S&P 500 Sektör Endekslerinin Fraktal Analizi

Bu çalışmada S&P 500 sektör endekslerinin çoklu fraktal özellikleri Çoklu Fraktal Eğilimden Arındırılmış Dalgalanma Analizi (ÇF-EADA) ile incelenmiştir. ÇF-EADA zaman serisi verilerinin çoklu fraktal özelliklerini tarif etmek için kullanılan bir sinyal işleme tekniğidir. Bu yöntem zaman serilerinin ölçekleme davranışını tahmin etmek için kullanılan Eğilimden Arındırılmış Dalgalanma Analizi (EADA) yönteminin bir uzantısıdır. ÇF-EADA yönteminin arkasında yatan temel fikir bir zaman serisini kaba ölçekli bir işlem kullanarak birden fazla ölçeğe ayırmak ve ardından EADA yöntemiyle her ölçeğin ölçeklenme davranışını tahmin etmektir. Bu, zaman serilerinin çok fraktal özelliklerini tanımlayan bir dizi ölçeklendirme üssü verir. ÇF-EADA sonuçlarımız, tüm S&P 500 sektör endekslerinde çoklu fraktalitenin varlığını göstermektedir. Bu indeksler çoklu fraktal olduğundan, ölçekleme değişkenliği, doğrusal olmayan dinamikler, kendine benzerlik, uzun menzilli bağımlılık, çok ölçekli korelasyonlar ve durağan olmama gibi özelliklere sahip oldukları sonucuna varabiliriz.

Anahtar Kelimeler:

Multifraktalite, MF-TADA, Multifraktal Trendden Arındırılmış Dalgalanma Analizi, Fraktal Teori, S&P 500

Fractal Analysis of S&P 500 Sector Indexes

In this study multifractal properties of S&P 500 sector indexes are investigated with Multifractal Detrended Fluctuation Analysis (MF-DFA). The MF-DFA is a signal processing technique that is used to describe the multifractal properties of a time series data. It is an extension of Detrended Fluctuation Analysis (DFA), which is a widely utilized method for estimating the scaling behavior of a time series. Main idea behind MF-DFA is to decompose a time series into multiple scales using a coarse-graining procedure, and then to estimate the scaling behavior of each scale using DFA. This gives a set of scaling exponents that describe the multifractal features of the time series. Our MF-DFA results indicates the presence of multifractality in all S&P 500 sector indexes. Since these indexes are multifractal, we can conclude that they possess properties such as scaling variability, nonlinear dynamics, self-similarity, long-range dependence, multiscale correlations and nonstationary.

Keywords:

Multifractality, MF-DFA, Multifractal Detrended Fluctuation Analysis, Fractal Theory, S&P 500,

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Ali, S., Shahzad, S. J. H., Raza, N. & Al-Yahyaee, K. H. (2018). Stock Market Efficiency: A Comparative Analysis of Islamic and Conventional Stock Markets. Physica A: Statistical Mechanics and Its Applications, 503, 139-153.
Cao, G., Cao, J. & Xu, L. (2013). Asymmetric Multifractal Scaling Behavior in The Chinese Stock Market: Based on Asymmetric MF-DFA. Physica A: Statistical Mechanics and Its Applications, 392(4), 797-807.
Degutis, A. & Novickytė, L. (2014). The Efficient Market Hypothesis: A Critical Review of Literature and Methodology. Ekonomika, 93, 7-23.
De Moura, E. P., Vieira, A. D. P., Irmao, M. A. S. & Silva, A. A. (2009). Applications of Detrended-Fluctuation Analysis to Gearbox Fault Diagnosis. Mechanical Systems and Signal Processing, 23(3), 682-689.
Duan, Q., An, J., Mao, H., Liang, D., Li, H., Wang, S. & Huang, C. (2021). Review About the Application of Fractal Theory in The Research of Packaging Materials. Materials, 14(4), 860.
Fama, E. F. (1970). Efficient Capital Markets: A Review of Theory and Empirical Work. The Journal of Finance, 25(2), 383-417.
Gilmore, M., Yu, C. X., Rhodes, T. L. & Peebles, W. A. (2002). Investigation of Rescaled Range Analysis, The Hurst Exponent, and Long-Time Correlations in Plasma Turbulence. Physics of Plasmas, 9(4), 1312-1317.
Hurst, H. E. (1951). Long-Term Storage Capacity of Reservoirs. Transactions of the American Society of Civil Engineers, 116(1), 770-799.
Hurst, H. E. (1957). A Suggested Statistical Model of Some Time Series Which Occur in Nature. Nature, 180(4584), 494-494.
Ivanova, K. & Ausloos, M. (1999). Application of the Detrended Fluctuation Analysis (DFA) Method for Describing Cloud Breaking. Physica A: Statistical Mechanics and Its Applications, 274(1-2), 349-354.
Kantelhardt, J. W., Zschiegner, S. A., Koscielny-Bunde, E., Havlin, S., Bunde, A. & Stanley, H. E. (2002). Multifractal Detrended Fluctuation Analysis of Nonstationary Time Series. Physica A: Statistical Mechanics and its Applications, 316(1-4), 87-114.
Kurnaz, M. L. (2004). Application of Detrended Fluctuation Analysis to Monthly Average of The Maximum Daily Temperatures to Resolve Different Climates. Fractals, 12(04), 365-373.
Kuznetsov, N. A. & Rhea, C. K. (2017). Power Considerations for The Application of Detrended Fluctuation Analysis in Gait Variability Studies. PLoS One, 12(3), e0174144.
Lo, A. W. (1991). Long-Term Memory in Stock Market Prices. Econometrica: Journal of the Econometric Society, 59(5), 1279-1313.
Makletsov, S. V., Opokina, N. A. & Shafigullin, I. K. (2019). Application of Fractal Analysis Method for Studying Stock Market. International Transaction Journal of Engineering, Management, & Applied Sciences & Technologies, 11(1).
Malkiel, B. G. (2003). The Efficient Market Hypothesis and Its Critics. Journal of Economic Perspectives, 17(1), 59-82.
Mandelbrot, B. B (1982). The Fractal Geometry of Nature (Vol. 1). New York: WH Freeman. Mensi, W., Tiwari, A. K. & Yoon, S. M. (2017). Global Financial Crisis and Weak-Form Efficiency of Islamic Sectoral Stock Markets: An MF-DFA Analysis. Physica A: Statistical Mechanics and Its Applications, 471, 135-146.
Mensi, W., Hamdi, A. & Yoon, S. M. (2018). Modelling Multifractality and Efficiency of GCC Stock Markets Using The MF-DFA Approach: A Comparative Analysis of Global, Regional and Islamic Markets. Physica A: Statistical Mechanics and Its Applications, 503, 1107-1116.
Milos, L. R., Haţiegan, C., Milos, M. C., Barna, F. M. & Boțoc, C. (2020). Multifractal Detrended Fluctuation Analysis (MF-DFA) of Stock Market Indexes. Empirical Evidence from Seven Central and Eastern European Markets. Sustainability, 12(2), 535.
Outcalt, S. I., Hinkel, K. M., Meyer, E. & Brazel, A. J. (1997). Application of Hurst Resecaling to Geophysical Serial Data. Geographical Analysis, 29(1), 72-87.
Peng, C. K., Buldyrev, S. V., Havlin, S., Simons, M., Stanley, H. E. & Goldberger, A. L. (1994). Mosaic Organization of DNA Nucleotides. Physical Review E, 49(2), 1685-1689.
Raimundo, M. S. & Okamoto Jr, J. (2018). Application of Hurst Exponent (H) and the R/S Analysis in the Classification of FOREX Securities. International Journal of Modeling and Optimization, 8(2), 116-124.
Resta, M. (2012). Hurst Exponent and Its Applications in Time-Series Analysis. Recent Patents on Computer Science, 5(3), 211-219.
Rizvi, S. A. R., Dewandaru, G., Bacha, O. I. & Masih, M. (2014). An Analysis of Stock Market Efficiency: Developed vs Islamic Stock Markets Using MF-DFA. Physica A: Statistical Mechanics and its Applications, 407, 86-99.
Ruan, Q., Zhang, S., Lv, D. & Lu, X. (2018). Financial Liberalization and Stock Market Cross-Correlation: MF-DCCA Analysis Based on Shanghai-Hong Kong Stock Connect. Physica A: Statistical Mechanics and Its Applications, 491, 779-791.
Shahzad, S. J. H., Nor, S. M., Mensi, W. & Kumar, R. R. (2017). Examining The Efficiency and Interdependence of US Credit and Stock Markets Through MF-DFA and MF-DXA Approaches. Physica A: Statistical Mechanics and its Applications, 471, 351-363.
Stošić, D., Stošić, D., Stošić, T. & Stanley, H. E. (2015). Multifractal Properties of Price Change and Volume Change of Stock Market Indices. Physica A: Statistical Mechanics and Its Applications, 428, 46-51.
Talkner, P. & Weber, R. O. (2000). Power Spectrum and Detrended Fluctuation Analysis: Application to Daily Temperatures. Physical Review E, 62(1), 150.
Tiwari, A. K., Aye, G. C. & Gupta, R. (2019). Stock Market Efficiency Analysis Using Long Spans of Data: A Multifractal Detrended Fluctuation Approach. Finance Research Letters, 28, 398-411.
Wang, L., Zeng, X., Yang, H., Lv, X., Guo, F., Shi, Y. & Hanif, A. (2021). Investigation and Application of Fractal Theory in Cement-Based Materials: A Review. Fractal and Fractional, 5(4), 247.
Ying, Q., Yousaf, T., Ain, Q. U., Akhtar, Y. & Rasheed, M. S. (2019). Stock Investment and Excess Returns: A Critical Review in The Light of The Efficient Market Hypothesis. Journal of Risk and Financial Management, 12(2), 97.
Zhu, H. & Zhang, W. (2018). Multifractal Property of Chinese Stock Market in The CSI 800 Index Based on MF-DFA Approach. Physica A: Statistical Mechanics and its Applications, 490, 497-503.