On forecasting exchange rates using neural networks

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  • This article was downloaded by: [Moskow State Univ Bibliote]On: 18 February 2014, At: 01:28Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

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    On forecasting exchange rates using neuralnetworksPhilip Hans Franses & Paul van HomelenPublished online: 07 Oct 2010.

    To cite this article: Philip Hans Franses & Paul van Homelen (1998) On forecasting exchange rates using neuralnetworks, Applied Financial Economics, 8:6, 589-596, DOI: 10.1080/096031098332628

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  • Applied Financial Economics, 1998, 8, 589 596

    On forecasting exchange rates using neuralnetworks

    PHILIP HANS FRANSES and PAUL VAN HOMELEN*

    Econometric Institute and Rotterdam Institute for Business Economic Studies, ErasmusUniversity Rotterdam, The Netherlands and *Institute for Research and InvestmentServices, Rotterdam, The Netherlands

    The paper considers the modelling, description and forecasting of four daily exchangerate returns relative to the Dutch guilder using arti cial neural network models(ANNs). Based on simulations it is argued (i) that neglected GARCH does not lead tospuriously successful ANNs and (ii) that if there is some form of nonlinearity otherthan GARCH, ANNs will exploit this for improved forecasting. For the sample data itis found that ANNs do not yield favourable in-sample ts or forecasting performance.These results are interpreted as indicating that the nonlinearity often found inexchange rates is most likely due to GARCH and therefore ANNs are recommendedas a diagnostic for mean nonlinearity.

    I . INTRODUCTION

    When daily or weekly nancial time series are analysedusing tests for generalized autoregressive conditional hete-roscedasticity (GARCH) or for other forms of nonlinearity,one typically nds empirical evidence for both features.Bollerslev et al. (1992) summarize many studies involvingGARCH, while for example Hsieh (1988 and 1989) andScheinkman and LeBaron (1989) nd substantial nonlin-earity in nancial data. The key di erence between GARCHmodels and many other nonlinear models is that theGARCH model usually cannot be exploited to generatenonzero out-of-sample forecasts for the returns of a nan-cial asset, while many nonlinear models can be used as such.In fact, the GARCH model is speci cally designed to gener-ate out-of-sample forecasts for the variance of a series ofreturns, and hence this model is often used for optionpricing, among other uses.Recent applications of arti cial neural network models

    (ANNs) to exchange rates in, for example, Kuan and Liu(1995), Swanson and White (1995) and Genay (1996) showthat these models can describe in-sample data rather well,and that they also sometimes generate good out-of-sampleforecasts, where g`ood is usually de ned in terms of smallmean squared prediction error or in terms of directionalaccuracy of the forecasts. An ANN can be viewed as a gen-eral nonlinear time series model, and hence the apparent

    nonlinear features in the daily exchange rate data analysedin the above studies seem to be picked up by the ANNs.Recent results documented in Brooks (1996), however, sug-gest that such nonlinear features, which are often elicited bystatistical tests and by the success of empirical nonlinearmodels tted to the data, may be due to neglected GARCH.In other words, it may be that GARCH in daily exchangerates leads to the nding of forms of nonlinearity which inturn may suggest that one can forecast the returns on theexchange rates themselves. Obviously, when GARCH leadsto the spurious application of nonlinear models such asneural networks, there may be no forecasting gain.In this paper we aim to contribute to the area of research

    using ANNs to describe and forecast daily exchange ratereturns by analysing the possible e ects of GARCH. Simu-lation experiments are used to investigate the possibilitythat ANNs mistakenly pick up GARCH properties of thedata, and hence mistakenly suggest that the returns series canbe forecasted. In the same set of simulation experiments wealso examine whether some speci c forms of nonlinearity,where two bilinear models and one exponential autoregres-sive (EAR) model are included, can be captured by the moregeneral (and possibly overparameterized) ANN. To mimica realistic practical situation we focus on the fraction oftimes that a model adequately forecasts the sign of thereturn on the following day. For four daily exchange rates,the US and Canadian dollars, the British pound and the

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  • Japanese Yen (all versus the Dutch guilder), the in sample t and out-of-sample predictive performance of variousANNs is evaluated. Several possible model variants con-cerning lag structures and the number of hidden layer cellsare considered since this allows one to evaluate whetheroften applied model selection criteria result in good fore-casting models.The outline of the paper is as follows. In the next section,

    the empirical strategy which is followed throughout thispaper is described. In Section III, the results of some simula-tion exercises are presented. In Section IV, the outcomes ofmodelling and forecasting four daily exchange rate series arepresented. In Section V it is concluded that these ndingsadd to recent work by Brooks (1996), in the sense thatapparent nonlinearity in exchange rates seems due toGARCH, and some suggestions for further research areprovided.

    II . EMPIRICAL APPROACH

    Consider a daily observed real exchange rate series xt, witht = 0, 1, 2, , n and de ne the returns yt as yt = log(xt/xt 1 ) for t = 1, 2, , n, where log denotes the natural log-arithm. In this paper, we aim to describe yt by the arti cialneural network model with a linear component, i.e.

    yt = m + / 1yt 1 + / 2yt 2 + + / pyt p

    +q

    +j = 1b jG (m j + / 1 jyt 1 + / 2 jyt 2 + + / pjyt p)

    + e t (1)

    with

    G (a) = (1 + exp( - a)) 1 (2)

    where e t is some error process, and where m , / 1 to / p, b j , m jand / 1 j to / pj are unknown parameters, for j = 1, 2, , q.Hence, the model contains (q + 1) (p + 2) - 1 parameters.The ANN model in Equations 1 2 contains p lags of yt andq so-called hidden layer cells. G (a) is the logistic activationfunction, which connects the p + 1 input components (i.e.p lagged yt variables and a constant) with the hidden layer.The hidden layer is connected with the output variableyt through the b j parameters. The model in Equation 1 canbe abbreviated as an ANN(p, q, 1) model. Model 1 2 is oftenused in the empirical analysis of economic and nancialtime series data (Swanson and White, 1995; Kuan and Liu,1995). One possible motivation for its frequent use is thatKuan and White (1994), for example show that with largeenough q, an ANN can approximate (with arbitrary pre-cision) any function f that connects yt with its past realiz-ations, i.e. any function f in yt = f (yt 1 , , yt p) + e t .This property of ANNs make them useful in recognizinga variety of patterns in time series data with great precision;

    see, for example, Ripley (1994), Bishop (1995) and Fransesand Draisma (1997). This also leads to our choice of anANN to describe and forecast potentially nonlinear dailyexchange rate data.The parameters in the ANN are estimated by minimizing

    the residual sum of squares. In the rst round of optimiza-tion, a penalty is imposed on the values of the parameters, asis advocated in Ripley (1994). In the last round, this penaltyis removed. The estimation procedure uses the simplexmethod of Nelder and Mead and the Broyden Fletcher Goldfarb Shanno algorithm, of which the latter is avail-able in the Gauss optimization toolbox (version 3.1.1).The Gauss programs are available from the authors onrequest.The parameters in Equation 1 are estimated for p, q

    {1, 2, 3, 4, 5}. Hence, we do not explicitly focus on the linearmodel (i.e. the case where q = 0) but allow for at least somedegree of nonlinearity in our simulated data and our sampleseries. This leads to the estimation of parameters in 25di erent ANN models. All these models will be used forout-of-sample forecasting. In doing so, we will be able toexamine whether certain in-sample model selection criteria,such as the Akaike and Schwarz information criteria, lead togood forecasting models.We focus on describing a time series of returns yt by the

    ANNmodel in Equations 1 and 2 for the rst n observationsand on forecasting the next m observations n + 1 to n + m.Then the estimation sample is shifted to m + 1 until n + m,and the observations in n + m + 1 to n + 2m are forecast. Intotal there are n + 4m observations, and hence four modelevaluation samples. Notice that the in-sample model selec-tion criteria to be calculated below are averages over thesefour cases. One-step ahead forecasting only is considered,i.e. the model structure and the values of the tted para-meters (obtained from the rst n data points) are heldconstant, while the truly observed data yn+ k to forecastyn+ k+ 1 , for k = 1, 2, , m - 1 are used. In simulation ex-periments, n was set at 750 and m at 250, since for dailyobserved returns this roughly corresponds to three years ofdata to t the model, and one year of data to evaluate theforecasts, respectively. For larger values of n and othervalues of m the results obtained were not qualitativelydi erent from those reported in the next two sections.There are several approaches to evaluate the in-sample

    and out-of-sample performance of time series models.For the in-sample performance the Akaike and Schwarzinformation criteria is used to indicate the best model.Additionally, and in order to mimic a practically realisticsituation, it is investigated whether or not the sign of the tted return yW t equals that of the truly observed return yt , fort = 1, 2, , n. For this purpose, the success ratio (SR) isde ned as

    SR = n 1n

    +t= 1

    It[yt yW t > 0]

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  • where It[ . ] is an indicator function that takes a value of1 when its argument is true and a value of 0 otherwise.The out-of-sample forecasts are also evaluated by com-

    paring the sign of the forecast observation yW n+ i with the truewithheld observation yn+ i , for i = 1, 2, , m. Again, weuse the SR measure, which for the out-of-sample data isgiven by

    SR = m 1m

    +i= 1

    Ii[yn+ i yW n+ i > 0] (3)

    For the out-of-sample forecasts, it is tested whether or notthe SR value in Equation 3 di ers signi cantly from an SRvalue that would be obtained in the case where the yn+ i andyW n+ i are independent. For this purpose, the test proposed inPesaran and Timmermann (1992) is used. De ne

    P = m 1m

    +i= 1

    Ii[yn+ i > 0]

    and

    P = m 1m

    +i= 1

    Ii[yW n+ i > 0]

    then the success rate in the case of independence (SRI) ofyn+ i and yW n+ i is given by

    SRI = PP + (1 - P) (1 - P )

    with variance

    var(SR) = m 1 SRI(1 - SRI). (4)

    The variance of SR in Equation 3 is given by

    var(SRI) = m 2 [m (2P - 1)2 P(1 - P) + m (2P - 1)2 P (1 - P )

    + 4PP (1 - P) (1 - P )]. (5)

    The directional accuracy (DA) test of Pesaran and Timmer-mann (1992) is now calculated as

    DA = [var(SR) - var (SRI)] 1 /2 (SR - SRI) (6)

    Pesaran and Timmermann (1992) show that under the nullhypothesis that yn+ i and yW n+ i are independently distributed,the DA test follows the standard normal distribution. In thispaper, the null hypothesis is evaluated in a two-sided testprocedure since we also want to allow for signi cantly poorforecasts.

    II I . SOME SIMULATION RESULTS

    Before analysing the set of daily exchange rate series, theproposed empirical approach is evaluated using some simu-lation experiments. The rst set of experiments is motivatedby the nding that many daily exchange rate series displayGARCH properties (Baillie and Bollerslev, 1989), and by the nding that tests for neural network nonlinearity havepower against GARCH (Tera svirta et al., 1993; Brooks,

    1996). Hence, it is investigated whether or not ANNs can be tted to GARCH data and the possibility that the successfulin-sample results in Kuan and Liu (1995) and Swanson andWhite (1995) are caused by neglected GARCH. A total of1750 data points are generated from

    yt = (ht)1 /2 h th t ~ N(0, 1)

    ht = 0.1 + 0.4ht 1 + 0.4y2t 1where h t and y0 are drawn from the standard normal distri-bution and h0 is set equal to 0.5. The parameters in theANNs are estimated for the yt series for 750 data points, and250 one-step-ahead forecasts are generated; then the sampleis shifted forward 250 observations, and so on. As notedabove, p and q are allowed to be elements of {1, 2, 3, 4, 5},i.e. 25 di erent ANNmodels are considered. The forecastingresults for the GARCH model are summarized in the left-hand side panel of Table 1.At the 5% level, only for one of the 25 cases it is found

    that the corresponding ANN model yields signi cantlymore accurate forecasts. At the 10% level, only four of the25 cases are found with better forecasting performance. It isobserved that the Akaike and Schwarz criteria both selectthe smallest ANN model. If the 25 out-of-sample SRs arecorrelated with the in-sample SRs a correlation coe cientof - 0.110 is obtained, indicating that there is almost nopredictive value of the in-sample t for the out-of-sampleperformance. In sum, this suggests that when the data aregenerated from a GARCH model, there is no evidence thatANNs can yield accurate forecasts, even though linearitytests would suggest nonlinearity; see Brooks (1996).The second set of experiments allows one to investigate

    whether ANN models are useful when the data are indeednonlinear; see Kuan and White (1994) for a similar exercisefor chaotic time series. On some occasions it may be di cultto select a speci c nonlinear model on the basis of teststatistics only, and hence one may maintain several ANNmodels, partly because of their ability to approximate anyfunction of past data rather well, given a su cient dimen-sion, and one can investigate whether any forecasting gaincan be obtained. In a rst set of experiments data aregenerated from the bilinear model:

    yt = 0.6yt 1 e t 1 + e twhere e t and y0 are drawn from a standard normal distribu-tion. The forecasting results are reported in the right-handside panel of Table 1. Clearly, the SRs often exceed 50, andthe DA test convincingly shows that for 22 of the 25 cases(at a 5% signi cance level) the ANN model results in goodforecasts. The correlation between the out-of-sample SRand the in-sample SR is 0.269, which suggests that thein-sample SR has some predictive power for out-of-sampleforecasting properties of the models. The models selected bythe Akaike and Schwarz criteria also point towards models

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  • Table 1. Performance of neural network models with q hidden cells and p lags, when data are generated by a GARCH anda bilinear model

    GARCH Bilinear

    Model Success ratio Success ratio

    q p In-sample Out-of-sample DA test In-sample Out-of-sample DA test

    1 1 51.28 h , h h 49.81 - 0.12 65.47 63.73 2.09**1 2 52.26 49.90 - 0.05 65.35 63.64 2.05**1 3 51.02 49.33 - 0.43 65.09 63.64 1.84*1 4 50.64 52.59 1.69* 65.35 62.97 1.081 5 52.94 50.77 0.49 65.73 63.35 1.95**

    2 1 50.90 52.78 1.87* 65.60 63.16 1.66*2 2 52.17 50.00 0.02 65.47 65.26 5.24**2 3 52.05 51.05 0.70 66.75 64.78 4.74**2 4 50.38 51.92 1.25 66.75 65.55 5.54**2 5 53.84 50.29 0.19 66.37 64.11 4.45**

    3 1 51.02 53.35 2.26** 65.35 64.40 5.12**3 2 54.22 47.70 - 1.51 67.39 64.88 5.50**3 3 51.02 49.71 - 0.18 68.16 64.40 4.78**3 4 52.17 51.82 1.19 67.65 65.84 7.06**3 5 52.94 52.20 1.42 68.80 64.05 4.94**

    4 1 51.92 50.10 0.07 66.11 64.40 5.51**4 2 53.71 50.10 0.08 67.77 h h 65.65 6.60**4 3 53.45 51.25 0.83 68.16 64.98 5.80**4 4 55.50 52.78 1.80* 67.01 65.29 6.14**4 5 55.12 52.20 1.44 68.16 65.36 5.95**

    5 1 51.92 51.92 1.29 66.37 62.87 3.52**5 2 51.41 50.38 0.25 66.88 65.36 7.00**5 3 52.69 52.20 1.43 67.01 h 66.22 6.93**5 4 51.02 52.01 1.30 65.09 66.32 7.32**5 5 51.02 52.01 1.30 69.44 63.92 5.44**

    ** Signi cant at the 5% level.* Signi cant at the 10% level.h Model selected using the minimum Akaike information criterion rule.h h Model selected using the minimum Schwarz information criterion rule.Success ratio is the percentage of correct sign forecasts. DA denotes the Pesaran and Timmermann (1992) two-sided testfor directional accuracy of the out-of-sample forecasts. Underlined are the maximum values of the success ratios.

    with signi cant DA test values. It should be noted herethat since almost all ANNs forecast well, the numberof parameters estimated in the ANNs does not seem tomatter much. For example, the ANN with p = 5 and q = 5forecasts well, even though both model selection criteria nd this model to have too many parameters relative toits t.Similar conclusions can be drawn from the simulation

    results given in Table 2 for the bilinear model

    yt = 0.6yt 2 e t 2 + e t

    where e t, y0 and y 1 are drawn from a standard normaldistribution, and the exponential AR (EAR) model (Hagganand Ozaki, 1981) is given by

    yt = a tyt 1 + e t

    with

    a t = 0.5 + 0.4 exp( - y2t 1 )

    where e t and y0 are drawn from a standard normal distribu-tion. The correlation between in-sample and out-of-sampleSR is 0.249 for the bilinear model (if the p = 1 cases aredeleted), while it is only 0.086 for the EAR model. Again,the model selected using the Akaike and Schwarz criteriayields good forecasts and again it does not seem to matterhow many parameters are included in the ANN models.Notice from the left-hand side panel of Table 2 that itis quite important to include the appropriate lags in theANN. In fact, for all p = 1, it is found that the ANNsperform poorly.To summarize the results of the simulation experiments

    in this section, it is concluded (i) that neglected GARCH

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  • Table 2. Performance of neural network models with q hidden cells and p lags, when data are generated by a bilinear modeland an exponential AR model

    Bilinear Exponential AR

    Model Success ratio Success ratio

    q p In-sample Out-of-sample DA test In-sample Out-of-sample DA test

    1 1 52.30 48.80 - 0.78 69.18 h h 70.53 12.76**1 2 59.21 59.71 6.50** 69.69 71.58 13.39**1 3 58.57 59.23 6.63** 69.18 70.24 12.52**1 4 58.82 59.04 6.41** 70.33 71.10 13.03**1 5 60.10 59.14 6.44** 69.18 70.14 12.44**

    2 1 52.94 49.28 - 0.42 69.18 71.20 13.02**2 2 61.76 61.72 7.70** 68.54 69.47 12.01**2 3 61.64 62.39 8.16** 68.80 70.14 12.43**2 4 61.38 60.67 7.00** 69.57 69.95 12.33**2 5 61.51 60.77 7.05** 69.95 69.57 12.05**

    3 1 51.15 49.47 - 0.30 69.18 70.14 12.52**3 2 61.25 62.30 8.06** 71.23 69.19 11.81**3 3 61.51 61.05 7.31** 69.18 69.47 11.99**3 4 62.79 61.63 7.59** 68.67 69.76 12.20**3 5 62.66 61.44 7.44** 70.08 71.00 13.04**

    4 1 50.00 51.20 0.92 69.05 70.33 12.58**4 2 60.74 61.44 7.54** 69.69 66.67 12.10**4 3 62.28 59.71 6.38** 69.18 68.71 11.52**4 4 63.55 h h 59.62 6.26** 68.67 69.28 11.81**4 5 62.92 61.44 7.44** 69.31 h 68.33 11.28**

    5 1 52.30 49.57 - 0.23 69.31 70.72 12.82**5 2 60.87 62.20 7.98** 69.82 70.43 12.71**5 3 63.68 60.19 6.66** 67.90 69.00 11.63**5 4 63.55 h 59.33 6.10** 69.95 68.71 11.43**5 5 62.66 60.77 7.01** 71.99 69.67 12.11**

    ** Signi cant at the 5% level.* Signi cant at the 10% level.h Model selected using the minimum Akaike information criterion rule.h h Model selected using the minimum Schwarz information criterion rule.Success ratio is the percentage of correct sign forecasts. DA denotes the Pesaran and Timmermann (1992) two-sidedtest for directional accuracy of the out-of-sample forecasts. Underlined are the maximum values of the successratios.

    does not lead to spuriously successful ANNs, neither forin-sample t nor for out-of-sample forecasts, and (ii) that ifthere is some form of nonlinearity, ANNs can exploit thisfor improved forecasting. In the latter case it is also foundthat model selection criteria can be useful to reduce com-putational e orts. In fact, when there is some nonlinearity inthe data, any of the ANNs considered performs well, eventhose which include the maximum number of 41 parameters(in the case p = 5 and q = 5).

    IV . EMPIRICAL RESULTS

    In this section the empirical approach is applied to the dailyobserved exchange rate returns for the US and Canadian

    Dollar, British Pound and the Japanese Yen, all in terms ofthe Dutch Guilder. Data cover the period 1986 to 1992.Standard statistical tests for ARCH and nonlinearity resultin p-values invariably below 0.05, and hence the data seemto have these properties. The ndings correspond with theempirical results in Brooks (1996) and others, where it isconvincingly shown that both features are usually found indaily exchange rates. Detailed test results can be obtainedfrom the authors on request.The same strategy is followed as in the previous section,

    i.e. ANNs are estimated for three years of daily data andforecast for one year. First, models for 1986 1988 are esti-mated and forecasts generated for 1989; next 1987 1989 isconsidered and 1990 forecast, and so on until 1992. In sodoing, the aim is to reduce the impact of the selected sample,

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  • Table 3. Performance of neural network models with q hidden cells and p lags of returns on USA Dollar/Guilder andCanadian Dollar/Guilder exchange rate returns

    US Dollar Canadian Dollar

    Model Success ratio Success ratio

    q p In-sample Out-of-sample DA test In-sample Out-of-sample DA test

    1 1 51.49 h h 49.26 - 1.14 52.38 h h 51.88 1.421 2 49.80 49.16 - 1.11 51.06 49.60 - 0.171 3 51.59 50.35 0.06 52.25 49.90 - 0.051 4 50.10 50.15 - 0.20 53.17 51.69 1.131 5 55.78 49.16 - 0.65 56.48 51.19 0.79

    2 1 52.19 50.84 0.16 52.12 50.40 0.312 2 51.39 46.57 - 3.05** 52.12 50.00 0.072 3 53.98 47.96 - 1.91* 53.97 49.01 - 0.642 4 53.98 49.26 - 1.03 53.44 49.50 - 0.322 5 53.49 49.95 - 0.35 56.08 49.11 - 0.60

    3 1 53.09 50.65 0.13 51.85 48.41 - 1.013 2 53.49 48.46 - 1.38 54.50 48.12 - 1.223 3 53.49 48.86 - 1.01 52.38 49.31 - 0.453 4 56.27 50.25 0.10 53.84 51.98 1.333 5 53.98 52.04 1.24 53.17 48.61 - 0.86

    4 1 51.29 49.95 - 0.46 53.70 52.08 1.414 2 52.89 50.45 0.06 51.59 h 51.88 1.254 3 55.68 48.56 - 1.22 54.63 50.00 0.024 4 55.68 48.56 - 1.11 58.60 49.60 - 0.304 5 55.98 47.07 - 2.10** 55.42 49.90 - 0.06

    5 1 54.38 50.25 - 0.11 50.00 52.28 1.585 2 54.78 49.35 - 0.14 53.17 50.50 0.325 3 55.28 47.57 - 1.74* 55.42 49.40 - 0.415 4 57.37 49.85 - 0.23 54.23 49.11 - 0.565 5 57.47 h 49.85 - 0.14 57.14 48.91 - 0.70

    ** Signi cant at the 5% level.* Signi cant at the 10% level.h Model selected using the minimum Akaike information criterion rule.h h Model selected using the minimum Schwarz information criterion rule.Success ratio is the percentage of correct sign forecasts. DA denotes the Pesaran and Timmermann (1992) two-sidedtest for directional accuracy of the out-of-sample forecasts. Underlined are the maximum values of the successratios.

    both for the in-sample t and out-of-sample forecasting. Theresults in Tables 3 and 4 are averages over all generatedforecasts. Again, p and q are allowed to take the values1, 2, 3, 4, and 5.Table 3 shows the results for the US and Canadian

    Dollar. For the US Dollar, it is found that the onlysigni cant DA values are negative. This may be a sign thatthe ANN models are overparameterized. For the CanadianDollar, no signi cant DA test statistics are obtained. Themodels that are selected using the Akaike or Schwarzinformation criteria do not lead to signi cantly betterforecasting models.In Table 4, the results for the British Pound and Japanese

    Yen are reported. Again, it is found that the ANNs seldom

    generate successful forecasting records. That is, only for theYen, where there are three NN models which yield slightlybetter forecasts (at the 10% level). For the Yen series, it isalso found that the Akaike criterion would select a modelwith reasonably good forecasts. In sum, the Akaike criteriongenerally selects the best in-sample model.These empirical results, in conjunction with those in Sec-

    tion III, lead to a conclusion that seems to corroborate theempirical results in Brooks (1996): that the often foundnonlinearity in daily exchange rates seems to be due toGARCH. Indeed, the forecasting results for the ANNs forthe four daily sample series clearly mimic the pattern foundfrom the simulations where a GARCH model was used togenerate the data.

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  • Table 4. Performance of neural network models with q hidden cells and p lags of returns on British Pound/Guilder andJapanese Y en/Guilder exchange rate returns

    British Pound Japanese Yen

    Model Success ratio Success ratio

    q p In-sample Out-of-sample DA test In-sample Out-of-sample DA test

    1 1 52.87 h h 51.85 1.09 55.21 h h 48.60 - 0.941 2 53.32 51.29 0.44 55.61 48.90 - 0.541 3 52.27 51.96 0.74 55.48 50.70 0.361 4 54.08 52.07 1.19 55.35 52.70 1.66*1 5 54.38 52.41 1.15 55.35 53.00 1.84*

    2 1 53.32 50.39 0.00 55.88 50.10 0.172 2 54.68 51.40 0.48 56.15 50.00 - 0.162 3 53.47 48.04 - 1.56 57.22 50.80 0.372 4 52.87 50.84 0.18 55.35 52.40 1.502 5 55.29 49.94 - 0.04 60.16 50.20 0.19

    3 1 52.72 49.50 - 0.54 55.08 48.50 - 0.773 2 53.63 48.60 - 1.04 55.21 50.30 0.013 3 53.47 51.18 - 0.50 56.82 49.80 - 0.463 4 54.08 50.95 0.79 56.42 49.90 - 0.103 5 54.08 51.18 0.84 57.89 50.70 0.42

    4 1 53.17 49.50 - 0.65 56.02 48.90 - 0.424 2 54.38 49.27 - 0.76 56.28 49.50 - 0.364 3 55.74 50.73 0.16 56.30 52.10 0.964 4 54.08 50.84 0.50 58.04 50.60 0.334 5 55.74 49.83 - 0.16 61.53 h 53.01 1.79*

    5 1 53.47 49.83 - 0.70 56.30 48.80 - 0.695 2 54.68 49.94 - 0.16 58.85 49.90 - 0.245 3 56.34 50.95 0.38 59.25 50.40 0.025 4 53.17 46.92 - 1.95* 58.71 51.80 1.035 5 59.37 h 50.39 0.32 55.75 48.95 - 0.77

    ** Signi cant at the 5% level.* Signi cant at the 10% level.h Model selected using the minimum Akaike information criterion rule.h h Model selected using the minimum Schwarz information criterion rule.Success ratio is the percentage of correct sign forecasts. DA denotes the Pesaran and Timmermann (1992) two-sided testfor directional accuracy of the out-of-sample forecasts. Underlined are the maximum values of the success ratios.

    V. CONCLUSION

    In this paper it has been shown through Monte Carlosimulations and the analysis of four sample series that theoften documented nonlinearity in the returns of daily ex-change rates may be due to GARCH. Hence, evidence basedon the use of arti cial neural networks for out-of-sampleforecasting adds to the recent ndings in Brooks (1996).Simulations showed that when the data are truly nonlinear(other that GARCH), even possibly overparameterizedANNs can exploit such nonlinearities for forecasting. Onlywhen the data are generated by a GARCHmodel, is it foundthat no gain can be obtained for out-of-sample forecasting.Our sample series clearly showed the same features, andthus we are led to the conclusion that mean nonlinearity inthese data is due to GARCH. For practical purposes the use

    of ANNmodels to examine possible nonlinearity in the datais recommended. Nonlinearity test statistics may be misledby GARCH, but the in-sample and out-of-sample perfor-mance of ANN models is not.There are several avenues for further research. Even

    though simulation results seem to suggest otherwise, it maystill be that ANNs are overparameterized and that moresubtle nonlinear models could be designed that can elicitnonlinearity and exploit it for forecasting. Further, it may bethat other model selection criteria (such as Rissanens pre-dictive stochastic complexity criterion) are more reliable inpicking the best model. One may also consider alternativemeasures of evaluating forecasts and alternative forecastinghorizons to those adopted in this paper. Finally, the resultsin Genay (1996), where ANNs are combined with technicaltrading rules, seem promising, and hence ANNs could

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  • perhaps be augmented by other explanatory variables onthe right-hand side.

    ACKNOWLEDGEMENTS

    The rst author thanks the Royal Netherlands Academy ofArts and Sciences for its nancial support. This paper drawsupon the MSc thesis of the second author. We thank Dickvan Dijk and an anonymous referee for many helpfulcomments.

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