A Comparative Study between Discrete Wavelet Transform and Maximal Overlap Discrete Wavelet Transform for Testing Stationarity

Authors: Amel Abdoullah Ahmed Dghais, Mohd Tahir Ismail

Abstract:

In this paper the core objective is to apply discrete wavelet transform and maximal overlap discrete wavelet transform functions namely Haar, Daubechies2, Symmlet4, Coiflet2 and discrete approximation of the Meyer wavelets in non stationary financial time series data from Dow Jones index (DJIA30) of US stock market. The data consists of 2048 daily data of closing index from December 17, 2004 to October 23, 2012. Unit root test affirms that the data is non stationary in the level. A comparison between the results to transform non stationary data to stationary data using aforesaid transforms is given which clearly shows that the decomposition stock market index by discrete wavelet transform is better than maximal overlap discrete wavelet transform for original data.

Keywords: Discrete wavelet transform, maximal overlap discrete wavelet transform, stationarity, autocorrelation function.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1089273

References:

[1] Bolzan, M., Guarnieri, FL Vieira and Paulo Cesar. Comparisons between two wavelet functions in extracting coherent structures from solar wind time series. Brazilian journal of physics, 39(1) (2009), 12-17.
[2] Stolojescu, C. R., Ion Moga, Sorin Isar and Alexandru, Comparison of wavelet families with application to WiMAX traffic forecasting. 12th International conference on optimization of electrical and electronic Equipment. IEEE. (2010). 932-937.
[3] Razak, A. A., Rasimah and Ismail, Mohd Tahir. Denoising Malaysian time series data: A comparison using discrete and stationary wavelet transforms. International conference on science and social research (CSSR), Kuala Lumpur, Malaysia, 2010. IEEE.
[4] Strang, G. Wavelets and dilation equations: A brief introduction. Society for Industrial and applied mathematics review 31(4) (1989), 614-627.
[5] Heil, C. E. and D. F. Walnut. Continuous and discrete wavelet transforms. SIAM review 31(4) (1989). 628-666.
[6] Percival, D. B. and A. T. Walden. Wavelet methods for time series analysis. Cambridge University Press. Cambridge, U.K. (2000).
[7] Nason, G. P. and B. W. Silverman. The stationary wavelet transform and some statistical applications. Wavelets and statistics, Springer New York (1995). 281-299.
[8] Liang, J. and T. W. Parks. A translation-invariant wavelet representation algorithm with applications. IEEE transactions on signal processing. 44(2) (1996). 225-232.
[9] Pesquet, J.C., Karim, H. and Carfantan, H. Time-invariant orthonormal wavelet representations. IEEE transactions on signal processing. 44(8) (1996). 1964-1970.
[10] Phillips, P. C. and P. Perron. Testing for a unit root in time series regression. Biometrika. 75(2) (1988). 335-346.
[11] Dickey, D.A., Fuller, W.A. Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica. 49 (1981). 1057–1072.
[12] Mallat, S. G. Multiresolution approximations and wavelet orthonormal bases of L2(R). Transactions of the American Mathematical Society .315(1) (1989). 69-87.