Int. Journal of Math. Analysis, Vol. 5, 211, no. 27, 1343-1352 MODWT Based Time Scale Decomposition Analysis of BSE and NSE Indexes Financial Time Series Anu Kumar 1* and Loesh K. Joshi 2 Department of Mathematics, Faculty of Science & Technology The ICFAI University, Dehradun-248197, India anu4march@gmail.com A. K. Pal 3 and A. K. Shula 4 Department of Mathematics, Statistics and Computer Science, College of Basic Sciences and Humanities, G. B. Pant University of Agriculture and Technology, Pantnagar-263145, U. S. Nagar, Uttarahand, India Abstract In this paper, wavelet based concepts have been employed to study two strongly correlated financial time series of BSE and NSE indexes using index data from April 199 to March 26 by decomposing index based financial time series into time-scale components using the MODWT (Maximal Overlap Discrete Wavelet Transform) analysis. The results have clearly established that MODWT based time scale decomposition analysis gives better results than the Fourier transform based spectral analysis of BSE and NSE indexes financial time series. Mathematics Subect Classification: 42C4, 37M1 Keywords: stoc marets, scaling, MODWT, financial time series, BSE & NSE indexes, spectral analysis 1. Introduction The past decades have witnessed different activities with regard to study of the nature of financial time series. Various new concepts and methods of both
1344 A. Kumar, L. K. Joshi, A. K. Pal and A. K. Shula Applied Mathematics and Economics have been suitably employed to study financial time series for long range and short range studies. The classical method used to investigate features in a time series to compute the covariance and correlation functions in the time domain and by the frequency decomposition of the time series in the frequency domain has been achieved through the Fourier analysis. In practice, we sometime consider time scale of often non-stationary processes and, therefore, time resolved methods become absolutely necessary. The time resolved method for Fourier analysis is a windowed Fourier analysis but this method has its own limitations which maes it less desirable for analysis of time series with specific characteristics lie the time series with sharp spies and discontinuities lie financial time series. Wavelet analysis provides a better approximation for time series with such characteristics. Moreover, it often extracts more information about the series than any other classical approach of analysis. The methodologies [4,6,8,9] usually employed in empirical studies may generally be stated only over a long time horizon that is only in the long-run as the time series analysis techniques may separate out in ust two time scales economic time series, i.e. the short run and the long run. But the stoc maret sets an example of a maret in which the agents involved consist of heterogeneous investors maing decisions over different time horizons (from minutes to years) and operating at each moment on different time scales. In this way, the nature of the relationship between stoc returns and growth rates of industrial production may well vary across time scales according to the investment horizon of the traders, as small time scales may be lined to speculative activity and coarse scales to investment activity. Thus, for example, if we thin that big institutional investors have long term horizons and, consequently, follow macroeconomic fundamentals, we expect the relationship between stoc returns and economic activity to be stronger at intermediate and coarsest time scales than at the finest ones. In such a context where both the time horizons of economic decisions and the strength and direction of economic relationships between variables according to the time scale analysis [3] may differ, the most appropriate choice of the analytical tool will be the wavelet analysis. In this paper, wavelet concepts have been employed to study two strongly correlated financial time series of Bombay stoc exchange (BSE) and National stoc exchange (NSE) indexes. Both these stoc exchanges belong to India and open and close at the same time i.e. they are synchronous. We have used the BSE and NSE indexes data from April 199 to March 26. Figure 1 and 2 exhibit BSE and NSE indexes financial time series. The structure of the paper will be as follows. Section 2 will describe briefly the methodology employed, i.e. spectral and wavelet analysis for the time series while Section 3 will deal with the empirical results from Fourier analysis and maximal overlap discrete wavelet transform analysis. Section 4 will conclude the paper.
MODWT based time scale decomposition analysis 1345 12 11 BSE Index Time Series 6 55 5 NSE Index Time Series 1 45 BSE Sensex Value 9 8 7 6 5 4 3 2 1 NSE Sensex Value 4 35 3 25 2 15 1 5 1-Nov-91 1-Jul-93 1-Mar-95 1-Nov-96 1-Jul-98 Time 1-Mar- 1-Nov-1 1-Jul-3 1-Mar-5 -- 1-Nov-91 1-Jul-93 1-Mar-95 1-Nov-96 1-Jul-98 Time 1-Mar- 1-Nov-1 1-Jul-3 1-Mar-5 -- Fig. (1) Fig. (2) 2. Methodology Both the series (BSE and NSE indexes time series) are filtered through wavelet transform technique a relatively new mathematical approach to decompose according to time scale components instead of frequencies as has been in case of Fourier approach. Wavelets use a similar strategy as Fourier analysis as they employ some basis functions (wavelets instead of sines and cosines) and use them to decompose the series. Wavelet analysis, in contrast to Fourier analysis, does not need any stationary assumption in order to decompose the series, as spectral decomposition methods is performed on global analysis whereas the wavelets method acts locally in time and so do not need stationary cyclical components. In this section, we shall provide a brief discussion on spectral analysis for time series and then we will present the method of wavelet analysis for time series under which we will go into the details of discrete wavelet transform (DWT) and maximal overlap discrete wavelet transform (MODWT). 2.1 Fourier Transform Based Spectral Analysis for Time Series The Fourier transform has long been applied for analysis of continuous and discrete signals and systems in many different fields. It decomposes a signal or a function into a sum of harmonic components of different frequencies via a linear combination of Fourier basic functions (sines and cosines). Thus, the Fourier transform is a frequency domain representation of a signal or a function containing
1346 A. Kumar, L. K. Joshi, A. K. Pal and A. K. Shula the same information of the original function, but summarized as a function of frequency. As a consequence, it may be interpreted as a decomposition of a signal on a frequency-by-frequency basis. Consider a finite time sequence u ( ), =,..., N 1 of length T = N δ t, where N is the number of data, and δ t is the sampling periodicity. The discrete Fourier transform (DFT) U ( ) of U ( ) and its inverse DFT for finite sequences are respectively defined by U U 1 1 N N = ( ) = u ( ) N 1 2 e ( ) = u ( ) N = 2 2π i N e 2π i N (1) (2) where [.] denotes largest integer smaller or equal than the operand and =,1,..., N 1. When we sample the series with finite period δ t, we limit the 1 1 1 spectrum of the study to the frequency band ω,, where is the 2δ t 2δ t 2δ t Nyquist frequency as frequency outside the range are folded inside by sampling, an effect nown as aliasing. Spectral Estimation The Fourier decomposition is a way of separating the time series into different frequency components to give more insight into the data. Consider a stationary time series { t } fˆ is called the spectrum of { } quantity ( ) f ( ) = δ t ( N 1) = ( N 1) γ e 2π i N with auto-covariance sequence γ ( ) t : Its estimator would be. The N ( ) 1 2 π = δ t γ + 2δ t γ ( ) cos (3) = 1 N ( N ) where ( ) ( ) ( )( ) = 1 γ = γ = t t + is a sample estimator at lag of N t ( N ) the auto-covariance sequence and δ t is the sampling periodicity. The spectrum is a real-valued function because the series is real-valued and the auto-covariance sequence is even. The spectrum thus defined above is an asymptotically unbiased estimator of a theoretical one.to construct spectral estimator which has a small, we use the technique of windowing. This method is variance compared to ( ) f
MODWT based time scale decomposition analysis 1347 employed both in time and frequency domain. We can smooth all abrupt variations and minimize the spurious fluctuations generated every time when the series gets truncated. The result of windowing a time series { t } with n observations is the estimated smoothed spectrum fˆ M 2 ( ) = t ω ( ) ˆ γ ( ) M = 2 M 2 π i N δ e (4) where the auto-covariance sequence is weighted by the lag window ( ) ω M of width M which is equivalent to splitting the series in n/m sub-series of length M. Alternatively, f ˆ ( ) can be obtained by the convolution of the expected ω through spectrum ( ) f with Fourier transform of ( ) M 2 ( ) = t f ( ) W ( ) fˆ δ (5) where ( ) M = 2 M W M is the spectral window of width M. Thus the smooth spectrum at is observed through a window opened on a convenient interval around. 2.2 Wavelet Analysis for Time Series Many economic and financial time series are nonstationary and exhibit changing frequencies over time. The usefulness of wavelet analysis is in its flexibility in handling a variety of nonstationary signals. Indeed, as wavelets are constructed over finite intervals of time and are not necessarily homogeneous over time, they are localized both in time and scale. Thus, two interesting features of wavelet time scale decomposition for economic variables are that, i) since the base scale includes any non-stationary components, the data need not be differenced, and ii) the nonparametric nature of wavelets taes care of potential nonlinear relationships without losing any detail [1]. Roughly, wavelet analysis decomposes a given series in orthogonal components as in the Fourier approach but according to time scale components instead of frequency components. Mathematically, we may note that if there are two basic wavelet functions: the father and the mother wavelets, φ () t and ψ ( t) respectively then the father wavelet is given by the function = t 2 φ 2 2 φ (6) 2 defined as non-zero over a finite time length support which corresponds to given mother wavelets\ M
1348 A. Kumar, L. K. Joshi, A. K. Pal and A. K. Shula = t 2 ψ 2 2 ψ (7) 2 With = 1,, J we have a J-level wavelets decomposition. The former integrates to 1 and reconstructs the longest time-scale component of the series (trend), while the latter integrates to (similar to sine and cosine) and is used to describe all deviations from trend. The mother wavelets, as defined above, play a role similar to sines and cosines in the Fourier decomposition. They are either compressed or dilated in time domain to generate cycles fitting to the actual data [5]. To compute the decomposition we have to calculate wavelet coefficients at all scales representing the proections of the time series onto the basis generated by the chosen family of wavelets i.e. f t () () t d = ψ s = f φ where the coefficients d and s are the wavelet transform coefficients representing, respectively, the proection onto mother and father wavelets. The orthogonal wavelet series approximates to a signal or function f () t in L 2 ( R) given by f t = sj, φ J, t + d J, ψ J, t +... + d ψ t +... + d1, ψ 1, t () ( ) ( ) ( ) ( ) (8) where J is the number of multiresolution components and ranges from 1 to the number of coefficients in the specified components. The multiresolution decomposition of the original signal f (t) is given by the sum of the smooth signal V J and the detail signals W, W J 1,..., W1 with V s, φ, () t and W d, ψ, ( t) ( = 1, 2,.., J). J = J J. J = J The sequence of the terms VJ, WJ,..., W,..., W1 in this equation represent a set of signals components which provide representations of the signal at the different resolution levels 1 to J, and the detail signals W provide the increments at each individual scale or resolution level. The restrictions of DWT on sample size multiple of 2 J and sensitivity to circular shifts due to the downsampling approach are overcome by the maximal overlap DWT (MODWT) and applies to any sample and is translation invariant, at the cost of giving up orthogonality. The maximal overlap discrete wavelet transform (MODWT) is a non-orthogonal variant of the classical discrete wavelet transform that unlie the orthogonal discrete wavelet transform is translate invariant because shifts in the signal do not change the pattern of coefficients. Application of a th order nondecimated version of the orthogonal DWT, i.e. the maximal overlap DWT
MODWT based time scale decomposition analysis 1349 ~ (MODWT), yields J vectors of wavelet filter coefficients W, t for = 1,., J and t = 1,.,N/2 ~, and one vector of wavelet filter coefficient V, t through ~ W ~ V t t = = where L 1 l= L 1 l= ~ h g~ ~ h, l l l f f ( t l) ( t l) and ~ are, respectively, the rescaled wavelet and scaling filter g, l coefficient from a Daubechies compactly supported wavelet family [7]. 3. Empirical Analysis 3.1 Fourier Transform Based Spectral Analysis of BSE and NSE indexes Financial Time Series Fourier transform based Spectral analysis is purely a descriptive technique. It is a tool for inspecting cyclic phenomena and highlighting lead-lag relations among time series in order to provide an accurate way to define each series components and also a reliable method by means of filtering. Cross spectral analysis allows the detailed study of the correlation among time series. Fig. (3) and (4) below shows the frequency spectra of BSE index financial time series in Fig. (1) and NSE index financial time series in Fig.(2). 6 3 Amplitude 4 2 Amplitude 2 1..1.2.3.4.5.6 Frequency (Hz)..1.2.3.4.5.6 Frequency (Hz) Fig. (3) Fig. (4)
135 A. Kumar, L. K. Joshi, A. K. Pal and A. K. Shula 3.2 MODWT Based Time Scale Decomposition Analysis of BSE and NSE indexes Financial Time Series The analysis has been conducted by using average of monthly data of Bombay Stoc Exchange index and National Stoc Exchange index of India between April 199 and March 26 (sourses:www.sebi.gov.in) and the averages are based on daily closing index. We have decomposed the two financial time series into their time-scale components using the MODWT which is a non-orthogonal variant of the classical discrete wavelet transform that unlie the orthogonal discrete wavelet transform is translation invariant, as shifts in the signal do not change the pattern of coefficients. BSE index financial time series NSE index financial time series 199 1994 1998 22 26 199 1994 1998 22 26 Fig. (5) BSE index financial time series (left) and NSE index financial time series (right)
MODWT based time scale decomposition analysis 1351 The wavelet filter used in the decomposition is the Daubechies least asymmetric (LA) wavelet filter of length L = 8, or LA(8) wavelet filter, based on eight non-zero coefficients [2], with periodic boundary conditions. Given that the maximum decomposition level J given by log 2 (N) we have applied the MODWT up to a level J = 5 which produces six wavelet and scaling filter sets of coefficients v5, w5, w4, w3, w2, w1. Given that the level of the transform defined the effective scale λ of the corresponding wavelet coefficients for all families of Daubechies compactly supported wavelets the level wavelet coefficients are associated with changes at scale 2 1. All computations have been performed using the Waveslim Matlab Pacage. Fig. (5) shows the MODWAT multiresolution decomposition analysis. 4. Conclusion On looing at the frequency spectrum of BSE index financial time series and NSE index financial time series in Fig. (3) and Fig. (4), we observe that the spectrum is dominated by higher frequency components. It does not clearly show the presence of other frequency components in BSE and NSE index financial time series so Fourier transform based spectral analysis is not the correct choice to analyze the BSE and NSE index financial time series which are non-stationary in nature. On the other hand wavelet transform (filter) depends on two parameters frequency and time, that provide the time and frequency information simultaneously. Hence it provides the socalled time-scale or time-frequency representation of the signal where the scale factor is inversely related to the frequency of the wavelet. Prior nowledge of which spectral components occur at which time interval in time series is of great importance when analyzing financial time series. So if we want to now that what spectral components occur and at which time interval in a financial time series, Fourier transform based spectral analysis is not the correct transform to use thus when time localization of the spectral component is needed, wavelet transform based time scale decomposition analysis of financial time series should be adopted. References 1. C. Schleicher, 22, An introduction to wavelets for economists, Ban of Canada Woring Paper, No. 2-3. 2. I. Daubechies, 1992, Ten lectures on wavelets, SIAM, Philadelphia. 3. J. B. Ramsey, C. Lampart, 1998a, The decomposition of economic relationship by time scale using wavelets: money and income, Macroeconomic Dynamics 2, pp. 49 71.
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