exchange rate forecasting using entropy optimized ......exchange rate forecasting using entropy...

Exchange rate forecasting using entropy optimized multivariate wavelet denoising model

He Kaijian Wang Lijun Zou Yingchao Lai Kin Keung

Published inMathematical Problems in Engineering

Published 01012014

Document VersionFinal Published version also known as Publisherrsquos PDF Publisherrsquos Final version or Version of Record

LicenseCC BY

Publication record in CityU ScholarsGo to record

Published version (DOI)1011552014389598

Publication detailsHe K Wang L Zou Y amp Lai K K (2014) Exchange rate forecasting using entropy optimized multivariatewavelet denoising model Mathematical Problems in Engineering 2014 [389598]httpsdoiorg1011552014389598

Citing this paperPlease note that where the full-text provided on CityU Scholars is the Post-print version (also known as Accepted AuthorManuscript Peer-reviewed or Author Final version) it may differ from the Final Published version When citing ensure thatyou check and use the publishers definitive version for pagination and other details

General rightsCopyright for the publications made accessible via the CityU Scholars portal is retained by the author(s) andor othercopyright owners and it is a condition of accessing these publications that users recognise and abide by the legalrequirements associated with these rights Users may not further distribute the material or use it for any profit-making activityor commercial gainPublisher permissionPermission for previously published items are in accordance with publishers copyright policies sourced from the SHERPARoMEO database Links to full text versions (either Published or Post-print) are only available if corresponding publishersallow open access

Take down policyContact lbscholarscityueduhk if you believe that this document breaches copyright and provide us with details We willremove access to the work immediately and investigate your claim

Download date 18072020

Research ArticleExchange Rate Forecasting Using Entropy OptimizedMultivariate Wavelet Denoising Model

Kaijian He12 Lijun Wang1 Yingchao Zou1 and Kin Keung Lai23

1 School of Economics and Management Beijing University of Chemical Technology Beijing 100029 China2Department of Management Sciences City University of Hong Kong Tat Chee Avenue Kowloon Tong Kowloon Hong Kong3 School of Economics and Management North China Electric Power University Beijing 102206 China

Correspondence should be addressed to Kaijian He kaijianhemycityueduhk

Received 29 December 2013 Accepted 19 March 2014 Published 17 April 2014

Academic Editor Fenghua Wen

Copyright copy 2014 Kaijian He et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Exchange rate is one of the key variables in the international economics and international trade Its movement constitutes one ofthe most important dynamic systems characterized by nonlinear behaviors It becomes more volatile and sensitive to increasinglydiversified influencing factors with higher level of deregulation and global integrationworldwide Facing the increasingly diversifiedand more integrated market environment the forecasting model in the exchange markets needs to address the individual andinterdependent heterogeneity In this paper we propose the heterogeneous market hypothesis- (HMH-) based exchange ratemodeling methodology to model the micromarket structure Then we further propose the entropy optimized wavelet-basedforecasting algorithmunder the proposedmethodology to forecast the exchange ratemovementThemultivariatewavelet denoisingalgorithm is used to separate and extract the underlying data components with distinct features which are modeled withmultivariate time series models of different specifications and parameters The maximum entropy is introduced to select the bestbasis and model parameters to construct the most effective forecasting algorithm Empirical studies in both Chinese and Europeanmarkets have been conducted to confirm the significant performance improvement when the proposed model is tested against thebenchmark models

1 Introduction

In the post-Bretton Woods era the worldwide exchangemarkets have shifted towards the more floating and volatileera which are characterized by high level of fluctuations andrisk exposures Given its role as one of the most importanteconomic factors for the national economy in the increas-ingly open and globalized economic system the accurateand reliable forecasting of the exchange rate movementhas profound impacts throughout different levels of theeconomy including government enterprises and academics[1]

Theoretically numerous empirical studies have been con-ducted to investigate the interrelationship and comovementbetween the exchange markets and other markets includingcrude oil market stock market and bond market For exam-ple Salisu and Mobolaji [2] found the statistical evidence ofbidirectional relationship between oil price and US-Nigeria

exchange rate [2] Chkili and Nguyen [3] use a regime-switching model to identify the dynamic linkages betweenthe exchange rates and stock market returns during bothcalm and turbulent periods [3] Hacker et al [4] have foundnew evidence of negative linkage between the exchangerate and interest rate differentials in the wallet time scaledomain [4] Meanwhile in the literature there is much lessattention paid to the exploration of linkage among differentexchange markets which represent the essential theoreticalchallenge For example Kim et al [5] find that the conditionalcorrelation between Japanese Yen and other Asian economiesis decoupled and insignificant due to liquidity deteriorationand elevated risk aversions in the international capital market[5] But recent empirical studies suggest that the comovementacross markets with transmission as one particular caseturns out to be more complicated than what is assumed inthe traditional linear framework This may stem from theloss of information using the low frequency data and the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 389598 9 pageshttpdxdoiorg1011552014389598

2 Mathematical Problems in Engineering

inferior methodology to analyze more accurately the crosscorrelations

Methodologically traditional linear structural modelshave been effective in the forecasting of the exchange ratemovement over the medium to long time horizon withacceptable approximation accuracy and computational effi-ciency where the aggregated price behavior is comparablystable and stationary These include models in both asset andmonetary views as well as Keynesian view regressionmodelscointegration and vector autoregressive models multivariatestochastic models and so forth [6] When it comes to theshorter time horizon these traditional models also havelargely failed to demonstrate the competent forecasting per-formance in the empirical studies [6] Part of the reasons canbe attributed to the nonlinear data characteristics revealedin the recent empirical studies [7 8] Different distributionfree artificial intelligence techniques such as neural networkand support vector regressions together with the innovativeoptimization methods have shown superior performanceunder different circumstances [7 9ndash11] But they are mainlyblack box in nature and offer little insights into the underlyingpatterns as well as supporting theories behind

Recent empirical studies on the fractal and multiscaledata characteristics indicate the emergence of multiscalemodeling as the important alternative [7] Wavelet analysisas one popular multiscale modeling technique has beenintroduced to model not only horizontal dependency inthe time domain such as volatility clustering (conditionalheteroscedasticity) and long memory (slow decaying auto-correlation) but also vertical dependency across time domainsimultaneously [12 13] For example Tiwari et al [14] identifythe comovement of oil price and Indian Rupee at highertime scales but not lower ones using wavelet analysis [14]Reboredo and Rivera-Castro [15] use the wavelet analysisto disentangle the oil price-exchange rate relationship inthe time scale domain [15] Orlov [16] tests for the timevarying exchange rate comovements at different time scales[16] Recently the emergence of this approach leads to theemergence of the heterogeneous market hypothesis (HMH)as the theoretical foundation to replace efficient markethypothesis (EMH) to model the heterogenous data charac-teristics Methodologically this approach helps explain theheterogeneity of the underlying data comovement and trans-mission mechanism behind the exhibited nonlinear datacharacteristics However in the exchange rate forecasting lit-erature we have only witnessed limited attempts in modelingthe correlations and comovements among exchange marketswhen constructing the forecasting algorithm

In the meantime the entropy theory has been used toanalyze the information content of the wavelet decomposedmultiscale data structure Wavelet entropy relative waveletentropy and many other variants have been proposed in theliterature to calculate the entropy of the energy distribution inthe typical wavelet decomposition as well as the cost functionfor the best basis algorithm to choose the optimal basis forwavelet packet transform [17 18] Xu et al [19] use the mod-ified wavelet entropy measure to differentiate between thenormal and hypertension states [19] Samui and Samantaray[20] incorporate the wavelet entropymeasure in constructing

the measuring index for islanding detection in distributedgeneration [20] Wang et al [21] use best basis-based waveletpacket entropy to extract feature in the decomposed structurefor the follow-up classification algorithm which performswell in EEG analysis for patient classification [21] In theforecasting field the entropy maximization has recently beenproposed to select the best forecasters but with much lessattention attracted in the literature For example Bessa et al[22] adopt the maximum entropy criteria in neural networktraining and find it to provide more superior performancethan traditional mean square error (MSE) criteria in windpower prediction [22]

In this paper we propose aninnovative entropy optimizedmultivariate wavelet denoising model Empirical studies areconducted in the closely related Chinese and Europeanexchange markets to evaluate the additional value offeredby the incorporation of nonlinear multiscale cross-marketscorrelations in the proposed algorithm Our contributionsare threefold Firstly we provide the empirical evidence ofmultiscale heterogeneous data characteristics distinguish-able by sizes Secondly we incorporate this stylized fact inthe construction of the innovative wavelet denoising-basedforecasting algorithms Thirdly we propose the maximumentropy as a measure for in-sample performance to selectthe best basis and decomposition level To the best of ourknowledge work in this paper is unique and amongst thefirst in introducing the maximum entropy in forecasting theexchange rate movement in the multiscale domain to selectthe best basis and parameters

The rest of the paper proceeds as follows Section 2briefly reviewed the two relevant theories that ismultivariatewavelet denoising theory and the entropy theory underlyingthe proposed model Section 3 proposes the multivariatewavelet analysis to analyze the time varying correlations Wefurther construct the multivariate wavelet denoising-basedexchange rate forecasting algorithm In Section 4 we conductexperiments to empirically test and confirm the performancesuperiority of the proposed algorithm against the benchmarkmodels Detailed analysis of experiment results is illustratedas well Section 5 concludes with summarizing remarks

2 Multivariate Wavelet-Based DenoisingTheory and Entropy Theory

The ultimate goal of denoising is to set the right boundaryand remove the noises while preserving major data featuresIn recent years wavelet denoising algorithm dominates moretraditional methods such as moving average filter exponen-tial smoothing filter linear Fourier smoothing and simplenonlinear noise reduction as it does not assume homoge-neous error structure For example Kwon et al [23] point outthe problems with existing denoising techniques as assuminghomogeneous error structure and they proposed waveletdenoising method incorporating a variance change pointdetection thresholdingmethod to deal with it in proteinmassspectroscopy applications [23] Boto-Giralda et al [24] usethe stationary wavelet-based denoising methods to improvethe performance of traffic volume prediction models in

Mathematical Problems in Engineering 3

intelligent transportation system [24] Gao et al [25] proposean adaptive denoising algorithm and contend it to be superiorto wavelet-based approaches when applied to analysis of elec-troencephalogram (EEG) signals contaminated with noises[25] Lotric and Dobnikar [26] and Lotric [27] integratethe neural network with the wavelet denoising method tooptimize the denoising parameters dynamically and find theperformance improvement in prediction accuracy [26 27]

Generally the multivariate wavelet denoising algorithminvolves the following procedures

(1) The original data series are projected into the generalhigher-dimensional space 119871

2(R119889) into different subspaces

characterized by scales using multivariate wavelet transformThese subspaces constitute a doubly infinite nested sequenceof subspaces pairs of both denoised and noise data compo-nents (119881DN

119895 119881119873

119895) respectively of 119871

2(R119889) as follows [28ndash30]

sdot sdot sdot sub (VDN2V1198732) sub (VDN

1V1198731) sub (VDN

0V1198730) sub (VDN

minus1V119873minus1)

sub (VDNminus2V119873minus2) sub sdot sdot sdot sub 119871

2(R119889) cap119895isinZ (V

DN119895V119873119895) = 0

cup119895isinZV119895 = 1198712 (R

119889)

(1)

in which

(VDN119895V119873119895) =

119889

⨂

119894=1

(VDN119895119894V119873119895119894) sub 1198712(R119889) (2)

where⨂ is the tensor product operatorThen for each (VDN

119895minus1V119873119895minus1) in (VDN

119895V119873119895) the orthogonal

complement space (WDN119895W119873119895) is defined as

(VDN119895minus1V119873119895minus1) = (VDN

119895V119873119895)⨁(WDN

119895W119873119895) (3)

For all 119891 119892 isin L2(R2) the decomposition and recon-struction in the two-dimensional case is defined as in (4)respectively

int

infin

0

119889119886

119886119899+1int

infin

minusinfin

119889119887 (119879wav119891) (119886 119887) (119879

wav119892) (119886 119887) = 119862120595⟨119891 119892⟩

119891 = 119862minus1

120595int

infin

0

119889119886

119886119899+1intR119899119889119887 (119879

wav119891) (119886 119887) 120595

119886119887

(4)

where (119879wav119891)(119886 119887) = ⟨119891 120595

119886119887⟩ 120595119886119887(119909) = 119886

minus1198992120595((119909 minus 119887)119886)

and 119886 isin R+ 119886 = 0 119887 isin R119899 In the 2-dimensional case three

orthonormal wavelet bases including horizontal waveletΨ

horizontal(119909 119910) vertical wavelet Ψvertical

(119909 119910) and diagonalwavelet Ψdiagonal

(119909 119910) are needed to produce subspaces [31]When the data are of finite support the discrete wavelet

transform would encounter the boundary distortion issue atthe edge of the data under analysis Padding that is addingextra data points to either left or last data is one approach tofacilitate the transform Different padding techniques existincluding zero padding and symmetric padding [30]

(2) The separation between subspaces pairs (119881DN119895

119881119873

119895)

and (119882DN119895

119882119873

119895) is achieved by applying the threshold chosen

specifically at different scales for different directions to eithersuppress or shrink the wavelet coefficients The denoised andnoise part are separated with finer details revealing patternsat more microscales

The dominant threshold selection rules include Univer-sal Minimaxi and Steins unbiased risk estimate (SURE)[32] Different threshold selection rules have different targetswhen setting the noise reduction target For example theuniversal threshold selection rule aims to reduce the noisesat maximum level possible The threshold is selected as 120578119880 =radic2 log119873 where 119873 is the number of wavelet coefficientsand is the estimate of the volatility level When the samplesize 119873 is large and the data series are normally distributedthe universal threshold gives the upper bound value to thenoise level in the data statistically However this methodachieves the maximal level of smoothness at the cost of lowergoodness-of-fitThe denoised data risk loses some importantdata features The Minimaxi threshold selection rule adoptsthe function fitness criteria such as MSE The denoised datarepresent the best fit approximation to the original dataretaining spikes and hikes However this is achieved at thecost of lower function smoothness

Themainstream shrinkage rules are hard and soft thresh-old selection rules [32 33] The hard threshold selectionrule is the high pass filter which suppresses the waveletcoefficients below the chosen threshold values and leaves therest coefficients intact as follows

120575119867

120578(119874119905) =

119874119905

if 10038161003816100381610038161198741199051003816100381610038161003816 gt 120578

0 otherwise(5)

where 119874119905refers to the wavelet coefficients and 120578 is the set

threshold value The soft shrinkage rules focus on the signalsmoothing It suppresses the wavelet coefficients below theset threshold value and subtracts the threshold value from theremaining wavelet coefficients Compared to hard thresholdselection rules the data processing following soft thresholdselection rules is smoother but loses the abrupt changes inthe original data It filters the signal as follows

120575119878

120578(119874119905) = sign (119874

119905) (1003816100381610038161003816119874119905

1003816100381610038161003816 minus 120578)+ (6)

where

sign (119874119905) =

+1 if 119874119905gt 0

0 if 119874119905= 0

minus1 if 119874119905lt 0

(119909)+=

119909 if 119909 ge 00 if 119909 lt 0

(7)

(3) Processed wavelet coefficients are reconstructed intothe unified data series using wavelet synthesis

During the denoising process there are unknown param-eters that have significant impacts on the denoising per-formance Other than the statistical approach to quantify


them the information theoretic approach can be broughtin to quantify them as well The entropy is a widely usedstatistical measure of disorder and uncertainty quantifyingthe data randomness [34] It also corresponds to a measureof information content For a stochastic time series systemthe classical Shannon entropy is defined as follows [35]

119878 = int

infin

0

minus119891 (119909) ln (119891 (119909)) 119889119909 (8)

The value of entropy lies between 0 and 1 The higher theentropy is the higher the level of disorder and uncertainty is

3 An Entropy Optimized Multivariate WaveletDenoising-Based Exchange Rate ForecastingModel (MWVAR)

Homogeneity and rationality are two basic assumptionsimposed in the traditional EMH behind major multivariateexchange rate forecasting algorithms To recognize the mul-tiscale properties in the high frequency data we proposethe HMH instead In HMH the heterogeneous marketmicrostructure is acknowledged explicitly by assuming dif-ferent investors strategy scale and time horizon just to namea few [36ndash42]

Following HMH framework the exchange marketreceives the joint influence frommarket agents with differentdefining characteristics including investment strategies timehorizons and investment scales

Based on the stylized facts we make some simplifyingassumptions (1) investment strategies within each timehorizon are homogeneous and (2) investment strategiesacross time scales are mutually independent Then basedon the aforementioned theoretical framework we proposethe entropy optimized multivariate wavelet denoising-basedexchange rate forecasting algorithm It involves the followingsteps

(1) Suppose that the return of the exchange markets arethe sum of common latent factors and the individual latentfactors the multivariate wavelet denoising algorithm is usedto separate data from noise using particular wavelet familiesBy decomposing data into the multiscale domain data andnoises are separated based on their different characteristicsacross scales with noise smaller in scales Thus more subtledistinction between data and noise can be set

(2) The denoised and noise data are supposed to befollowing some particular stochastic processThe conditionalmean matrix for the denoised data and noise data is modeledby employing the particular conditional time series modelsIn this paper we adopt vector autoregressive (VAR) processesas follows

119910119905= 120575119905+

119898

sum

119894=1

120601119894119910119905minus119894+

119899

sum

119895=1

120579119895120576119905minus119895

+ 120576119905 (9)

where 119910119905is the conditional mean at time 119905 119910

119905minus119894is the lag 119898

returns with parameter 120601119894 and 120576

119905minus119895is the lag 119899 residuals in

the previous period with parameter 120579119895

VAR is chosen over more theoretically sound vectorautoregressive moving average (VARMA) model in this

paper due to the following reasons firstly there is lack ofauthoritative methodology to uniquely identify and estimateVARMA model although some initial attempts have beenmade [43] VAR is still by far the most well establishedand applied multivariate time series models in the literatureSecondly any invertible vector ARMA can be approximatedby VAR with infinite order [44] We use the informationcriteria (IC) to determine the optimal specification for theVAR model Typical IC includes akaike information criteria(AIC) and Bayesian information criteria (BIC)

(3) Using the in-sample data different criteria such asMSE and entropy can be used to determine the modelspecifications and parameters The minimization of MSEcorresponds to the minimization of error variance Theentropy maximization corresponds to the maximization ofinformation content in the predictors and higher general-izability Given the predicted random variable isin 119877

119899generated with the unknown data generating process (DGP)with unknown parameters and the observation 119884 isin 119877

119899 theShannon entropy of predictor is defined as follows

119867() = 119864 [minus log119901 ()] = minusint119877119899

119901 (119884) log119884 (119884) 119889119909 (10)

where 119867() refers to the Shannon entropy of the predictor 119901(119909) refers to the probability density function (PDF) Theobjective is tomaximize the119867() of themeasurable function119884 by adjusting different parameters of forecasting algorithmthat produces

(4)With the chosenmodel specifications and parametersthe forecast matrix is reconstructed from the individualforecasts both denoised and noise parts Thus the meanmatrix can be aggregated from the individual mean matrixforecasts

4 Empirical Studies

41 Experiment Settings We choose the US Dollar againstChinese Renminbi (RMB) and US Dollar against EuropeanEuro (Euro) exchange rates to construct the data set forthe empirical studies The dataset in the empirical studiesextends from 23 July 2007 to 30August 2013The starting dateis chosen as the Chinese government changes its exchangerate policy from the fixed pegging to US Dollar to a basketof currency This has significant impact on the exchangerate movement shifting into a different regime and evolvingbased on different underlying mechanism with wider bandsas well as higher level of fluctuations The end date is setto include the latest data available when the research wasconducted The dataset is preprocessed for recording errorsand to remove the records when the readings are interrupteddue to holiday breaks in either market This results in 1533daily observations Since there is no consensus on the divisionof dataset either in the machine learning or econometricliterature [45] when dividing the dataset we follow thecommon criteria that reserves at least 60data as the trainingset and retain sufficiently large size of the test set for the resultsto be statistically valid [46] The dataset is divided into threesubdataset that is the training set for the proposed wavelet


Table 1 Descriptive statistics and statistical tests

Statistics 119875Euro 119875RMB 119903Euro 119903RMB

Mean 13706 01503 0 00001Max 16010 01636 00461 00059Min 11957 01315 minus00303 minus00087Std 00877 00079 00071 00011Skewness 05550 minus02730 02650 minus01982Kurtosis 26664 24969 56767 94280119901JB 00010 00010 00010 00010

denoising VAR model (36) the model tuning set to selectthe best basis and relevant parameters (24) and the testset for the out-of-sample test to evaluate the performance ofdifferent models (40) We perform one day ahead forecastusing rolling-window method

Descriptive statistics for data characteristics are listed inTable 1

Where 119875Euro and 119875RMB refer to the price of both euroand RMB 119901JB refers to the P value of the JB test statisticsDescriptive statistics in Table 1 show some interesting styl-ized facts The market exhibits considerable fluctuations assuggested by the significant volatility level The distributionof the market price is fat-tail and leptokurtic as suggestedby significant skewness and kurtosis levels There is alsohigh level of market risk exposure due to extreme eventsin the market as reflected in the significant kurtosis levelThemarket return also deviates from the normal distributionand exhibits nonlinear dynamics this is further confirmedby the rejection of Jarque-Bera test of normality [47 48] Asautocorrelation and partial autocorrelation function indicatethe trend factors the daily prices are log differenced at the firstorder to remove them as in 119910

119905= ln(119901

119905(119901119905minus1)) We further

calculate the descriptive statistics on the return data andfind it to approximate the normal distribution as indicatedby the four moments The kurtosis appears to deviate fromthe normal level which indicates that the market exhibitssignificant abnormal return changes event Besides since thenull hypothesis of JB test is rejected this further indicates thatthe market return contains unknown nonlinear dynamicsnot easily captured by traditional linear models

42 Empirical Analysis of Dynamic Behaviors The exchangemarkets are subject to increasingly frequent and abruptexternal shocks such as the subprime crisis and Europeandebt crisis which are illustrated in the plot of returns of bothChinese Renminbi and European Euro as in Figures 1 and 2

This is where 119903rmb119894 119894 = 1 2 3 refers to the RMB returnsover period 119894 119903euro119894 119894 = 1 2 3 refers to the Euro returns overperiod 119894 The three periods refer to July 2007 to May 2009May 2009 to December 2011 and December 2011 to August2013 It can be seen from Figures 1 and 2 and Table 2 thatreturns in both markets share some common features whiledemonstrating their unique characteristics Both markets aresubject to the influences of the subprime crisis fromFebruary2007 and spread to May 2009 and European debt crisis fromDecember 2009 to December 2011 Meanwhile individually

July 23 May 01 Dec 01 Aug 30minus001

minus0008minus0006minus0004minus0002

00002000400060008

001

Date

Retu

rn

USDRenminbi

13 11 09 07

Figure 1 Returns of USDRenminbi

July 23 May 01 Dec 01 Aug 30minus004minus003minus002minus001

0001002003004005

Date

Retu

rn

USDEuro

11 13 09 07

Figure 2 Returns of USDEuro

each market is subject to the influences of some uniqueexternal forces As shown in Figure 1 Renminbi market hasmuch lower level of fluctuations than the Euro marketssubject to tighter control of the central bank As shown inFigure 2 the European debt crisis has much stronger impacton the Euro markets than Renminbi market Generally theimpact of subdebt crisis is larger than the European debt crisisfor both markets Meanwhile as shown in both figures bothmarkets are subject to some common latent factors Bothmarkets demonstrate the obvious jump behaviours around2008 when they are subject to the subprime crisis

We further calculate the cross correlations between twoexchange markets using the roll windows of size 252 andplotting the results in Figure 3

It can be seen from Figure 3 that the correlations aredynamically changing over different periods at the signifi-cant level Taking subprime crisis and European debt crisisas the marking events we further calculated the descriptivestatistics of correlations for three periods and listed the resultsin Table 3

It can be seen from both Figure 3 and Table 3 that forperiods 1 and 3 the correlation is positive and relatively stableThe correlations turn negative for period 2 accompanied bysignificant fluctuationsWithin period two there are frequentregime-switching behaviors For example the correlationfrom November 2007 to August 2009 is mainly affectedby subprime crisis The other result from European debtcrisis is as follows We see that from November 2009 toDecember 2010 the correlation stays at the positive levelsubject to the positive measures taken by the EuropeanUnion Then during December 2010 and August 2011 the



Statistics 119903rmb1 119903rmb2 119903rmb3 119903euro1 119903euro2 119903euro3

Minimum minus00087 minus00060 minus00029 minus00303 minus00237 minus00158Maximum 00046 00059 00039 00461 00238 00196Mean 00002 00001 00001 minus00001 0 0Standard deviation 00012 00011 000094435 00087 00070 00054Kurtosis 92450 107865 51886 62843 34396 37204Skewness minus07236 minus00094 04462 05609 minus00970 01224119901JB 00010 00010 00010 00010 00434 00109

July 23 07ndashJuly 22 08 May 01 09ndashMay 03 10 Dec 01 11ndashNov 30 12 minus1


002040608

1

Time

Cor

relat

ion

Figure 3Dynamic correlations betweenUSDRMBandUSDEuro

Table 3 Descriptive statistics of cross correlations

Statistics 1205881

1205882

1205883

Minimum minus06171 minus04471 03941Maximum 09417 08911 07987Mean minus00175 03139 06762Standard deviation 05574 04575 00972Kurtosis 17873 16746 30009Skewness 05717 minus02462 minus09368119901JB 00019 0001 00011

correlation becomes negative again because of the secondwave of European debt crisis In general we find thatthe correlations of Renminbi and Euro exhibit significantfluctuating behaviors over different periods They are stableover relatively short periods of time and shift frequentlybetween different regimes This observation also implies thatthe time varying features of correlations are the results fromthe joint influences of common latent factors such as theglobal financial crisis as well as the individual risk factor suchas Chinarsquos tight exchange rate bands

43 Experiment Results To evaluate the performance of theproposed algorithm against the benchmark ones we useMSEto measure its predicative accuracy and Clark West test ofequal predictive accuracy to test the statistical significance ofthe predictive accuracy [49 50]

MSEs for the benchmark vector random walk (VRW)and VAR model are 38774

(times10minus5)and 38737

(times10minus5) Then

the wavelet denoising VAR model is applied to the testingdata to investigate the effects of different parameters on the

model performance Different combinations of parameterschoices are pooled into the parameters pool includingtwo shrinkage rules (ie hard and soft shrinkage rules)decomposition levels up to scale 9 and 2 wavelet familiesincludingDaubechies 2 andCoiflet 2The universal thresholdis used during the denoising process The model orders forVAR (119903 119898) processes are determined following AIC and BICminimization principle

Experiment results in Table 4 further confirm that theforecasting accuracy of the proposed wavelet denotingmodelis sensitive to the wavelet parameters used to denoise theoriginal data Some of the wavelet parameters can improvethe forecasting performance to the level that beats thetraditional benchmark models significantly For example thewavelet denoising VAR outperforms both VRW and VARmodel when Coiflet 2 wavelet is used for both hard and softshrinkage rules

Different criteria can then be used to determine themodel specifications and parameters Following the MSEminimization principle when identifying the appropriatemodel specifications the chosen model specifications fromresults in Table 4 are Coiflet 2 wavelet family with hardthreshold strategy at decomposition level 7

Meanwhile we propose the entropy maximization prin-ciple as the criteria when identifying the appropriate modelspecifications Experiment results for different model specifi-cations and parameters are listed in Table 5

From results in Table 5 the chosen model specificationsare Coiflet 2 with hard threshold strategy at decompositionlevel 3

Experiment results in Table 6 show that the proposedalgorithm outperforms the benchmark VRW and VARmodel in terms of predictive accuracyThe ClarkWest test ofequal predictive accuracy suggests that the performance gapis significant at 7 confidence level against the VRW modeland 13 confidence level against VAR model Meanwhileexperiment results also show that adopting the proposedentropy measure during the model parameter determina-tion would lead to lower MSE and higher P value for itsstatistical significance performance out-of-sample comparedto the currently dominant MSE measure Out-of-sampleperformance comparisons for the proposed MWVAR usingdifferent criteria during the model parameter determinationin-sample suggest that using the proposed entropy measureleads to improved model performance out-of-sample The


Table 4 MSE of MWVAR with different wavelet parameters andmodel specifications

Wavelets 1198631198872119867

1198631198872119878

1198621199001198941198911198971198901199052119867

1198621199001198941198911198971198901199052119878

(times10minus5) (times10

minus5) (times10

minus5) (times10

minus5)

1 38863 38766 38660 387042 38757 38696 38524 386223 38771 38650 38529 386044 38814 38688 38531 386115 38816 38726 38524 386046 38817 38728 38528 385947 38818 38729 38517 385998 38819 38741 38519 385949 38818 38746 38519 38573Average 38810 38719 38539 38612

Table 5 Entropy of MWVAR prediction with different waveletparameters and model specifications

Wavelets 1198631198872119867

1198631198872119878

1198621199001198941198911198971198901199052119867

1198621199001198941198911198971198901199052119878


minus5) (times10

minus5) (times10

minus5)

1 33462 19278 05385 099222 37502 30319 21602 236013 39625 31196 25694 270954 38269 31158 25220 259555 38084 32815 24967 315386 38023 34627 24291 323427 38048 35441 25636 344398 38031 36604 25646 349299 37963 37501 25565 36166

Table 6 Performance evaluation based on MSE

MSE (times10minus5) 119875CWRW 119875CWVAR

VRW 29557 NA 01017VAR 29518 NA NAMWVARMSE 29481 00796 01537MWVAREntropy 29472 00671 01222

proposedMWVAR algorithm has achieved the lower MSE ingeneral compared with the algorithm performance using in-sample MSE as the criteria

The performance improvement is attributed to the anal-ysis of latent risk structure in the multiscale domain usingwavelet analysis aswell as the best basis andparameters selec-tion based on entropy maximization principle These resultsfurther imply that the exchange rate data is complicatedprocesses with a mixture of underlying DGPs of differentnaturesTheremay be redundant representation of the under-lying latent structure since there lacks explicit and analyticsolutions The determination of best basis and parametersheterogeneous in nature hold the key to further performanceimprovement andmore thorough understanding of theDGPsduring the modeling process

5 Conclusions

In this paper we propose the HMH-based theoretical frame-work for exchange rate forecasting Under the proposedtheoretical framework we further propose the multivariatewavelet-based exchange rate forecasting algorithm as oneparticular implementation We find that the exchange ratebehaviors are affected by noises and main trends which havedifferent characteristics The separation of noises and dataneed to be conducted in a multiscale manner to recover theuseful data for further modeling by VARmodel Results fromempirical studies using the USD against RMB and Euro asthe typical pair confirm the performance improvement of theproposedmodels against the benchmarkmodelsWork donein this paper suggests that the more accurate separation ofdata and noises leads to better behaved data and higher levelof model generalizability

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthorswould like to express their sincere appreciation tothe editor and the three anonymous referees for their valuablecomments and suggestions which helped improve the qualityof the paper tremendously This work is supported by theCityU Strategic Research Grant (no 7004135) the NationalNatural Science Foundation of China (NSFC nos 71201054and 91224001) and the Fundamental Research Funds for theCentral Universities (no ZZ1315)

References

[1] L Sarno and G Valente ldquoEmpirical exchange rate models andcurrency risk some evidence from density forecastsrdquo Journal ofInternational Money and Finance vol 24 no 2 pp 363ndash3852005

[2] A A Salisu and H Mobolaji ldquoModeling returns and volatilitytransmission between oil price and US-Nigeria exchange raterdquoEnergy Economics vol 39 pp 169ndash176 2013

[3] W Chkili and D K Nguyen ldquoExchange rate movements andstock market returns in a regime-switching environment evi-dence for BRICS countriesrdquo Research in International Businessand Finance vol 31 pp 46ndash56 2014

[4] R S Hacker H K Karlsson and KMansson ldquoAn investigationof the causal relations between exchange rates and interest ratedifferentials using waveletsrdquo International Review of Economicsamp Finance vol 29 pp 321ndash329 2014

[5] B H Kim H Kim and H G Min ldquoReassessing the linkbetween the Japanese yen and emerging Asian currenciesrdquoJournal of International Money and Finance vol 33 pp 306ndash326 2013

[6] R A Meese and K Rogoff ldquoEmpirical exchange rate models ofthe seventies do they fit out of samplerdquo Journal of InternationalEconomics vol 14 no 1-2 pp 3ndash24 1983


[7] F Wen Z Li C Xie and D Shaw ldquoStudy on the fractal andchaotic features of the Shanghai composite indexrdquo Fractals-Complex Geometry Patterns and Scaling in Nature and Societyvol 20 no 2 pp 133ndash140 2012

[8] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science andComplexity vol 22 no 3 pp 360ndash371 2009

[9] W Huang K K Lai Y Nakamori S Wang and L Yu ldquoNeuralnetworks in finance and economics forecastingrdquo InternationalJournal of Information TechnologyampDecisionMaking vol 6 no1 pp 113ndash140 2007

[10] L Yu S Y Wang and K K Lai ldquoA novel nonlinear ensembleforecasting model incorporating GLAR and ANN for foreignexchange ratesrdquo Computers amp Operations Research vol 32 no10 pp 2523ndash2541 2005

[11] F Wen and Z Dai ldquoModified Yabe-Takano nonlinear conjugategradient methodrdquo Pacific Journal of Optimization vol 8 no 2pp 347ndash360 2012

[12] E Peters Fractal Market Analysis Applying Chaos Theory toInvestment and Economics John Wiley amp Sons London UK1994

[13] R Gencay F Selcuk and A Ulugulyagci ldquoHigh volatility thicktails and extreme value theory in value-at-risk estimationrdquoInsurance Mathematics and Economics vol 33 no 2 pp 337ndash356 2003

[14] A K Tiwari A B Dar and N Bhanja ldquoOil price and exchangerates a wavelet based analysis for Indiardquo Economic Modelingvol 31 pp 414ndash422 2013

[15] J C Reboredo andM A Rivera-Castro ldquoA wavelet decomposi-tion approach to crude oil price and exchange rate dependencerdquoEconomic Modeling vol 32 pp 42ndash57 2013

[16] A G Orlov ldquoA cospectral analysis of exchange rate comove-ments during Asian financial crisisrdquo Journal of InternationalFinancialMarkets Institutions andMoney vol 19 no 5 pp 742ndash758 2009

[17] S Blanco A Figliola R Q Quiroga O A Rosso and ESerrano ldquoTime-frequency analysis of electroencephalogramseries III Wavelet packets and information cost functionrdquoPhysical Review E vol 57 no 1 pp 932ndash940 1998

[18] R R Coifman and M Wickerhauser ldquoEntropy-based algo-rithms for best basis selectionrdquo IEEE Transactions on Informa-tion Theory vol 38 no 2 pp 713ndash718 1992

[19] P Xu X Hu andD Yao ldquoImproved wavelet entropy calculationwith window functions and its preliminary application to studyintracranial pressurerdquo Computers in Biology and Medicine vol43 no 5 pp 425ndash433 2013

[20] A Samui and S R Samantaray ldquoWavelet singular entropy-based islanding detection in distributed generationrdquo IEEETransactions on Power Delivery vol 28 no 1 pp 411ndash418 2013

[21] D Wang D Miao and C Xie ldquoBest basis-based wavelet packetentropy feature extraction and hierarchical EEG classificationfor epileptic detectionrdquo Expert Systems with Applications vol38 no 11 pp 14314ndash14320 2011

[22] R J Bessa V Miranda and J Gama ldquoEntropy and correntropyagainst minimum square error in offline and online three-dayahead wind power forecastingrdquo IEEE Transactions on PowerSystems vol 24 no 4 pp 1657ndash1666 2009

[23] D Kwon M Vannucci J J Song J Jeong and R MPfeiffer ldquoA novel wavelet-based thresholding method for thepre-processing of mass spectrometry data that accounts forheterogeneous noiserdquo Proteomics vol 8 no 15 pp 3019ndash30292008

[24] D Boto-Giralda F J Diaz-Pernas D Gonzalez-Ortega etal ldquoWavelet-based denoising for traffic volume time seriesforecasting with self-organizing neural networksrdquo Computer-Aided Civil and Infrastructure Engineering vol 25 no 7 pp530ndash545 2010

[25] J Gao H Sultan J Hu and W-W Tung ldquoDenoising nonlineartime series by adaptive filtering and wavelet shrinkage acomparisonrdquo IEEE Signal Processing Letters vol 17 no 3 pp237ndash240 2010

[26] U Lotric and A Dobnikar ldquoPredicting time series usingneural networks with wavelet-based denoising layersrdquo NeuralComputing amp Applications vol 14 no 1 pp 11ndash17 2005

[27] U Lotric ldquoWavelet based denoising integrated intomultilayeredperceptronrdquo Neurocomputing vol 62 no 1ndash4 pp 179ndash1962004

[28] B Vidakovic Statistical Modeling by Wavelets Wiley Series inProbability and Statistics Applied Probability and StatisticsJohn Wiley amp Sons London UK 1999

[29] M Jansen Second Generation Wavelets and ApplicationsSpringer London UK 2005

[30] G P Nason Wavelet Methods in Statistics with R SpringerLondon UK 2008

[31] IDaubechiesTenLectures onWavelets SIAMPhiladelphia PaUSA 1992

[32] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 no 432 pp 1200ndash1224 1995

[33] D Donoho and I Johnstone ldquoIdeal denoising in an orthonor-mal basis chosen from a library of basesrdquo Comptes Rendus delrsquoAcademie des Sciences Serie I Mathematique vol 319 no 12pp 1317ndash1322 1994

[34] J E Bates and H K Shepard ldquoMeasuring complexity usinginformation fluctuationrdquo Physics Letters A vol 172 no 6 pp416ndash425 1993

[35] C E Shannon ldquoA mathematical theory of communicationrdquoACM SIGMOBILE Mobile Computing and CommunicationsReview vol 5 no 1 pp 3ndash55 2001

[36] W A Brock and C H Hommes ldquoA rational route to random-nessrdquo Econometrica vol 65 no 5 pp 1059ndash1095 1997

[37] W A Brock and A W Kleidon ldquoPeriodic market closure andtrading volume a model of intraday bids and asksrdquo Journal ofEconomic Dynamics and Control vol 16 no 3-4 pp 451ndash4891992

[38] W A Brock and B D LeBaron ldquoA dynamic structural modelfor stock return volatility and trading volumerdquo The Review ofEconomics and Statistics vol 78 no 1 pp 94ndash110 1996

[39] C H Hommes ldquoFinancial markets as nonlinear adaptiveevolutionary systemsrdquo Quantitative Finance vol 1 no 1 pp149ndash167 2001

[40] M M Dacorogna R Gencay U A O Muller O V Pictetand R B Olsen An Introduction to High-Frequency FinanceAcademic Press San Diego Calif USA 2001

[41] J K Goeree and C Hommes ldquoHeterogeneous beliefs and thenon-linear cobweb modelrdquo Journal of Economic Dynamics andControl vol 24 no 5ndash7 pp 761ndash798 2000

[42] J Farmer ldquoMarket force ecology and evolutionrdquo Industrial andCorporate Change vol 11 no 59 pp 895ndash953 2002

[43] G Athanasopoulos and F Vahid ldquoVARMA versus VAR formacroeconomic forecastingrdquo Journal of Business amp EconomicStatistics vol 26 no 2 pp 237ndash252 2008


[44] S T Rachev S Mittnik F J Fabozzi S M Focardi and T JasicEds Financial Econometrics FromBasics to AdvancedModelingTechniques John Wiley amp Sons Chichester UK 2007

[45] P R Hansen and A Timmermann ldquoChoice of sample split inout-of-sample forecast evaluationrdquo Economics Working PapersECO201210 European University Institute 2012

[46] H Zou G Xia F Yang and H Wang ldquoAn investigation andcomparison of artificial neural network and time series modelsfor Chinese food grain price forecastingrdquo Neurocomputing vol70 no 16ndash18 pp 2913ndash2923 2007

[47] C M Jarque and A K Bera ldquoA test for normality of observa-tions and regression residualsrdquo International Statistical Reviewvol 55 no 2 pp 163ndash172 1987

[48] W A Broock J A ScheinkmanWD Dechert and B LeBaronldquoA test for independence based on the correlation dimensionrdquoEconometric Reviews vol 15 no 3 pp 197ndash235 1996

[49] A Alves Portela Santos N Carneiro Affonso da Costa Jr andL dos Santos Coelho ldquoComputational intelligence approachesand linear models in case studies of forecasting exchange ratesrdquoExpert Systems with Applications vol 33 no 4 pp 816ndash8232007

[50] R S Mariano and F X Diebold ldquoComparing predictiveaccuracyrdquo Journal of Business amp Economic Statistics vol 13 no3 pp 253ndash263 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Research ArticleExchange Rate Forecasting Using Entropy OptimizedMultivariate Wavelet Denoising Model

Kaijian He12 Lijun Wang1 Yingchao Zou1 and Kin Keung Lai23

1 School of Economics and Management Beijing University of Chemical Technology Beijing 100029 China2Department of Management Sciences City University of Hong Kong Tat Chee Avenue Kowloon Tong Kowloon Hong Kong3 School of Economics and Management North China Electric Power University Beijing 102206 China

Correspondence should be addressed to Kaijian He kaijianhemycityueduhk

Received 29 December 2013 Accepted 19 March 2014 Published 17 April 2014

Academic Editor Fenghua Wen

Copyright copy 2014 Kaijian He et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Exchange rate is one of the key variables in the international economics and international trade Its movement constitutes one ofthe most important dynamic systems characterized by nonlinear behaviors It becomes more volatile and sensitive to increasinglydiversified influencing factors with higher level of deregulation and global integrationworldwide Facing the increasingly diversifiedand more integrated market environment the forecasting model in the exchange markets needs to address the individual andinterdependent heterogeneity In this paper we propose the heterogeneous market hypothesis- (HMH-) based exchange ratemodeling methodology to model the micromarket structure Then we further propose the entropy optimized wavelet-basedforecasting algorithmunder the proposedmethodology to forecast the exchange ratemovementThemultivariatewavelet denoisingalgorithm is used to separate and extract the underlying data components with distinct features which are modeled withmultivariate time series models of different specifications and parameters The maximum entropy is introduced to select the bestbasis and model parameters to construct the most effective forecasting algorithm Empirical studies in both Chinese and Europeanmarkets have been conducted to confirm the significant performance improvement when the proposed model is tested against thebenchmark models

1 Introduction

In the post-Bretton Woods era the worldwide exchangemarkets have shifted towards the more floating and volatileera which are characterized by high level of fluctuations andrisk exposures Given its role as one of the most importanteconomic factors for the national economy in the increas-ingly open and globalized economic system the accurateand reliable forecasting of the exchange rate movementhas profound impacts throughout different levels of theeconomy including government enterprises and academics[1]

Theoretically numerous empirical studies have been con-ducted to investigate the interrelationship and comovementbetween the exchange markets and other markets includingcrude oil market stock market and bond market For exam-ple Salisu and Mobolaji [2] found the statistical evidence ofbidirectional relationship between oil price and US-Nigeria

exchange rate [2] Chkili and Nguyen [3] use a regime-switching model to identify the dynamic linkages betweenthe exchange rates and stock market returns during bothcalm and turbulent periods [3] Hacker et al [4] have foundnew evidence of negative linkage between the exchangerate and interest rate differentials in the wallet time scaledomain [4] Meanwhile in the literature there is much lessattention paid to the exploration of linkage among differentexchange markets which represent the essential theoreticalchallenge For example Kim et al [5] find that the conditionalcorrelation between Japanese Yen and other Asian economiesis decoupled and insignificant due to liquidity deteriorationand elevated risk aversions in the international capital market[5] But recent empirical studies suggest that the comovementacross markets with transmission as one particular caseturns out to be more complicated than what is assumed inthe traditional linear framework This may stem from theloss of information using the low frequency data and the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 389598 9 pageshttpdxdoiorg1011552014389598

















119895 119881119873




1V1198731) sub (VDN

0V1198730) sub (VDN



2(R119889) cap119895isinZ (V

DN119895V119873119895) = 0

cup119895isinZV119895 = 1198712 (R

119889)

(1)

in which

(VDN119895V119873119895) =

119889

⨂

119894=1

(VDN119895119894V119873119895119894) sub 1198712(R119889) (2)






119895V119873119895)⨁(WDN

119895W119873119895) (3)


int

infin

0

119889119886

119886119899+1int

infin

minusinfin

119889119887 (119879wav119891) (119886 119887) (119879

wav119892) (119886 119887) = 119862120595⟨119891 119892⟩

119891 = 119862minus1

120595int

infin

0

119889119886

119886119899+1intR119899119889119887 (119879

wav119891) (119886 119887) 120595

119886119887

(4)

where (119879wav119891)(119886 119887) = ⟨119891 120595

119886119887⟩ 120595119886119887(119909) = 119886

minus1198992120595((119909 minus 119887)119886)








119881119873

119895)

and (119882DN119895

119882119873





120575119867

120578(119874119905) =

119874119905

if 10038161003816100381610038161198741199051003816100381610038161003816 gt 120578

0 otherwise(5)



120575119878

120578(119874119905) = sign (119874

119905) (1003816100381610038161003816119874119905

1003816100381610038161003816 minus 120578)+ (6)

where

sign (119874119905) =

+1 if 119874119905gt 0

0 if 119874119905= 0

minus1 if 119874119905lt 0

(119909)+=

119909 if 119909 ge 00 if 119909 lt 0

(7)





119878 = int

infin

0

minus119891 (119909) ln (119891 (119909)) 119889119909 (8)








119910119905= 120575119905+

119898

sum

119894=1

120601119894119910119905minus119894+

119899

sum

119895=1

120579119895120576119905minus119895

+ 120576119905 (9)


119905minus119894is the lag 119898










119901 (119884) log119884 (119884) 119889119909 (10)



4 Empirical Studies









119905= ln(119901

119905(119901119905minus1)) We further






00002000400060008

001

Date

Retu

rn

USDRenminbi

13 11 09 07



0001002003004005

Date

Retu

rn

USDEuro

11 13 09 07












002040608

1

Time

Cor

relat

ion



Statistics 1205881

1205882

1205883
















Wavelets 1198631198872119867

1198631198872119878

1198621199001198941198911198971198901199052119867

1198621199001198941198911198971198901199052119878


minus5) (times10

minus5) (times10

minus5)

1 38863 38766 38660 387042 38757 38696 38524 386223 38771 38650 38529 386044 38814 38688 38531 386115 38816 38726 38524 386046 38817 38728 38528 385947 38818 38729 38517 385998 38819 38741 38519 385949 38818 38746 38519 38573Average 38810 38719 38539 38612


Wavelets 1198631198872119867

1198631198872119878

1198621199001198941198911198971198901199052119867

1198621199001198941198911198971198901199052119878


minus5) (times10

minus5) (times10

minus5)

1 33462 19278 05385 099222 37502 30319 21602 236013 39625 31196 25694 270954 38269 31158 25220 259555 38084 32815 24967 315386 38023 34627 24291 323427 38048 35441 25636 344398 38031 36604 25646 349299 37963 37501 25565 36166






5 Conclusions




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























119895 119881119873




1V1198731) sub (VDN

0V1198730) sub (VDN



2(R119889) cap119895isinZ (V

DN119895V119873119895) = 0

cup119895isinZV119895 = 1198712 (R

119889)

(1)

in which

(VDN119895V119873119895) =

119889

⨂

119894=1

(VDN119895119894V119873119895119894) sub 1198712(R119889) (2)






119895V119873119895)⨁(WDN

119895W119873119895) (3)


int

infin

0

119889119886

119886119899+1int

infin

minusinfin

119889119887 (119879wav119891) (119886 119887) (119879

wav119892) (119886 119887) = 119862120595⟨119891 119892⟩

119891 = 119862minus1

120595int

infin

0

119889119886

119886119899+1intR119899119889119887 (119879

wav119891) (119886 119887) 120595

119886119887

(4)

where (119879wav119891)(119886 119887) = ⟨119891 120595

119886119887⟩ 120595119886119887(119909) = 119886

minus1198992120595((119909 minus 119887)119886)








119881119873

119895)

and (119882DN119895

119882119873





120575119867

120578(119874119905) =

119874119905

if 10038161003816100381610038161198741199051003816100381610038161003816 gt 120578

0 otherwise(5)



120575119878

120578(119874119905) = sign (119874

119905) (1003816100381610038161003816119874119905

1003816100381610038161003816 minus 120578)+ (6)

where

sign (119874119905) =

+1 if 119874119905gt 0

0 if 119874119905= 0

minus1 if 119874119905lt 0

(119909)+=

119909 if 119909 ge 00 if 119909 lt 0

(7)





119878 = int

infin

0

minus119891 (119909) ln (119891 (119909)) 119889119909 (8)








119910119905= 120575119905+

119898

sum

119894=1

120601119894119910119905minus119894+

119899

sum

119895=1

120579119895120576119905minus119895

+ 120576119905 (9)


119905minus119894is the lag 119898










119901 (119884) log119884 (119884) 119889119909 (10)



4 Empirical Studies









119905= ln(119901

119905(119901119905minus1)) We further






00002000400060008

001

Date

Retu

rn

USDRenminbi

13 11 09 07



0001002003004005

Date

Retu

rn

USDEuro

11 13 09 07












002040608

1

Time

Cor

relat

ion



Statistics 1205881

1205882

1205883
















Wavelets 1198631198872119867

1198631198872119878

1198621199001198941198911198971198901199052119867

1198621199001198941198911198971198901199052119878


minus5) (times10

minus5) (times10

minus5)

1 38863 38766 38660 387042 38757 38696 38524 386223 38771 38650 38529 386044 38814 38688 38531 386115 38816 38726 38524 386046 38817 38728 38528 385947 38818 38729 38517 385998 38819 38741 38519 385949 38818 38746 38519 38573Average 38810 38719 38539 38612


Wavelets 1198631198872119867

1198631198872119878

1198621199001198941198911198971198901199052119867

1198621199001198941198911198971198901199052119878


minus5) (times10

minus5) (times10

minus5)

1 33462 19278 05385 099222 37502 30319 21602 236013 39625 31196 25694 270954 38269 31158 25220 259555 38084 32815 24967 315386 38023 34627 24291 323427 38048 35441 25636 344398 38031 36604 25646 349299 37963 37501 25565 36166






5 Conclusions




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119878 = int

infin

0

minus119891 (119909) ln (119891 (119909)) 119889119909 (8)








119910119905= 120575119905+

119898

sum

119894=1

120601119894119910119905minus119894+

119899

sum

119895=1

120579119895120576119905minus119895

+ 120576119905 (9)


119905minus119894is the lag 119898










119901 (119884) log119884 (119884) 119889119909 (10)



4 Empirical Studies









119905= ln(119901

119905(119901119905minus1)) We further






00002000400060008

001

Date

Retu

rn

USDRenminbi

13 11 09 07



0001002003004005

Date

Retu

rn

USDEuro

11 13 09 07












002040608

1

Time

Cor

relat

ion



Statistics 1205881

1205882

1205883
















Wavelets 1198631198872119867

1198631198872119878

1198621199001198941198911198971198901199052119867

1198621199001198941198911198971198901199052119878


minus5) (times10

minus5) (times10

minus5)

1 38863 38766 38660 387042 38757 38696 38524 386223 38771 38650 38529 386044 38814 38688 38531 386115 38816 38726 38524 386046 38817 38728 38528 385947 38818 38729 38517 385998 38819 38741 38519 385949 38818 38746 38519 38573Average 38810 38719 38539 38612


Wavelets 1198631198872119867

1198631198872119878

1198621199001198941198911198971198901199052119867

1198621199001198941198911198971198901199052119878


minus5) (times10

minus5) (times10

minus5)

1 33462 19278 05385 099222 37502 30319 21602 236013 39625 31196 25694 270954 38269 31158 25220 259555 38084 32815 24967 315386 38023 34627 24291 323427 38048 35441 25636 344398 38031 36604 25646 349299 37963 37501 25565 36166






5 Conclusions




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119905= ln(119901

119905(119901119905minus1)) We further






00002000400060008

001

Date

Retu

rn

USDRenminbi

13 11 09 07



0001002003004005

Date

Retu

rn

USDEuro

11 13 09 07












002040608

1

Time

Cor

relat

ion



Statistics 1205881

1205882

1205883
















Wavelets 1198631198872119867

1198631198872119878

1198621199001198941198911198971198901199052119867

1198621199001198941198911198971198901199052119878


minus5) (times10

minus5) (times10

minus5)

1 38863 38766 38660 387042 38757 38696 38524 386223 38771 38650 38529 386044 38814 38688 38531 386115 38816 38726 38524 386046 38817 38728 38528 385947 38818 38729 38517 385998 38819 38741 38519 385949 38818 38746 38519 38573Average 38810 38719 38539 38612


Wavelets 1198631198872119867

1198631198872119878

1198621199001198941198911198971198901199052119867

1198621199001198941198911198971198901199052119878


minus5) (times10

minus5) (times10

minus5)

1 33462 19278 05385 099222 37502 30319 21602 236013 39625 31196 25694 270954 38269 31158 25220 259555 38084 32815 24967 315386 38023 34627 24291 323427 38048 35441 25636 344398 38031 36604 25646 349299 37963 37501 25565 36166






5 Conclusions




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















002040608

1

Time

Cor

relat

ion



Statistics 1205881

1205882

1205883
















Wavelets 1198631198872119867

1198631198872119878

1198621199001198941198911198971198901199052119867

1198621199001198941198911198971198901199052119878


minus5) (times10

minus5) (times10

minus5)

1 38863 38766 38660 387042 38757 38696 38524 386223 38771 38650 38529 386044 38814 38688 38531 386115 38816 38726 38524 386046 38817 38728 38528 385947 38818 38729 38517 385998 38819 38741 38519 385949 38818 38746 38519 38573Average 38810 38719 38539 38612


Wavelets 1198631198872119867

1198631198872119878

1198621199001198941198911198971198901199052119867

1198621199001198941198911198971198901199052119878


minus5) (times10

minus5) (times10

minus5)

1 33462 19278 05385 099222 37502 30319 21602 236013 39625 31196 25694 270954 38269 31158 25220 259555 38084 32815 24967 315386 38023 34627 24291 323427 38048 35441 25636 344398 38031 36604 25646 349299 37963 37501 25565 36166






5 Conclusions




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Wavelets 1198631198872119867

1198631198872119878

1198621199001198941198911198971198901199052119867

1198621199001198941198911198971198901199052119878


minus5) (times10

minus5) (times10

minus5)

1 38863 38766 38660 387042 38757 38696 38524 386223 38771 38650 38529 386044 38814 38688 38531 386115 38816 38726 38524 386046 38817 38728 38528 385947 38818 38729 38517 385998 38819 38741 38519 385949 38818 38746 38519 38573Average 38810 38719 38539 38612


Wavelets 1198631198872119867

1198631198872119878

1198621199001198941198911198971198901199052119867

1198621199001198941198911198971198901199052119878


minus5) (times10

minus5) (times10

minus5)

1 33462 19278 05385 099222 37502 30319 21602 236013 39625 31196 25694 270954 38269 31158 25220 259555 38084 32815 24967 315386 38023 34627 24291 323427 38048 35441 25636 344398 38031 36604 25646 349299 37963 37501 25565 36166






5 Conclusions




Acknowledgments


References




























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









exchange rate forecasting using entropy optimized ......exchange rate forecasting using entropy...

Documents