# Forecasting exchange rates using local regression

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Applied Economics LettersPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/rael20

Forecasting exchange rates using local regressionMarcos Alvarez-Diaz a & Alberto Alvarez ba Centre de Recerca Econmica (UIBSa Nostra), Carrer del Ter , 16-Poligon Son Fuster,07009 Palma de Mallorca, Islas Baleares, Spainb Instituto Mediterrneo de Estudios Avanzados-IMEDEA (CSIC-UIB) , c/ Miquel Marqus, 21,070190, Esporles, Islas Baleares, SpainPublished online: 09 May 2008.

To cite this article: Marcos Alvarez-Diaz & Alberto Alvarez (2010) Forecasting exchange rates using local regression, AppliedEconomics Letters, 17:5, 509-514, DOI: 10.1080/13504850801987217

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Applied Economics Letters, 2010, 17, 509514

Forecasting exchange rates

using local regression

Marcos Alvarez-Diaza,* and Alberto Alvarezb

aCentre de Recerca Economica (UIBSa Nostra), Carrer del Ter,16-Poligon Son Fuster, 07009 Palma de Mallorca, Islas Baleares, SpainbInstituto Mediterraneo de Estudios Avanzados-IMEDEA (CSIC-UIB),

c/ Miquel Marques, 21, 070190, Esporles, Islas Baleares, Spain

In this article we use a generalization of the standard nearest neighbours,

called local regression (LR), to study the predictability of the yen/US$ and

pound sterling/US$ exchange rates. We also compare our results with

those previously obtained with global methods such as neural networks,

genetic programming, data fusion and evolutionary neural networks. We

want to verify if we can generalize to the exchange rate forecasting problem

the belief that local methods beat global ones.

I. Introduction

For a long time, modelling and forecasting

exchange rates has become a puzzle difficult of

deciphering. Many academic researchers and practi-

tioners have tried to explain and predict its

complex, erratic and apparently random behaviour

using both a structural and a univariant perspec-

tive. From a structural point of view, the majority

of the models have assumed a linear relationship

between exchange rate dynamic and some funda-

mental variables such as GDP, interest rate,

inflation rate, money supply or current account

balance. However, the existing difficulties of estab-

lishing an adequate relationship among variables

have led to the presence of no statistical significant

estimated coefficients, many times with incorrect

signs, and a scarce or null forecasting ability. On

the other hand, from a univariate perspective, the

models only use historical values of the own

analysed exchange rates. Nevertheless, adopting

both a univariant and a structural perspective do

not allow any forecasting improvement regarding to

the random walk model. In the well-known

competition realized by Meese and Rogoff (1983),

it was shown that neither structural nor univariant

models could not improve the nave random walk

model. Therefore, the best predictor can be

obtained considering exclusively the previous value

of the exchange rate. Other many researchers have

also concluded that the exchange rates, just like

other financial time series, can be well approxi-

mated by a random walk model (Mussa, 1979;

Newbold et al., 1998).Nowadays, it is widely admitted the possibility

that the exchange rates evolve in a nonlinear

fashion. Therefore, it is possible the existence of

nonlinear dependence between observations, even

though they are linearly uncorrelated (Hsieh, 1989).

As we can find in Brooks (1996), many tools have

been developed and applied to verify the existence

of hidden nonlinear components in exchange rates,

including chaos. All these tests allow us to know

and understand more about the exchange rate

dynamic and, implicitly, they represent an

incentive to employ, improve and develop nonlinear

forecasting techniques such as nearest neighbour

(Diebold and Nason, 1990), artificial neural

*Corresponding author. E-mail: malvarezd@cre.sanostra.es

Applied Economics Letters ISSN 13504851 print/ISSN 14664291 online 2010 Taylor & Francis 509http://www.informaworld.com

DOI: 10.1080/13504850801987217

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mailto:malvarezd@cre.sanostra.eshttp://www.informaworld.com

networks (Franses and Homelen, 1998), geneticalgorithms (Alvarez-Daz and Alvarez, 2003),Markov switching regimes (Kirikos, 1998) orESTAR models (Kilian and Taylor, 2003).However, many times, these tests only can detectsome kind of nonlinearities which are not useful ifwe want to predict the expected value of theexchange rates (e.g. presence of a nonlinearstructure in variance).

In this article we use a generalization of thestandard nearest neighbours, called local regression(LR), to study the predictability of the yen/US$ andpound sterling/US$ exchange rates. As a localforecasting method, LR does not try to find aglobal model to the whole time series, but uses onlylocal information about the points to be predicted.Our goal is two-fold. Firstly, analysing one-period-ahead forecasting, we compare the LR results andthose obtained with global methods such as neuralnetworks (feedforward backprogation neural net-work; FBNN), genetic programming (GP), datafusion (DF) and evolutionary neural network(EANN). We want to verify if we can generalize tothe exchange rate forecasting problem the belief thatlocal methods beat global methods (Gencay, 1999).Secondly, we follow the procedure developed bySugihara and May (1990) to detect the possibleexistence of short-term predictable structures in theconsidered exchange rates. In order to test thestatistical significance of our predictions, we alsoapply the surrogate method to construct empiricalconfidence intervals.

The article is structured in four sections. Afterthis introductory section, the LR method isexplained. In Section III, the out-of-sample pre-dictive ability of the LR is evaluated and comparedwith some global forecasting tools. We also test theexistence of significant predictable structures.Lastly, we conclude with a summary of the mainfindings and results.

II. Nearest Neighbour: Local Regression

Nearest Neighbour is one of the nonlinear techniquesmost widely used for nonlinear financial predictionand, specifically, for exchange rates forecasting(Diebold and Nason, 1990). The method is inspiredby the predictions of nonlinear dynamic systems(Farmer and Siderowich, 1987) and seeks to predictthe future dynamics of a time series by analysing howit has evolved in similar situations in the past before.

In our application, we use a generalization of the

method known as LR. Briefly, the procedure can be

described by a series of steps. First of all, the

trajectory matrix is constructed from the time series

frtgTt1:

MTm1m

M1

M2

MTm1

0BBBBBBBB@

1CCCCCCCCA

r1 r2 rmr2 r3 rm1 : : :

rTm1 rTm2 rT

0BBBBBBBB@

1CCCCCCCCA1

Each row of the trajectory matrix is made up of

vectors of the following form

Mi ri, ri1, . . . , rmi1 2

defining a vector space whose dimension (m) is

called embedding dimension. According to the

Takens theorem (1981), the geometrical trajectory

of this sequence of vectors forms a multi-dimen-

sional object at

reflects the value to which each of the K vectorsevolves period-ahead

NKm1

N1

N2

NK

0BBBBBBBB@

1CCCCCCCCA

k11 k12 k1mk21 k22 k2m

kK1 kK2 kKm

0BBBBBBBB@

1CCCCCCCCA;

EK1

E1

E2

EK

0BBBBBBBB@

1CCCCCCCCA

4

For example, the vector N1 has evolved to a return E1

at periods in the future, while the vector NK hascreated a return EK. The predicted value of the futurereturns (rT ) from the vector M

Tm1 will bedetermined by the regression model:

rT b0 b1 rTm1 b2 rTm2 bm rT5

where the coefficients bi have been estimated byordinary least squares, using the matrices N andE(b N0N1N0E).

A crucial aspect using LR is to determine appro-priately the embedding dimension (m) and thenumber of nearest neighbours (K). The success ofthe prediction depends on the right choice of theseparameters. In spite of its importance, there is no asingle rule for choosing these parameters which hasbeen generally accepted in the literature. However, itis very common to select them using a trial-and-errorprocess. We try with different values of K and m, andwe select the combination which optimizes a given fitcriterion in a specific sub-sample (selection period).We follow the recommendations given by Hsieh(1991) analysing a number of nearest neighbour from10% of all observations up to 90%, increasing insteps of 10%. For the case of the embeddingdimension, we consider values from 2 to 10.

III. Results

In this forecasting study we employed weeklyexchange rates data of Japanese yen and Britishpound against the American dollar. A week

periodicity allows avoiding possible biases inherentto daily data and, moreover, it contains sufficientinformation to be able to accurately reflect thedynamics of exchange rates (Yao and Tan, 2000).As usual in exchange rates forecasting, we considerthe difference of the exchange rate logarithm,

xt logyt logyt1 6

where yt is the exchange rate under analysis, log(yt) isits logarithmic transformation and xt is its return. Ifthe exchange rates followed a random walk, thesequence fxtgTt1 would be random and, in conse-quence, unpredictable.

The sample period starts on the first week ofJanuary 1973 and finishes on the last week of July2002, comprising a total of 1541 observations. It wasdivided into three sub-periods: training, selection andout-of-sample. The first one, composed by the first1080 observations, is reserved as history of the timeseries. The selection period, which covers the 306following observations, is used to determine theoptimal embedding dimension and the optimalnumber of neighbours. Finally, we have reservedthe last 155 observations to validate the predictiveability of the proposed technique.

In order to choose the optimal combination ofparameters and judge the out-of-sample results, weconsider as fit criterion the normalized mean squareerror (NMSE) defined by the expression

NMSE 1Varxt

XMtm1 xt xt

2

M7

where Var(xt) is the variance of the time series is thetotal number of observations in the specific sub-sample, and xt and xt are the predicted and the actualvalues, respectively. This fit criterion, which has beenrecommended by Casdagli (1989) and widelyemployed in exchange rate forecasting, comparesthe errors of the forecasting method and the errorsobtained by considering the sample mean as naivepredictor. Therefore, a NMSE value lower than/equal/higher than one would imply a forecastingability better than/equal to/worse than the mean aspredictor.

Figure 1 shows the sensitivity of LR to differentembedding dimensions, in terms of the NMSEobtained in the selection period. As we can observe,both exchange rates show certain stability. However,as previously mentioned, we have chosen the m whichminimizes the fit criterion. Table 1 depicts theoptimum combination of K and m finally chosen,and the out-of-sample results for one period ahead.

Forecasting exchange rates using local regression 511

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- In both cases, the out-of-sample NMSE is
increasing the forecast horizon. However, if theexistence of short, nonlinear predictable dynamicswas important, we should observe that the accuracyof the nonlinear forecast falls off with increasingprediction-time interval. Figure 2 shows how themost accurate predictions are achieved for one periodahead and, for more periods ahead, the out-of-sampleNMSE increases and fluctuates around one. Thischaracteristic seems to indicate the existence of aslightly and significant short-term predictable patternin the studied exchange rates returns.

IV. Conclusion

In this letter we have used LR to verify three aspectsregarding to exchange rate forecasting for theJapanese yen and the British pound against USdollar. Firstly, we analyse their predictability dis-covering the existence of a short-term predictablestructure in the temporal evolution of both

currencies. Secondly, we confirm the homogeneitybehaviour in terms of forecasting for weekly exchangerates and, finally, we also verify that local methods donot always beat to the global ones in an exchange rateforecasting exercise.

Acknowledgements

Marcos Alvarez-Daz gratefully acknowledgesMinisterio de Educacion y Ciencia (GrantMTM2005-01274, FEDER funding included) for itsfinancial support, and Pacific Exchange Rate Servicefor providing the data.

References

Alvarez-Daz, M. and Alvarez, A. (2007) Forecastingexchange rates using an evolutionary neural network,Applied Financial Economics Letters, 3, 59.

British pound/$ exchange rate

1 2 3 4 5 6 7 8 9 100.85

0.9

0.95

1

1.05

1.1

1.15

Forecast horizon

NM

SE

out

-of-

sam

ple

MeanNearest neighbourIC 0.99

1 2 3 4 5 6 7 8 9 10 0.85

0.9

0.95

1

1.05

1.1

1.15

Forecast horizon

NM

SE

out

-of-

sam

ple

Mean

IC 0.99

Yen/$ exchange rate

Nearest neighbour

Fig. 2. Prediction to different horizons.

Table 2. Comparison among different methods

Normalized mean square error

Evolutionary neuralnetwork (EANN)

Feedforwardbackpropagationneural network (FBNN)

Geneticprogramming (GP)

Data fusion(DF)

Exchangerates

Selectionperiod

Out-of-sampleperiod

Selectionperiod

Out-of-sampleperiod

Selectionperiod

Out-of-sampleperiod

Selectionperiod

Out-of-sampleperiod

Yen/$ 0.9159 0.939 0.9225 0.9329 0.9051 0.9313 0.9051 0.9233British pound/$ 0.9643 0.9223 0.9479 0.9261 0.9591 0.9190 0.9591 0.9189

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Alvarez-Daz, M. and Alvarez, A. (2005) Geneticmulti-model composite forecast for non-linearforecasting of exchange rates, Empirical Economics,30, 64363.

Alvarez-Daz, M. and Alvarez, A. (2003) Forecastingexchange rates using genetic algorithms, AppliedEconomics Letters, 10, 31922.

Brooks, C. (1996) Testing for non-linearity in daily sterlingexchange rates, Applied Financial Economics, 6,30717.

Casdagli, M. (1989) Nonlinear prediction of chaotic timeseries, Physica D, 35, 33556.

Diebold, F. X. and Nason, J. A. (1990) Nonparametricexchange rate prediction?, Journal of InternationalEconomics, 28, 31532.

Farmer, D. and Siderowich, J. (1987) Predicting chaotictime series, Physical Review Letters, 59, 8458.

Franses, P. H. and Homelen, P. V. (1998) On forecastingexchange rates using neural networks,Applied FinancialEconomics, 8, 58996.

Gencay, R. (1999) Linear, non-linear and essentialforeign exchange rate prediction with simple technicaltrading rules, Journal of International Economics, 47,91107.

Hsieh, D. A. (1989) Testing for nonlinear dependence indaily foreign exchange rates, Journal of Business, 62,32968.

Hsieh D. A. (1991) Chaos and nonlinear dynamics:applications to financial markets. Journal of Finance,46, 183977.

Kilian, L. and Taylor, M. P. (2003) Why is sodifficult to beat the random walk forecast ofexchange rates? Journal of International Economics,60, 85117.

Kirikos, D. G. (1998) Forecasting exchange rates out ofsample: random walk vs Markov switching regimes,Applied Economics Letters, 7, 13336.

Meese, R. and Rogoff, K. (1983) Empirical exchange ratemodels of the 1970s: do they fit out of sample?,Journal of International Economics, 14, 324.

Mussa, M. (1979) Empirical Regularities in the Behavior ofExchange Rates and Theories of the Foreign ExchangeMarket, in Policies for Employment, Prices andExchange Rates, Vol. 11 (Eds) K. Brunner andA. H. Meltzer, Carnegie-Rochester Conference Serieson Public Policy, North-Holland, Amsterdam,pp. 957.

Newbold, P., Rayner, T., Kellard, N., Ennew, C., et al.(1998) Is the dollar/ECU exchange rate a randomwalk?, Applied Financial Economics, 8, 5538.

Sugihara, G. and May, R. M. (1990) Nonlinear forecastingas a way of distinguishing chaos from measurementerror in time series, Nature, 344, 73441.

Takens, F. (1981) Detecting strange attractors in turbu-lence, in Dynamical Systems and Turbulence (Eds)D. A. Rand and L. S. Young, Springer-Verlag, Berlin,pp. 36681.

Yao, J. and Tan, C. L. (2000) A case study on using neuralnetworks to perform technical forecasting of Forex,Neurocomputing, 34, 7998.

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