A wavelet-based approach of the Athens stock market predictability

A WAVELET-BASED APPROACH OFATHENS STOCK MARKET PREDICTABILITYEleni Avaritsioti 1Centre for Quantitative FinanceTanaka Business SchoolImperial College**of** Science, Technology and MedicineTel: +30 6977455672E-mail: eleni.avaritsioti03@imperial.ac.ukAbstractA process has been developed, in **the** framework **of** my PhD work, for financialtime series prediction which employs **the** Discrete Wavelet Transform (DWT).The predicted **wavelet** coefficients at each decomposition level, are combinedthrough **the** Inverse DWT to calculate future values **of** **the** time series. Two**wavelet** filters have been used: **the** Haar and **the** Daubechies-4. The results frombacktesting experiments using historical data **of** four **of** **the** new sector-**based**indices **of** ATHEX are compared to **the** ones obtained from an AR predictionmodel with an optimum number **of** lagged coefficients, according to **the** AkaikeInformation Criterion. Hit rate and RMSE have been used as measures **of**forecasting accuracy. Hit rates above 51% are reported with **the** Db-4 filter with amaximum **of** 60% for **the** Banks Index.IntroductionA number **of** models have been developed for time series prediction **based** onhistorical data as discussed and described by De Gooijer and Hyndman,2006.The most popular are **the** linear regression methods like autoregressive (AR),Autoregressive Moving Average (ARMA) and **the** Autoregressive Integratedmoving Average (ARIMA) discussed by Box and Jenkins, 1976 and Diebold,2004. An important requirement for **the** successful application **of** **the**se models is**the** existence **of** stationarity. Financial time series, however, exhibit complicatedpatterns and abrupt changes may be observed over time, because **of** trends,volatility clustering, etc. and **the**ir transformation to stationary ones requires ausually lengthy trial and error process and application **of** special tests. As analternative, **the** **wavelet** transform may be used given that many non-stationarytime series become stationary after **wavelet** transformations, see Marsy, 1993.Multi-resolution decomposition techniques such as **the** **wavelet** transform,(Ramsey, 1999) decompose a time series into different time scales as discussedby Gencay et al, 2002, so that **the**y reveal seasonalities, discontinuities, volatilityclustering and **the**y identify local and global dynamic properties **of** **the** process.Forecasting techniques have been proposed that use **wavelet** transforms toprocess **the** historical data; Fryzlewicz, 2003 employed **the** Maximal OverlapDiscrete Wavelet Transform (MODWT) and **the** Haar filter and Renaud et al,2005, proposed an “a trous” **wavelet** transform, which in essence is a non-1 Address for correspondence: 84, Salaminos Str., Brilissia 152 35, A**the**ns, Greece.1

decimated DWT, **based** on **the** Haar **wavelet** with an arbitrary selection **of** detailcoefficients to be predicted. In this paper, we propose a new combinedprediction and filtering method, which can be seen as a bridge between **the****wavelet** de-noising techniques and **the** **wavelet** predictive methods, using ARmodels to predict future values **of** detail coefficients from **the** values **of** **the** onesobtained by **the** multi-scale decomposition **of** a time series provided by **the** Haarand Daubechies-4 **wavelet**s.The paper is organized as follows: a synopsis is made on **the** Discrete WaveletTransform (DWT) which decomposes a time series into **wavelet** coefficientswhich are subsequently thresholded for **the** de-noising or smoothing **of** financialtime series. The proposed prediction methodology that has been developed in**the** **wavelet** domain is briefly described. Finally, **the** results **of** **the** backtestingexperiments, for three rolling window widths, obtained with especially developedC++ codes, are presented in terms **of** RMSE and Hit Rate values for **the** case **of**four financial indices **of** ATHEX.The Discrete Wavelet Transform (DWT)A dyadic “discretised” **wavelet** is given by:−m/ 2 −mψm, n( t) = 2 ψ (2 t − n), (1)where **the** control parameters m and n are contained in **the** set **of** all integers Z2(both positive negative and zero) and ψm, n∈ L ( R)is a real-valued function.Wavelets **of** this type are chosen to be orthonormal i.e. both orthogonal to eacho**the**r and normalized to have unit energy. The requirement **of** orthogonality isnecessary in order to diminish redundancy and to ensure that **the** resultsobtained from a **wavelet** transform **of** a signal are uncorrelated.Scaling functions are associated to **the** **wavelet**s, which have **the** same form, i.e.−m/ 2 −mφm, n( t) = 2 φ(2 t − n), (2)2where φ,∈ L ( R ) is a real-valued function.m nMallat,1989 proved that at scale index m+1 **the**re are coefficientshkandwhich satisfy **the** following identities with n ∈Z and for arbitrary integer values **of**m:gk,φψ1( t) = ∑ h φ ( t)2m+ 1, n k m,2n+kk∈Z1( t) = ∑ g φ ( t)2m+ 1, n k m,2n+kk∈Z(3)(4)The first **of** **the**se equations means that **the** scaling function at any arbitrary scaleis expressed in terms **of** a weighted sum **of** shifted scaling functions **of** **the**2

previous scale factored by **the**ir respective coefficients hk. These equations form**the** basis for a decomposition algorithm as shown next. The approximationcoefficients cm,2n+ kfor each k and hence **the** approximation coefficients at scalem+1 can be calculated from **the** scaling coefficients **of** **the** previous scale m as:1cm+ 1, n= ∑ hk −2 ncm,k(5)2 kSimilarily, **the** **wavelet** coefficients at scale m+1 can be calculated from **the**approximation coefficients **of** scale m as:1dm+ 1, n= ∑ gk −2 ncm,k(6)2 kSo, given **the** approximation coefficients at a scale m, **the** approximation anddetail coefficients at all scales can be calculated by repeating **the** two equationsabove. Mallat’s multiresolution analysis, represented by Eqs. (5) and (6), issimilar to **the** sub-band coding algorithm in signal processing. The sequencegk2n{ −( ) } is known as **the** high-pass or high-band filter followed by decimation2h −k 2nand **the** sequence { ( ) } is known as **the** low-pass or low band filter followed2by down-sampling represented by **the** factor 2n. Both filters belong to **the** FiniteImpulse Response (FIR) filters and **the**y constitute Quadrature Mirror Filters.Repeating this process on **the** approximation coefficients, for several scales, isequivalent to **the** so-called cascading **of** filter banks.A signal reconstruction algorithm can be implemented by repeating **the**seequations in reverse order, i.e. calculating cm,nfrom cm + 1, nand dm + 1, nas:1 1c = h c + g d2 2∑ ∑ (7)m−1, n n−2 k m, k n−2 k m,kkkThe Haar **wavelet** system can be seeing as a filter **of** length L=2 that can bedefined by its scaling (low-pass) filter coefficients and by its **wavelet** (high-pass)filter coefficients, respectively as:h02and12h 0g g and12 2The Daubechies 4 is a filter **of** length L=4 with low-pass filter coefficients:and high-pass filter coefficients :h01+3 3 + 3 3−3= , h 1= , h2= , h3=4 441−34g1−33 − 33+31+3= h = , g1 = − h2= − , g2 = h1= , g3 = − h0= − .44440 33

Within-Scale De-correlation **of** Detail CoefficientsApplication **of** **the** DWT to **the** historical data decomposes **the**m into variousresolution levels that reveal **the**ir underlying structure and it produces detailcoefficients at each one **of** **the** five decomposition leves which show a smallwithin-scale correlation that is reduced with increasing decomposition level, asshown if Figure 1. This has been used for **the** calculation **of** future values **of** detailcoefficients with an AIC optimised AR model.Figure 1: Auto-correlation **of** Detail Coefficients as a Function **of** DecompositionLevel for **the** BANKS IndexWavelet “De-noising”A time series may be de-noised by thresholding **the** detail coefficients at alldecomposition levels. The choice **of** **the** value **of** threshold is a very importantproblem given that a large value will destroy all significant features **of** **the**decomposed signal leading thus to a reconstructed signal that does not resemble**the** original one. The “optimum” value **of** **the** threshold level, λ , is calculatedaccording to Donoho, 1995 as:λ2= 2σlog( N)(8)2where N is **the** length **of** **the** decomposed vector, and σ is **the** variance **of** **the**noise, which is estimated from **the** variance **of** **the** detailed coefficients at **the** firstdecomposition level. It is worth noting that **the** threshold level is a function **of** **the**volatility exhibited by **the** time series under consideration; when volatility is zero**the** threshold level is zero and as volatility increases **the** threshold level increasestoo. A second important point is that thresholding **of** **the** **wavelet** detailcoefficients does not remove **the** volatility but it suppresses it to a certain degreeso that **the** majority **of** **the** underlying features is preserved as it may beconfirmed by examining **the** descriptive statistics **of** **the** resulting residuals.Wavelet de-noising does not require any assumptions about **the** nature **of** **the**series and allows for discontinuities and variations.The resulting ( n in number) thresholded detail coefficients (at decompositions**of**tlevel j ) are given by: d [ n] = sign( d [ n]) ( d [ n] − λ).j j j4

Figure 2: Original and De-noised SeriesBanksInsuranceIndex Value5.5005.0004.5004.0003.5003.0002.5002.0001 52 103 154 205 256 307 358 409 460 511Number **of** PointsPrediction in **the** Wavelets DomainThe 512 daily values **of** **the** four new-sector indices **of** ATHEX, from 17/12/2003to 30/12/2005, are used as historical data. The first step **of** **the** predictionprocedure is to apply **the** DWT to **the** historical data with predefined a)rollingwindow width, b)number **of** decomposition levels, and c)Haar or **the** Daubechies-4 filter. The second step involves **the** estimation, at **the** jth decomposition level, **of**one future value from **the** approximate coefficients using linear regression andone future value from **the** detail coefficients using an AR model. The third stepinvolves **the** estimation **of** **the** appropriate (at each level) number **of** future values**of** detail coefficients, and **the** fourth step involves **the** application **of** **the** IDWT to**the** estimated future values.Forecasting Accuracy MeasuresThe forecasting accuracy has been evaluated with RMSE (a measure verysensitive to outliers in innovations) and Hit Rate (a practical measure) which arerespectively defined as:RMSE =1pp∑( actual(t)− predicted(t))t = 12whereTestSize1HR = ∑ SameDirection( ∆Ppred , t, ∆Pactual , t)TestSize t=2⎧1 if ∆Ppred , t≤ 0 and ∆Pactual , t≤ 0⎫⎪⎪SameDirection( ∆Ppred , t, ∆ Pactual , t) = ⎨1 if ∆Ppred , t≥ 0 and ∆Pactual , t≥ 0⎬⎪0o**the**rwise⎪⎩⎭with ∆ Ppred , t= Ppred , t− Ppred , t−1∆ P = P − P−actual, t actual, t actual, t 1Prediction ResultsRMSE values, shown in Table 1, are relatively high due to **the** outliers existing inall **of** **the** four series. Hit rate values show a strong dependence on **the** rollingwindow length. This may be attributed to **the** relation **of** **the** window length to **the**period **of** **the** oscillations that exist in **the** data. Specifically, in **the** case **of**5

Insurance Index, window length **of** 128 working days, which is comparable to **the**strong oscillations present, gives a hit rate **of** 58% a value that is reduced to 53%when different window lengths are used. On **the** contrary, in **the** case **of** BanksIndex which exhibits weaker oscillations **of** shorter period, as shown in Figure 2,**the** hit rate value **of** 60% has been obtained with **the** relatively wider rollingwindow length **of** 256.ATHEXIndexTable 1: Results from **the** Backtesting **of** **the** Proposed Prediction ProcessHaar Filter Db-4 Filter ARwindow 64 window 128 window 256 window 64 window 128 window 256 window 64RMSE HR RMSE HR RMSE HR RMSE HR RMSE HR RMSE HR RMSE HRBanks 58 0.54 54 0.55 55 0.56 49 0.52 49 0.52 52 0.60 84 0.000Telecoms 57 0.47 52 0.45 54 0.53 49 0.52 48 0.51 52 0.53 79 0.002Oil & Gas 66 0.50 62 0.52 69 0.53 55 0.52 55 0.52 64 0.52 97 0.000Insurance 82 0.52 74 0.54 75 0.51 69 0.53 69 0.58 73 0.54 123 0.002ConclusionsA forecasting process is proposed which in combination with **the** Daubechies 4filter predicts with a relative high degree **of** success all **of** **the** four sector-**based**indices **of** ATHEX studied. The AIC optimised AR model used as benchmarkshowed zero **predictability**. The prediction success depends on **the** rollingwindow length as it was expected. Extension **of** **the** proposed process with **the**use **of** Independent Component Analysis for **the** between-scale decorrelation **of**detail coefficients is expected to fur**the**r improve **the** forecasting success.ReferencesBox, G.P. and G. M. Jenkins (1976). Time Series Analysis: Forecasting andControl, Holden-Day.De Gooijer, J.G. and R. J. Hyndman. (2006). 25 years **of** time series Forecasting.Int. Journal. **of** Forecasting, 22, 443-473.Diebold, Francis X. (2004). Elements **of** Forecasting. 3d Edition, Thomson South-Western.Donoho, D. L. (1995). Denoising via s**of**t thresholding. IEEE transactions onInformation Theory, 41, 613-627.Fryzlewicz, P. (2003). Wavelet Techniques for Time Series and Poison Data. PhDDissertation, University **of** Bristol, U.K.Gencay, R., F. Selcuk and B. Whitcher (2002). An Introduction to Wavelets andO**the**r Filtering Methods in Finance and Economics. Academic Press.Mallat, S.G. (1989). A **the**ory for Multiresolution Signal Decomposition: TheWavelet Representation. IEEE Trans. On Pattern Analysis and MachineIntelligence, 11 (7), 674-692.Marsy, E. (1993).The **wavelet** transform **of** stochastic processes with stationaryincrements and its application to fractional Brownian motion. IEEETransactions on Information Theory, 39, 260-264.Ramsey, J. B. (1999). The contribution **of** **wavelet**s to **the** analysis **of** economic andfinancial data. Phil.Trans. R. Soc. Lond. A 357, 2593-2606.Renaud, O., J.-L. Starck, and F. Murtagh. (2005). Wavelet-Based Combined SignalFiltering and Prediction. IEEE Transactions Systems, Man and Cybernetics,part B, 35(6), 1241-1251.6