuition is provided by the VAR’s in-sample performance in our experiment.

Second, the fact that both AA’s and the VAR’s forecasts fared significantly better than the one of the theoretical fixed rules also deserves some attention. AA and the VAR share an ‘atheoretical’ attitude towards the market. They are given ‘theoretical information’ but with no a priori structure imposed upon the relevant variables. Now, look at Fig. 5a: from the first to the 150th month (i.e from 1979 to mid-1985) the Random Walk and the Eclectic Keynesian rules follow roughly the same path, but then they diverge and the Keynesian rule is overtaken until the end of the period. This ‘switching of rules’ exactly happens after a sharp change in the dollar-mark trend occurred, and it seems to support the view that structural changes in the data-generating process are mostly responsible for the breakdown of orthodox exchange-rate forecast models. Note that in the same period of time, AA undergoes sharp losses, but then it recovers quickly. Hence it seems that adaptiveness on a ‘goodness-of-fit’ basis, rather than being stuck to a priori structures, allows for greater ability to ‘understand’ environmental structural changes and to switch to new ‘good’ rules appro- priately.

Third, AA’s performance as out-of-sample forecaster was not as good as it was as in-sample pattern detector. This was caused by a marked inertia in the rules-actions used by AA in the new setup with respect to the one where it was trained. The rules created during the training runs failed to give correct re- sponses to the new signals more than half of the time. Though the VAR showed similar problems, our finding suggests that the recursive learning process may in fact induce some overfitting of the CS’s rules. However, behind this failure we find an interesting hint. The point we wish to make is that the genetics of the rules, i.e. the mechanism behind adaptive learning, should not be ‘frozen’, unless the enoironment is stable. Arthur (1992) draws attention to the same point, reporting that in an experiment with interacting AAs in an artificial financial market, ‘we find no evidence that market behaviour ever settles down (. . .) When we ‘freeze’ a successful agent’s predictors early on and inject the agent into the system later, the formerly successful agent is now a dinosaur’ (p. 24). In markets which operate far from structural stability, i.e. rational-expectations equilibria, learning and forecasting cannot be separate activities; learning-to-forecast is an ever going challenge.

On normative grounds, our results may add support to the view that human learning techniques, though sharply different from formal models of optimal information processing, may be quite effective, on a ‘goodness-of-fit’ basis, even in highly noisy market environments. An entirely open issue that we have not addressed here is how markets peopled with such human learners work, in particular whether markets will allow for some stable action-rule pattern of agents, and will eventually retain optimal allocative properties. A promising

L. Behametti et al. 1 Journal of Economic Dynamics and Control 21 (1997) 1543-1575 1573

research strategy opened by artificial intelligence methods consists of creating groups of interacting artificial agents (see again Arthur, 1992; Palmer et al., 1994).

Appendix. The data set

All the data were taken from the International Statistic Yearbook (KY) databank on CD-Rom and are described as follows:

_ dollar-mark and dollar-yen exchange rates: bilateral exchange rates (monthly series - source: Eurostat)

- US, German and Japanese money supply: Ml (monthly series - source: Eurostat)

_ US, German and Japanese prices: CPI (monthly series - source: Eurostat) - US and German interest rates: three months treasury bill interest rates

(monthly series - source: Eurostat) _ Japanese interest rates: three months deposit rates (monthly series: source:

IMF) _ US, German and Japanese GDP: real GDP (quaterly series: source: Eurostat)

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