New Criterion For Fractal Parameter In Financial Time Series
محورهای موضوعی : Financial and Economic Modelling
Mehrzad Alijani
1
(Department of Management and Economics, Science and Research Branch, Islamic Azad University, Tehran, Iran)
bahman banimahd
2
(Associate Professor in Accounting, Head of Accounting Department, Islamic Azad University- Karaj Branch, Iran)
Ahmad Yaghobnezhad
3
(Department of Economic and Accounting, Islamic Azad University of Central Tehran Branch, Tehran, ‎Iran ‎)
کلید واژه: R, Simulation, ARFIMA time series, fractal dimension, Hausdorff measure,
چکیده مقاله :
Since calculating the amount of fractal in the ARFIMA time series and increasing its ‎accuracy and bring it closer to reality is very important, this article intends to ‎investigate the possibility of modifying this computational formula by changing the ‎focus criterion and using simulation. In the present paper, by analysing and ‎simulating the fractal parameter for time series ARFIMA model and redefining and ‎reviewing the Fractal mathematical, a fractal calculus and dimension in ‎comparison ‎with Euclidean norms introduced. In this regard, first, a new criterion about fractal ‎or Hausdorff ‎component for measuring the forms of fractal time series introduced, ‎then the effects and functional ‎inquiries using simulation data searched, and some ‎mathematical proofs through simulation of ‎data achieved. The findings showed that, ‎the deviation of the new estimator from the simulated initial value is less, and closer ‎to reality as this new criterion introduced by changing the focus criterion and ‎replacing the mean with the median due to less sensitivity to out-dated data. The ‎new criterion is better for determining the fractal parameter and identifying its ‎degree of effectiveness. Finally, the findings empirically indicated that the proposed ‎criterion is more efficient and better ‎than the others for calculating fractal ‎dimensions.‎
[1] Abdon, A., Anum, S., Differential and integral operators with constant fractional order and variable fractional dimension, Chaos, Solutions and Fractals, 2019, 127, P. 226-243, Doi: 10.1016/j.chaos.2019.06.014.
[2] Alijani, M., Banimahd, B., Madanchi, M., Study and Research on the Six-Year Process of Bitcoin Price and Return, Advances in Mathematical Finance and Applications, 2019, 4(1), P.45-54.
Doi: 10.22034/amfa.2019.577434.1126.
[3] Bhardwaj, G., Swanson, N. R., An Empirical Investigation of the Usefulness of ARFIMA Models for Predicting Macroeconomic and Financial Time Series, Journal of Econometrics, 2004, 5, P.539-578. Doi: v131y2006i1-2p539-578.html
[4] Box, G. E. P., Jenkins, G. M., Time Series Analysis: Forecasting and Control, 1976, Revised Edition. Holden-Day.
[5] Bartels, R., The Rank Version of von Neumann’s Ratio Test for Randomness, Journal of the American Statistical Association, 1982, 77, P. 40-46. Doi: abs/10.1080/01621459.1982.10477764.
[6] Broock, W. A., Scheinkman, J. A., Dechert, W. D., and LeBaron, B., A Test for Independence Based on the Correlation Dimension, Econometric reviews, 1996, 15, P. 197-235. Doi:abs/10.1080/07474939608800353.
[7] Chattfield, C., The Analysis of Time Series. An introduction, 1975, New York, Chapman and Hall.
[8] Contreras, R. J. E., Palma, W., Statistical analysis of autoregressive fractionally integrated moving average models in R, Computer Stat, 2013, 28, P. 2309–2331. Doi:10.1007/s00180-013-0408-7.
[9] Dacorogna, M., Aste, T., Di Matteo, T., Long-term memories of developed and emerging markets: Using the scaling analysis to characterize their stage of development, Journal of Banking and Finance, 2005, 29, P. 827–851. Doi: 10.1016/j.jbankfin.2004.08.004.
[10] Dickey, D., Fuller, W., Distribution of the Estimators for Autoregressive Time Series with a Unit Root, Journal of the American Statistical Association, 1979, 44, P. 427-431. Doi: abs/10.1080/01621459.1979.10482531.
[11] Diebold, F.X., Inoue, A., Long Memory and Regime Switching, Journal of Econometrics, 2001, 105, P.131-159. Doi: org/10.1023/B:CSEM.0000026794.43145.fc.
[12] Eom, C., Oh, J. Jung, W, Relationship between efficiency and predictability in stock price change, Physica, 2008, 387, P. 5511–5517. DOI: 10.1016/j.physa.2008.05.059.
[13] Goodness, C., Aye, M. B., Rangan, G., Nicholas, K., Amandine, N. and Siobhan, R, Predicting BRICS stock returns using ARFIMA models, Applied Financial Economics, 2014, 24, P. 1159-1166. Doi:abs/10.1080/09603107.2014.924297.
[14] Granger, C.W. J., Joyeux, R, an introduction to long–range time series models and fractional differencing Journal of Time Series Analysis, 1980, 1, P. 15–30. Doi: org/10.1111/j.1467-9892.1980.tb00297.
[15] Ghoreishi, S. K., Alijani, M., Dynamic association modeling in 2× 2 contingency tables. Statistical Methodology, 2011, 8(2), P. 242-255. Doi.org/10.1016/j.stamet.2010.10.002.
[16] Granger, C., Hyung, N., Occasional Structural Breaks and Long Memory with an Application to the S&P 500 Absolute Stock Returns, Journal of Empirical Finance, 2004, 11, P.399-421. Doi:10.1.1.464.9432&rep.rep1&type.pdf.
[17] Guangxi, C., Yingying, S., Simulation analysis of multifractal detrended methods based on the ARFIMA process, Chaos, Solutions and Fractals, 2017, 105, P. 235-243. Doi:10.1.1.464.9432&rep.rep1&type.pdf.
[18] Hang, C. N., Palma, W., Estimation of Long-Memory Time Series Models: A Survey of Different Likelihood-Based Methods, Advances in Econometrics, 2005, P. 89-121. Doi 10.1.1.1072.2042&rep.rep1&type.pdf
[19] Hurst, H.E., The Long-Term Storage Capacity of Reservoir. Transactions of the American Society of Civil Engineers, 1951, 11, P. 89-121.
[20] Khodayari, M., Yaghobnezhad, A., Khalili Eraghi, K, A Neural-Network Approach to the Modeling of the Impact of Market Volatility on Investment, Advances in Mathematical Finance and Applications, 2020, 5(4), P. 569-581. Doi: 10.22034/amfa.2020.674953.
[21] Jiti, G., Qiying, W. and Jiying, Y., Long-range dependent time series specification, Bernoulli, 2013, 19, P. 1714-1749. Doi: 10.3150/12-BEJ427.
[22] Lo, A.W, Long Term Memory in Stock Market Prices, Econometrica, 1991, 59, P.1719-1739.
Doi: 10.3150/12-BEJ427.
[23] Lieberman, O., Phillips, P. C. B., Expansions for the distribution of the maximum likelihood estimator of the fractional difference parameter, Econometric Theory, , 2004, 20, P. 464–484. Doi: 10.1017/S0266466604203024.
[24] Linton, O. B., Estimating additive nonparametric models by partial Lq norm: the curse of fractionality, Econometric Theory, 2001, 17, P. 1037–1050. Doi: S0266466601176012/type/journal_article.
[25] Mandelbrot, B., Wallis, J., Noah, Joseph and operational hydrology, Water Resources Research, 1968, 4, P. 909–918. Doi:10.1029/WR004i005p00909.
[26] Mototsugu, S., Oliver, L., Nonparametric Neural Network Estimation of Lyapunov Exponents and a Direct Test for Chaos, Journal of Econometrics, 2003, 120, P. 1-33. Doi: 10.1016/S0304-4076(03)00205-7.
[27] Mandelbrot, B., Van, Ness, J.W., Fractional Brownian Motion, Fractional Noises and Application, SIAM Review, 1968, 10. P. 20-35. Doi: abs/10.1137/1010093.
[28] Matilla, G, M., Marín, M. R., Dore, M. I. Ojeda, R. B., Nonparametric Correlation Integral–Based Tests for Linear and Nonlinear Stochastic Processes, Decisions in Economics and Finance, 2014, 37, P. 181-193. Doi: 10.1007/s10203-013-0143-0
[29] Man, K. S., Long memory time series and short term forecasts, International Journal of Forecasting, 2003, 35, P. 477-491. Doi:10.1016/S0169-2070(02)00060-2.
[30] Mandelbrot, B. B., When can price be arbitraged efficiently? A limit to the validity of the random walk and martingale models, Review of Economics and Statistics, 1971, 23, P. 225-236. Doi.org/10.2307/1937966.
[31] Neama, I., Abdulkadhim, F., Simulation Study for some estimators of Exponential Distribution, International Journal of Mathematics of Trend and Technology, 2014, 23, P. 93-98. Doi: 10.14445/22315373/IJMTT-V10P515.
[32] Nasr, N., Farhadi Sartangi, M., Madahi, Z., A Fuzzy Random Walk Technique to Forecasting Volatility of Iran Stock Exchange Index, Advances in Mathematical Finance and Applications, 2019, 4(1), P. 15-30. Doi: 10.22034/amfa.2019.583911.1172.
[33] Régis, B., Magda, M., ARFIMA Process : Tests and Applications at a White Noise Process, A Random Walk Process and the Stock Exchange Index CAC 40, Journal of Economic Computation and Economic Cybernetics Studies and Research, Academy of Economic Studies, Bucharest, 2012, 46 (1), P. 22-39. Doi: /123456789/9331.
[34] Roel, F. Ceballos, F., On The Estimation of the Hurst Exponent Using Adjusted Rescaled Range Analysis, Detrended Fluctuation Analysis and Variance Time Plot: A Case of Exponential Distribution, Imperial Journal of Interdisciplinary Research, 2017, 3, P. 424-434. Doi: /abs/1805.08931
[35] Ray, B., Long Range Forecasting of IBM Product Revenues Using a Seasonal Fractionally Differenced ARMA Model, International Journal of Forecasting, 1993, 9, P. 22-50. Doi: 10.1016/0169-2070(93)90009-C.
[36] Robinson, P. M., The distance between rival nonstationarity fractional processes, Journal of Econometrics, 128, P. 283–300. Doi: 10.1016/j.jeconom.2004.08.015.
[37] Saad, K.M., New fractional derivative with non-singular kernel for deriving Legendre spectral collocation method, Alexandria Engineering Journal, 2019, P. 22–23. Doi: 10.1016/j.aej.2019.11.017.
[38] Sania, Q., Abdullahi, Y., Modeling chickenpox disease with fractional derivatives: From caputo to atangana-baleanu, Chaos, Solitons and Fractals, 2019, 122, P. 111-118. Doi: 10.1016/j.chaos.2019.03.020.
[39] Sowell, F., Maximum likelihood estimation of stationary univariate fractionally integrated time series models, Journal of Econometrics, 1992, 53, P. 165–188. Doi:10.1016/0304-4076(92)90084-5.
[40] Sheinkman, J.A. Leparon, B., Nonlinear Dynamics and stock returns, Journal of Business, 1989, 62, P. 311-338. Doi: 10.1016/j.jeconom.2004.09.014.
[41] Shimotsu, K., and Phillips, P. C. B., Local Whittle estimation of fractional integration and some of its variants, Journal of Econometrics, 2006, 130, P. 209–233. Doi: 10.1016/j.jeconom.2004.09.014.
[42] Sowell, F., Maximum Likelihood Test of Stationary Univariate Fractionally Integrated Time Series Models, Journal of Econometrics, 1992, 53(1), P. 165-188. Doi: 10.1016/0304-4076(92)90084-5.
[43] Tilmann, G., Hana, Š. and Donald, B., Percival Estimators of Fractal Dimension, Assessing the Roughness of Time Series and Spatial Data, 2012, 27, P. 247-277. Doi: 10.1214/11-STS370.
[44] Taherinia, M., The Impact of Investment Inefficiency and Cash Holding on CEO Turnover, Advances in Mathematical Finance and Applications, 2020, 5(4), P. 469-478. Doi: 10.22034/amfa.2020.674945.
[45] Valderio, R., Bovas, A. and Silvia, L., Estimation OF Parameters in ARFIMA Processes: A Simulation Study, Communications in Statistics - Simulation and Computation, 2001. 30(4), P. 787-803.
[46] Viano, M., Deniau, C., Oppenheim, G., Continuous–time fractional ARMA processes, Statistics and Probability Letters, 1994, 21, P. 323– 336. Doi.org/10.1155/2014/264217.
[47] Wang, W., Khan, M.A., Analysis and numerical simulation of fractional model of bank data with fractal fractional Atangana–Baleanu derivative, Journal of Computational and Applied Mathematics, 2020, 369, P. 11-40. Doi: 10.1016/j.cam.2019.112646.
[48] Wantin, W., Muhammad, A., Fatmawatic, P., Kumamde,T., A comparison study of bank data in fractional calculus, Chaos, Solitons and Fractals, 2019, 126, P. 369-384. Doi: 10.1016/j.chaos.2019.07.025.
[49] Zhu, L., Nonparametric Monte Carlo Tests and Their Applications. Lecture Notes in Statistics, Springer, New York 2005. Doi: 10.1007/0-387-29053-2
