Forecasting exchange rates using local regression
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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
To link to this article: http://dx.doi.org/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.
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: email@example.com
Applied Economics Letters ISSN 13504851 print/ISSN 14664291 online 2010 Taylor & Francis 509http://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
r1 r2 rmr2 r3 rm1 : : :
rTm1 rTm2 rT
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
k11 k12 k1mk21 k22 k2m
kK1 kK2 kKm
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.
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
XMtm1 xt xt
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
- 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.
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.
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.
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
MeanNearest neighbourIC 0.99
1 2 3 4 5 6 7 8 9 10 0.85
Yen/$ exchange rate
Fig. 2. Prediction to different horizons.
Table 2. Comparison among different methods
Normalized mean square error
Evolutionary neuralnetwork (EANN)
Feedforwardbackpropagationneural network (FBNN)
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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