Forecasting daily foreign exchange rates using genetically optimized neural networks

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  • Journal of ForecastingJ. Forecast. 21, 501511 (2002)Published online 12 July 2002 in Wiley InterScience ( DOI: 10.1002/for.838

    Forecasting Daily Foreign Exchange RatesUsing Genetically Optimized NeuralNetworks

    ASHOK K. NAG AND AMIT MITRA*Department of Statistical Analysis & Computer Services, ReserveBank of India, India

    ABSTRACTForecasting currency exchange rates is an important financial problem thathas received much attention especially because of its intrinsic difficulty andpractical applications. The statistical distribution of foreign exchange ratesand their linear unpredictability are recurrent themes in the literature ofinternational finance. Failure of various structural econometric models andmodels based on linear time series techniques to deliver superior forecasts tothe simplest of all models, the simple random walk model, have promptedresearchers to use various non-linear techniques. A number of non-lineartime series models have been proposed in the recent past for obtaining accu-rate prediction results, in an attempt to ameliorate the performance of simplerandom walk models. In this paper, we use a hybrid artificial intelligencemethod, based on neural network and genetic algorithm for modelling dailyforeign exchange rates. A detailed comparison of the proposed method withnon-linear statistical models is also performed. The results indicate supe-rior performance of the proposed method as compared to the traditionalnon-linear time series techniques and also fixed-geometry neural networkmodels. Copyright 2002 John Wiley & Sons, Ltd.

    KEY WORDS artificial intelligence; autoregressive conditionalheteroscedasticity; backpropagation; foreign exchange rate;feedforward network; genetic algorithm; neural network;random walk; recurrent network


    Building a forecasting model for exchange rates has always been a challenging area of research toapplied econometricians and statisticians. Pointing out the difficulties involved, Meese and Rogoff(1983) observed that forecasts arising out of several important structural models based on monetaryand asset theories of exchange rate determination perform no better than (in terms of out-of-sampleforecasting ability) ones generated by the simplest of all models, i.e. the simple random walk model.

    * Correspondence to: Amit Mitra, Assistant Adviser, Operational Analysis Division, DESACS, Reserve Bank of India, C-86th Floor Bandra Kurla Complex, Bandra (East), Mumbai-400 051, India. E-mail:

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  • 502 A. K. Nag and A. Mitra

    Boothe and Glassman (1987) also confirmed these findings for a number of key exchange rates overthe period 19761984, i.e. the Bretton Woods era. Alexander and Thomas (1987), and Wolff (1987)further showed that these econometric models are outperformed by the random walk model evenwhen time-varying parameters are incorporated into the models. Wolff (1988) reached a similarconclusion using time-varying autoregressive models. Although these results are supportive of theefficient-market hypothesis, researchers have continued to explore various alternate techniques formodelling of exchange rates.

    Since these empirical studies mainly rely on linear time series techniques, it is not unreason-able to conjecture that the linear unpredictability of exchange rates may be due to limitations oflinear models. Although changes in exchange rates are linearly uncorrelated, several studies (seee.g. Hsieh,1989; Papel and Sayers, 1990; Diebold and Pauly, 1988) find evidence of non-linearbehaviour. Meese and Rose (1991) also show that exchange rates exhibit significant non-linearities.However, after a fairly extensive empirical search they still could not find a non-linear modelthat dominates the out-of-sample performance of the random walk. Chinn (1991) also providessome support for non-linear exchange rate models by using the alternating conditional expectationsmodel.

    Many recent articles have presented evidence of existence of autoregressive conditional het-eroscedasticity (ARCH) effect in exchange rate series; see e.g. Bollerslev et al. (1992) for a review.ARCH models and later variants of it like ARCH-M and GARCH-M account for the most appar-ent type of non-linear structure in financial market prices, namely that small (large) price changesare followed by small (large) changes of either sign. ARCH models were introduced into theeconometric literature by Engel (1982) and subsequently generalized by Bollerslev (1986) throughGeneralized ARCH (GARCH) models. There is also recent evidence that non-linear dependenciesare present in the conditional mean of exchange rate changes. Conditional variance of the errorterm is used as one of the regressors explaining the conditional mean in these models. There areresults to show that accounting for conditional heteroscedasticity leads to better forecasts of monthlyexchange rate as compared to random walk and other non-linear models like the bilinear model(see Nachne and Ray, 1993).

    Use of neural network-based models is an alternative option available to researchers for capturingthe underlying non-linearity in the exchange rate series. There are several features of the artificialneural network (ANN) based models that make them attractive as a forecasting tool. First, asopposed to the traditional model-based methods, ANN-based models are data-driven and self-adaptive. Second, ANNs are universal function approximators. It has been shown that a networkcan approximate any continuous function to any desired accuracy (Hornik, 1991). Finally, ANNs arenon-linear models. The fact that real-world systems are often non-linear has led to the developmentof several non-linear time series models during the last decade. They include models such as thebilinear model (Granger and Anderson, 1978), the threshold autoregressive model (Tong and Lim,1980) and the autoregressive conditional heteroscedastic (ARCH) model (Engel, 1982). The non-linear modelling capability of the ANN has a distinct advantage over these models in the sensethat, under ANN, it is not necessary to specify the functional relationship between input and outputvariables. In recent years, ANN has been used as a successful tool for forecasting. Zhang et al.(1998) provide a comprehensive review of the current status of research in this area.

    Exchange rate forecasting using artificial neural networks (ANN) provides evidence that they aresignificantly better than existing statistical methods in terms of out-of-sample forecasting ability.Refenes (1996) observes with hourly exchange rate data that ANN models perform much better thanexponential smoothing and autoregressive integrated moving average (ARIMA) models. For daily

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  • Forecasting Daily Foreign Exchange Rates Using Neural Networks 503

    frequency exchange rate data, Weigend et al. (1992) proved that ANN is significantly better than therandom walk model. With weekly exchange rate data, Hann and Steurer (1996) showed that ANNoutperforms the linear models. Recently, Lisi and Schiavo (1999) performed a detailed comparisonof neural network models and chaotic models for predicting monthly exchange rates. They foundthat neural networks compare favourably with chaotic models and both perform significantly betterthan random walk models.

    The success of ANN-based methods for exchange rate forecasting at various frequencies is thusa well-established fact. It is, however, well known (Refenes, 1996) that a solution obtained throughstandard neural network training algorithms like backpropagation suffers from serious drawbacks.These include the possibility of convergence to a local minima and a search for optimum networkarchitecture through experimentation. Since there are no a priori realistic estimates for parametersof the network architecture, network design heavily depends on manual experimentation and fine-tuning of the learning parameters. Thus, although ANN models for exchange rates report superiorperformance, there is still room for further improvement.

    The purpose of this paper is to build ANN exchange rate models that do not suffer from the draw-backs of the traditional fixed-geometry ANN models and perform significantly better. The proposedANN model uses a genetic algorithm search procedure for optimizing the ANN architectural param-eters. Furthermore, the use of this procedure helps to avoid solutions being trapped in local minima.Using the proposed procedure, we construct artificial intelligence models for major internationallytraded daily foreign exchange rates, namely, the deutsche mark/US dollar, the Japanese yen/USdollar and the US dollar/British pound rates. A detailed comparison of these models with the classof conditional heteroscedastic models (ARCH, GARCH, AGARCH, EGARCH and GARCH-M) isalso performed.

    The rest of the paper is organized as follows. The proposed genetic neural network algorithm forforecasting daily foreign exchange rates is presented in the next section. Details of the constructedmodels and empirical findings are reported in the third section. Conclusions emerging from theempirical exercises are discussed in the final section.


    ANN is a field of research aimed at building a computationally feasible machine-based cognitivesystem that tries to capture key aspects of the human cognition process. Although in terms ofspeed of processing, the human brain is much inferior to modern microprocessors, its superioritylies in its organization of the processing of a high-dimensional array of input variables. Paral-lelism or connectionism, adaptability and self-organization are the main attributes of the brainssignal-processing mechanism. ANN technically is an information-processing technology that triesto incorporate these three features of the brain and nervous system and in a way attempts to mimicthe functioning of the human brain. From a statisticians point of view, ANN may be described asa non-linear, non-parametric, multivariate and completely data-driven inference procedure.

    An iterative procedure, called training of the network, is used to obtain the optimum set ofweights of a network. A training algorithm, like backpropagation, is typically used for trainingof a feedforward network. Asymmetric recurrent networks are trained using an algorithm calledrecurrent backpropagation. Apart from the weights of a network, there are several other unknownparameters of ANN architecture. These include the number of input units, the optimum combinationof input units, the number of hidden layers, the number of hidden neurons for each layer, the type

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  • 504 A. K. Nag and A. Mitra

    of transfer function (for hidden units and output units), the value of the learning rate and themomentum rate parameter (the latter two parameters are typically for a backpropagation trainingalgorithm). In the absence of any a priori information about these parameters, the choice of this israther subjective and depends on the experience of the experimenter. As a result of this subjectivechoice of structural parameters of the ANN, there is always a risk that the solution will be trappedin a local minimum. Ideally, the network architecture for a particular problem has to be optimizedover the entire parameter space of such parameters. We use a genetic algorithm search procedure ofthe survival of the fittest for arriving at the optimum values of these ANN architecture parameters.

    The proposed procedure using genetic algorithms for improving the performance of the usualfixed-geometry neural networks is presented in this section. Genetic algorithms (GA) are globaloptimization algorithms based on the mechanism of natural selection and genetics. A standard GAoptimization problem is formulated as

    Optimize FxSubject to x 2 D f0, 1gn


    The function to be optimized, F: !

  • Forecasting Daily Foreign Exchange Rates Using Neural Networks 505

    (2) Train and test these networks to determine how fit they are for solving the problem. Calculatethe fitness measure of each trained network in the current population.

    (3) Rank the networks according to their fitness value and select the best networks through design-ing a probability experiment.

    (4) Create the next generation by pairing up the genetic material representing the weights, inputsand neural structure of these networks. Refilling is done by mating the selected members byexchanging genes of chromosomes.

    (5) Apply mutation in a random fashion according to a preassigned mutation probability.(6) Go back to stage (2) of the training/testing cycle until the optimum population is reached.

    The process is continued generation after generation until an optimum (according to some prede-termined criteria) network architecture is reached. Through this process, the better networks surviveand their features are carried forward into future generations and are combined with others to findbetter networks for the particular application. This genetic search method is much more effectivethan random searching, as the genetic process of recombining features vastly improves the speed ofidentifying highly fit networks. It also has a potential advantage over using only personal experi-ence in building neural networks, as new and potentially better solutions may be found through thisprocess which might otherwise be overlooked because of almost unavoidable assumptions made bythe user.


    In this section we present the performance of the proposed genetically engineered neural net-work models developed for the three major internationally traded exchange rates, namely, deutschemark/US dollar, Japanese yen/US dollar and US dollar/British pound rates. The target variablein all these case studies is Closing exchange rate at the end of the next day. A comparison ofthe proposed method with the usual fixed-geometry neural networks and various types of con-ditional heteroscedastic models is also carried out. The daily spot market exchange rate dataon the three series from January 1992 to May 1998 have been taken from the Reuters datas-tream.

    Different types of neural networks constructed for the present study are:

    (1) Genetic algorithm neural networks (GANN): GANN are constructed using the methodologydescribed in the previous section. Both genetic feedforward (GFF) and genetic feedback (GFB)networks have been considered. Results for the usual square error loss (SEL) and the absoluteerror loss (AEL) functions are reported. The initial parameter space on which GA is appliedfor optimization is described below: Input layer neurons: lagged dependent variable and technical indicators. Hidden layers: maximum two. Hidden layer neurons: maximum 16. Activation functions: sigmoid, tanh and linear. Size of initial population: 100. Learning rate parameter range: 0.1 to 0.85 Momentum rate parameter range: 0.1 to 0.70 Network architecture: feedforward and feedback.We have reported performances of the following four types of GANN:

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  • 506 A. K. Nag and A. Mitra

    (a) Genetic feedforward network using absolute error loss function (GFF, AEL)(b) Genetic feedback network using absolute error loss function (GFB, AEL)(c) Genetic feedforward network using square error loss function (GFF, SEL)(d) Genetic feedback network using square error loss function (GFB, SEL)

    (2) Fixed geometry neural networks (FGNN): For the purpose of comparison of the proposedGANN networks with the fixed-geometry networks we report the performance of eight dif-ferent types of networks. The networks are different in terms of the number of hidden neu-rons (8 or 16), the type of loss functions (AEL or SEL) and the type of activation function(sigmoid or tanh). We consider networks with all possible combination of these parame-ters.

    Conditional heteroscedastic modelsVarious conditional heteroscedastic models have also been built for comparison with the ANN mod-els. In this paper, we construct the following ARCH type models for comparison with the ANNmodels: ARCH(1), GARCH(1,1), Absolute GARCH(1,1), exponential GARCH(1,1) and GARCH-in mean (1,1) or GARCH-M(1,1). The rationale for using these models is same as those discussedin Nachne and Ray (1993).

    Members of the GARCH and GARCH-M class of models can be written as

    yt D 0xt C h2t C t 3

    where h2t is the conditional variance of t with respect to the information set t1 at time (t 1)and is given by

    h2t D Vartjt1 D E2t jt1 D 0 Cq


    2ti C


    ih2ti 4

    The xt vector contains values of independent variables, which may also include lagged yt values.For D 0 in (3) and p D 0 and q D 1 in (4), we obtain the GARCH(0,1) or ARCH (1) model.For GARCH(p, q) models, D 0 and p 0 and q 0 in (3) and there is no effect of condi-tional variance in the conditional expectation equation (3). For general GARCH-M models, 6D 0;p 0 and q 0.

    Two other types of GARCH models considered for comparisons are absolute value GARCH(AGARCH) and exponential GARCH (EGARCH). AGARCH models have been introduced in theliterature by Heutschel (1991). For an AGARCH model, the conditional standard error of t isgiven by

    ht D

    Vartjt1 D 0 Cq

    iD1ijtij C


    ihti 5

    EGARCH models are due to Nelson (1991). For these models, the logarithm of the conditionalvariance of the errors has the following specification:

    log h2t D 0 Cq






    ( tihti



    i log h2ti 6

    Copyright 2002 John Wiley & Sons, Ltd. J. Forecast. 21, 501511 (2002)

  • Forecasting Daily Foreign Exchange Rates Using Neural Networks 507

    To evaluate the forecasting performance of the different models, we reserve the last 150 obser-vations as out-of-sample testing sample points. The out-of-sample forecasting performance of theartificial intelligence and conditional heteroscedastic models are evaluated in terms of the followingcriteria: average absolute error (AAE), mean absolute percentage error (MAPE), mean square error(MSE), maximum absolute error (MaxAE) and R-square (RSQ). The results of the ANN modelsare presented in Tables IIII. Results for the German deutsche mark/US dollar rate are given inTable I. Table II gives the results for the US dollar/British pound rate. Results for the Japaneseyen/US dollar rate are reported in Table III and those for the conditional heteroscedastic modelsare presented in Tables I(a), II(a) and III(a).

    It is observed from the case studies of the three exchange rates that all the genetically opti-mized neural networks, in general, perform much better than the fixed-geometry networks. Theperformance of the different fixed-geometry networks clearly indicates the difficulty in locating theoptimal topology and network architecture for a particular problem. A comparison of the resultsof the genetic neural networks with the fixed-geometry networks suggests that in most cases thefixed-geometry networks are trapped around a local minimum. On the other hand, the use of geneticalgorithms for arriving at the optimal topology is able to locate the global minimum of the errorsurface, the ultimate goal for any model-building application.

    Among the different genetically engineered neural networks, the networks using robust cost func-tions (absolute error loss) perform better (in terms of AAE and MAPE) than those using the usualsquare error loss functions. Another point that emerges from the present exercise is that there is not

    Table I. Out-of-sample results for German deutsche mark/US dollar ANN models




    GANN GAFB,AEL 6.066E-3 0.33635 7.494E-5 0.02781 0.86485GAFF,AEL 6.082E-3 0.33725 7.592E-5 0.26845 0.86209GAFB,SEL 6.186E-3 0.34273 7.433E-5 0.02532 0.86536GAFF,SEL 6.226E-3 0.34500 7.549E-5 0.02629 0.86275

    FGNN 8H,AEL,SIG 7.033E-3 0.38981 8.925E-5 0.03295 0.836548H,AEL,TANH 6.937E-3 0.38464 8.633E-5 0.03204 0.843208H,SEL,SIG 7.184E-3 0.39834 9.216E-5 0.03481 0.833438H,SEL,TANH 7.405E-3 0.40986 8.971E-5 0.03021 0.8393816H,AEL,SIG 7.258E-3 0.40265 9.585E-5 0.03471 0.8323616H,AEL,TANH 6.594E-3 0.36568 8.809E-5 0.03204 0.8534816H,SEL,SIG 7.677E-3 0.42503 9.979E-5 0.03499 0.8174316H,SEL,TANH 7.064E-3 0.39119 8.517E-5 0.02892 0.84721

    Table I(a). Out-of-sample results for German deutsche mark/US dollar conditionalheteroscedastic models

    Model AAE MAPE MSE Max AE R-SQARCH(1) 6.204E-3 0.34353 7.967E-5 0.02606 0.85948GARCH(1,1) 6.213E-3 0.34435 7.960E-5 0.02573 0.85987AGARCH(1,1) 6.203E-3 0.34348 7.970E-5 0.02604 0.85958EGARCH(1,1) 6.211E-3 0.34417 7.959E-5 0.02569 0.85983GARCH(1,1)-M 6.203E-3 0.34430 7.953E-5 0.02568 0.85995

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  • 508 A. K. Nag and A. Mitra

    Table II. Out-of-sample results for US dollar/British pound ANN models




    GANN GAFB,AEL 5.113E-3 0.30965 5.312E-5 0.02818 0.86033GAFF,AEL 5.162E-3 0.31259 4.963E-5 0.02681 0.86593GAFB,SEL 5.173E-3 0.31667 4.803E-5 0.02416 0.87057GAFF,SEL 5.116E-3 0.30980 4.799E-5 0.02451 0.87003

    FGNN 8H,AEL,SIG 5.767E-3 0.34902 5.841E-5 0.02575 0.847558H,AEL,TANH 5.636E-3 0.34119 5.577E-5 0.02549 0.858618H,SEL,SIG 5.701E-3 0.34523 5.718E-5 0.02598 0.856668H,SEL,TANH 5.657E-3 0.34261 5.598E-5 0.02391 0.8600816H,AEL,SIG 5.535E-3 0.33500 5.818E-5 0.02797 0.8534416H,AEL,TANH 5.523E-3 0.33431 5.516E-5 0.02536 0.8618616H,SEL,SIG 5.675E-3 0.34355 5.624E-5 0.02443 0.8570716H,SEL,TANH 5.747E-3 0.34794 5.764E-5 0.02488 0.85843

    Table II(a). Out-of-sample results for US dollar/British pound conditional heteroscedasticmodels

    Model AAE MAPE MSE Max AE R-SQARCH(1) 5.235E-3 0.31710 4.892E-5 0.02280 0.87020GARCH(1,1) 5.294E-3 0.32065 5.022E-5 0.02354 0.86731AGARCH(1,1) 5.327E-3 0.32267 5.080E-5 0.02385 0.86598EGARCH(1,1) 5.254E-3 0.31821 4.968E-5 0.02349 0.86809GARCH(1,1)-M 5.340E-3 0.32348 5.058E-5 0.02384 0.86732

    Table III. Out-of-sample results for Japanese yen/US dollar ANN models




    GANN GAFB,AEL 0.63067 0.49687 0.73081 3.99584 0.94422GAFF,AEL 0.63261 0.49781 0.72854 4.04459 0.94445GAFB,SEL 0.64438 0.50758 0.72852 3.75515 0.94435GAFF,SEL 0.63857 0.50205 0.72196 3.90782 0.94476

    FGNN 8H,AEL,SIG 0.69294 0.54423 0.83299 4.10183 0.937578H,AEL,TANH 0.69166 0.54440 0.85113 3.74998 0.940588H,SEL,SIG 0.68275 0.53619 0.78697 3.88950 0.939938H,SEL,TANH 0.69654 0.54606 0.85483 3.51927 0.9391516H,AEL,SIG 0.68006 0.53386 0.78729 3.72621 0.9399816H,AEL,TANH 0.67258 0.52757 0.81443 3.71148 0.9402216H,SEL,SIG 0.71689 0.56306 0.84009 3.60085 0.9371416H,SEL,TANH 0.67619 0.53112 0.80164 3.56012 0.94064

    much to differentiate between the genetic feedforward and genetic feedback networks. A possiblereason for this is that while constructing the genetic feedforward networks we have included enoughlags and moving averages as input variables for populating the initial population, the starting pointfor the genetic search procedure.

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  • Forecasting Daily Foreign Exchange Rates Using Neural Networks 509

    Table III(a). Out-of-sample results for Japanese yen/US dollar conditionalheteroscedastic models

    Model AAE MAPE MSE Max AE R-SQARCH(1) 0.66056 0.51976 0.74205 3.78740 0.94344GARCH(1,1) 0.66002 0.51933 0.74468 3.75438 0.94358AGARCH(1,1) 0.66100 0.52009 0.74238 3.78579 0.94343EGARCH(1,1) 0.66132 0.52034 0.74271 3.78216 0.94344GARCH(1,1)-M 0.66018 0.51947 0.74178 3.78532 0.94343

    The performance of the best-performing genetic neural networks is quite encouraging. Forthe deutsche mark/US dollar case study, the out-of-sample average absolute error for the best-performing genetic neural network is of the order 6.06 103, the mean square error is of theorder 7.49 105 and the mean absolute percentage error is around 0.33%. For the yen/US dollarrate, the best-performing genetic neural network recorded an average absolute error of 0.63, a meansquare error of 0.73 and a mean absolute percentage error of 0.49%. For the US dollar/Britishpound rates, the out-of-sample average absolute error for the best-performing genetic neural net-work is of the order 5.1 103, the mean square error is of the order 4.8 105 and the meanabsolute percentage error is 0.31%. All these models registered fairly low maximum absolute errorand reasonably high R-squared values.

    We also observe that the conditional heteroscedastic models perform better in most cases thanthe fixed-geometry ANN models that are trapped in local minima. The best-performing geneticallyoptimized ANN models, however, clearly outperform all the different types of conditional het-eroscedastic models in all the cases considered.


    There has been a growing interest in modelling and forecasting foreign exchange rate movementsover the past couple of decades. As changes in exchange rates are known to exhibit non-lineardependencies, ANN techniques have been successfully used by researchers for forecasting. Resultsfrom these studies indicate a superior performance of ANN models as compared to traditional sta-tistical models. A major problem under the traditional ANN approach is that the network builderhas to decide subjectively about the values of a number of network parameters such as the numberof hidden layers, the number of nodes in each hidden layer, the number of input nodes, etc. As aresult, there exists a substantial risk of convergence to a local minimum rather than to the globalminimum.

    With a view to exploring the non-linearity and overcoming the shortcomings of the traditionalANN models, we provide an alternate approach using ANN and genetic algorithm optimizationtechniques for building forecasting models for exchange rates. The out-of-sample forecasts obtainedusing the proposed approach are significantly better than those obtained using a traditional ANNand statistical time series modelling approach. An exhaustive comparison of the proposed techniquewith fixed-geometry neural networks proves that the proposed approach is better suited to obtainthe optimal topology of a neural network. The best experimental results are achieved with geneticneural networks using robust cost functions.

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    The Authors would like to thank Dr R. B. Barman, Executive Director, Reserve Bank of India forhis guidance and encouragement. The views expressed in this paper are those of authors own andnot of the institution to which they belong.


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    Authors biographies:Ashok K. Nag is currently Director, Department of Statistical Analysis & Computer Services, Reserve Bankof India. He received his BStat, MStat and PhD degrees from the Indian Statistical Institute, Calcutta. Hisresearch interests include the application of statistical and artificial intelligence techniques to economic andfinancial data, data warehousing and national income accounting. Dr Nag is a member of the editorial boardof the Reserve Bank of India Bulletin. He is also a member of the Indian Science Congress Association andthe Indian Association for Research in National Income and Wealth.

    Amit Mitra received his BSc and MSc degrees in Statistics from the University of Calcutta and the IndianInstitute of Technology, Kanpur, respectively, securing first position in both. He obtained his PhD in thefield of Statistical Signal Processing from the Indian Institute of Technology, Kanpur. He worked as a guestresearcher at the Department of Systems & Control, Uppsala University, Sweden, for a period of one year.Dr Mitra received the M. N. Murthy Young Scientist award from the Indian Statistical Institute for the bestresearch work in the area of applied statistics. Currently he is working as Assistant Adviser in the Departmentof Statistical Analysis & Computer Services, Reserve Bank of India. Dr Mitra is the author and co-authorof several articles in international journals in statistics and signal processing. His research interests includestatistical signal processing, computational finance and neural networks.

    Authors address:Ashok K. Nag and Amit Mitra, Department of Statistical Analysis & Computer services, Reserve Bank ofIndia, C-8 6th Floor Bandra Kurla Complex, Bandra (East), Mumbai-400 051, India.

    Copyright 2002 John Wiley & Sons, Ltd. J. Forecast. 21, 501511 (2002)


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