Forecasting exchange rates using local regression

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<ul><li><p>This article was downloaded by: [Western Kentucky University]On: 28 October 2014, At: 19:34Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK</p><p>Applied Economics LettersPublication details, including instructions for authors and subscription information:</p><p>Forecasting exchange rates using local regressionMarcos Alvarez-Diaz a &amp; 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.</p><p>To cite this article: Marcos Alvarez-Diaz &amp; Alberto Alvarez (2010) Forecasting exchange rates using local regression, AppliedEconomics Letters, 17:5, 509-514, DOI: 10.1080/13504850801987217</p><p>To link to this article:</p><p>PLEASE SCROLL DOWN FOR ARTICLE</p><p>Taylor &amp; Francis makes every effort to ensure the accuracy of all the information (the Content) containedin the publications on our platform. However, Taylor &amp; Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor &amp; Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.</p><p>This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms &amp; Conditions of access and use can be found at</p><p></p></li><li><p>Applied Economics Letters, 2010, 17, 509514</p><p>Forecasting exchange rates</p><p>using local regression</p><p>Marcos Alvarez-Diaza,* and Alberto Alvarezb</p><p>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),</p><p>c/ Miquel Marques, 21, 070190, Esporles, Islas Baleares, Spain</p><p>In this article we use a generalization of the standard nearest neighbours,</p><p>called local regression (LR), to study the predictability of the yen/US$ and</p><p>pound sterling/US$ exchange rates. We also compare our results with</p><p>those previously obtained with global methods such as neural networks,</p><p>genetic programming, data fusion and evolutionary neural networks. We</p><p>want to verify if we can generalize to the exchange rate forecasting problem</p><p>the belief that local methods beat global ones.</p><p>I. Introduction</p><p>For a long time, modelling and forecasting</p><p>exchange rates has become a puzzle difficult of</p><p>deciphering. Many academic researchers and practi-</p><p>tioners have tried to explain and predict its</p><p>complex, erratic and apparently random behaviour</p><p>using both a structural and a univariant perspec-</p><p>tive. From a structural point of view, the majority</p><p>of the models have assumed a linear relationship</p><p>between exchange rate dynamic and some funda-</p><p>mental variables such as GDP, interest rate,</p><p>inflation rate, money supply or current account</p><p>balance. However, the existing difficulties of estab-</p><p>lishing an adequate relationship among variables</p><p>have led to the presence of no statistical significant</p><p>estimated coefficients, many times with incorrect</p><p>signs, and a scarce or null forecasting ability. On</p><p>the other hand, from a univariate perspective, the</p><p>models only use historical values of the own</p><p>analysed exchange rates. Nevertheless, adopting</p><p>both a univariant and a structural perspective do</p><p>not allow any forecasting improvement regarding to</p><p>the random walk model. In the well-known</p><p>competition realized by Meese and Rogoff (1983),</p><p>it was shown that neither structural nor univariant</p><p>models could not improve the nave random walk</p><p>model. Therefore, the best predictor can be</p><p>obtained considering exclusively the previous value</p><p>of the exchange rate. Other many researchers have</p><p>also concluded that the exchange rates, just like</p><p>other financial time series, can be well approxi-</p><p>mated by a random walk model (Mussa, 1979;</p><p>Newbold et al., 1998).Nowadays, it is widely admitted the possibility</p><p>that the exchange rates evolve in a nonlinear</p><p>fashion. Therefore, it is possible the existence of</p><p>nonlinear dependence between observations, even</p><p>though they are linearly uncorrelated (Hsieh, 1989).</p><p>As we can find in Brooks (1996), many tools have</p><p>been developed and applied to verify the existence</p><p>of hidden nonlinear components in exchange rates,</p><p>including chaos. All these tests allow us to know</p><p>and understand more about the exchange rate</p><p>dynamic and, implicitly, they represent an</p><p>incentive to employ, improve and develop nonlinear</p><p>forecasting techniques such as nearest neighbour</p><p>(Diebold and Nason, 1990), artificial neural</p><p>*Corresponding author. E-mail:</p><p>Applied Economics Letters ISSN 13504851 print/ISSN 14664291 online 2010 Taylor &amp; Francis 509</p><p>DOI: 10.1080/13504850801987217</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Wes</p><p>tern</p><p> Ken</p><p>tuck</p><p>y U</p><p>nive</p><p>rsity</p><p>] at</p><p> 19:</p><p>34 2</p><p>8 O</p><p>ctob</p><p>er 2</p><p>014 </p><p>mailto:malvarezd@cre.sanostra.es</p></li><li><p>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).</p><p>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.</p><p>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.</p><p>II. Nearest Neighbour: Local Regression</p><p>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.</p><p>In our application, we use a generalization of the</p><p>method known as LR. Briefly, the procedure can be</p><p>described by a series of steps. First of all, the</p><p>trajectory matrix is constructed from the time series</p><p>frtgTt1:</p><p>MTm1m </p><p>M1</p><p>M2</p><p>MTm1</p><p>0BBBBBBBB@</p><p>1CCCCCCCCA</p><p>r1 r2 rmr2 r3 rm1 : : :</p><p>rTm1 rTm2 rT</p><p>0BBBBBBBB@</p><p>1CCCCCCCCA1</p><p>Each row of the trajectory matrix is made up of</p><p>vectors of the following form</p><p>Mi ri, ri1, . . . , rmi1 2</p><p>defining a vector space whose dimension (m) is</p><p>called embedding dimension. According to the</p><p>Takens theorem (1981), the geometrical trajectory</p><p>of this sequence of vectors forms a multi-dimen-</p><p>sional object at </p></li><li><p>reflects the value to which each of the K vectorsevolves period-ahead</p><p>NKm1 </p><p>N1</p><p>N2</p><p>NK</p><p>0BBBBBBBB@</p><p>1CCCCCCCCA</p><p>k11 k12 k1mk21 k22 k2m </p><p>kK1 kK2 kKm</p><p>0BBBBBBBB@</p><p>1CCCCCCCCA;</p><p>EK1 </p><p>E1</p><p>E2</p><p>EK</p><p>0BBBBBBBB@</p><p>1CCCCCCCCA</p><p>4</p><p>For example, the vector N1 has evolved to a return E1</p><p>at periods in the future, while the vector NK hascreated a return EK. The predicted value of the futurereturns (rT ) from the vector M</p><p>Tm1 will bedetermined by the regression model:</p><p>rT b0 b1 rTm1 b2 rTm2 bm rT5</p><p>where the coefficients bi have been estimated byordinary least squares, using the matrices N andE(b N0N1N0E).</p><p>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.</p><p>III. Results</p><p>In this forecasting study we employed weeklyexchange rates data of Japanese yen and Britishpound against the American dollar. A week</p><p>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,</p><p>xt logyt logyt1 6</p><p>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.</p><p>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.</p><p>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</p><p>NMSE 1Varxt</p><p>XMtm1 xt xt </p><p>2</p><p>M7</p><p>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.</p><p>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.</p><p>Forecasting exchange rates using local regression 511</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Wes</p><p>tern</p><p> Ken</p><p>tuck</p><p>y U</p><p>nive</p><p>rsity</p><p>] at</p><p> 19:</p><p>34 2</p><p>8 O</p><p>ctob</p><p>er 2</p><p>014 </p></li><li>In both cases, the out-of-sample NMSE is </li><li><p>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.</p><p>IV. Conclusion</p><p>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</p><p>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.</p><p>Acknowledgements</p><p>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.</p><p>References</p><p>Alvarez-Daz, M. and Alvarez, A. (2007) Forecastingexchange rates using an evolutionary neural network,Applied Financial Economics Letters, 3, 59.</p><p>British pound/$ exchange rate</p><p>1 2 3 4 5 6 7 8 9 100.85</p><p>0.9</p><p>0.95</p><p>1</p><p>1.05</p><p>1.1</p><p>1.15</p><p>Forecast horizon</p><p>NM</p><p> SE</p><p> out</p><p>-of-</p><p>sam</p><p>ple</p><p>MeanNearest neighbourIC 0.99</p><p>1 2 3 4 5 6 7 8 9 10 0.85</p><p>0.9</p><p>0.95</p><p>1</p><p>1.05</p><p>1.1</p><p>1.15</p><p>Forecast horizon</p><p>NM</p><p> SE</p><p> out</p><p>-of-</p><p>sam</p><p>ple</p><p>Mean</p><p>IC 0.99</p><p>Yen/$ exchange rate</p><p>Nearest neighbour</p><p>Fig. 2. Prediction to different horizons.</p><p>Table 2. Comparison among different methods</p><p>Normalized mean square error</p><p>Evolutionary neuralnetwork (EANN)</p><p>Feedforwardbackpropagationneural network (FBNN)</p><p>Geneticprogramming (GP)</p><p>Data fusion(DF)</p><p>Exchange...</p></li></ul>


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