Abstract
A key feature of agent-based modeling is the understanding of the macroscopic behavior based on data at the microscopic level. In this respect, financial market models are requested to replicate, at the aggregate level, the stylized facts of empirical data. Among them, a remarkable role is played by the long term behavior. Indeed, the study of the long-term memory is relevant, in that it describes if and how past events continue to maintain their influence for the future evolution of a system. In economic applications, this is relevant for understanding the reaction of the system to micro- and macro-economic shocks. Moreover, further information on the long-term memory properties of a system can be obtained by analyzing agents heterogeneity and the outcome of their aggregation. The aim of this paper is to review a few techniques—though the most relevant in our opinion—for studying the long-term memory as emergent property of systems composed by heterogeneous agents. Theorems relevant to the present analysis are summarized and their applications in four structural models with long-term memory are shown. This property is assessed through the analysis of the functional relation between model parameters.
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Notes
For the sake of simplicity, we will denote hereafter the entire time series or process as \(x_t\) instead of \(\{x_t\}\).
The discrete case can be seen as a particular case of the continuous one, where \(\alpha _i\) and \(\beta _i\) are a particular sample from the same distribution.
see the Appendix for the definition of Beta distribution, generalized beta distribution and the related normalization coefficient \(\beta (p,q)\).
This hypothesis is not strictly necessary, but it simplifies the calculus.
\(\delta _{i,j}\) is the usual Kronecker symbol, e.g. \(\delta _{i,j}=1\) for \(i=j;\, \delta _{i,j}=0\) for \(i \not =j\).
For the sake of simplicity, we will denote hereafter the entire time series (or process) as \(x_{t}\,\, (\mathrm{or}\,\, X_{t})\) instead of \(\{x_t\}\,\, (\mathrm{or}\,\, \{X_{t}\})\).
References
Alfarano, S., Lux, T., Wagner, F.: Time variation of higher moments in a financial market with heterogeneous agents: an analytical approach. J. Econ. Dyn. Control 32, 101–136 (2008)
Ausloos, M., Ivanova, K.: Low \(q\)-moment multifractal analysis of gold price, Dow Jones Industrial Average and BGL–USD exchange rate. Eur. Phys. J. 8, 665–669 (1999)
Ausloos, M., Ivanova, K.: Introducing False EUR and False EUR exchange rates. Phys. A 286, 353–366 (2000)
Ausloos, M., Ivanova, K., Vandervalle, N.: Crashes: symptoms, diagnosies and remedies. The advent of econophysics. In: Takayasu, H. (ed.) Empirical Sciences of Financial Fluctuations. Springer, Berlin (2002)
Avery, C., Zemsky, P.: Multidimensional uncertainty and herd behavior in financial markets. Am. Econ. Rev. 88(4), 724–748 (1988)
Battacharya, R.N., Waymire, E.C.: Stochastic processes with applications. SIAM 61 (2009)
Bianchi, S.: Autocorrelazione delle serie finanziarie e non robustezza del range standardizzato. In: Atti della Giornata di Studio “Aspetti scientifici e didattici della teoria del rischio”, vol. 1, pp. 31–44. Universit degli Studi del Molise, Campobasso, 18/06/1997, CAMPOBASSO: Uniservice (1997)
Bianchi, S.: A new distribution-based test of self-similarity. Fractals 3, 331–346 (2004)
Bianchi, S., Pantanella, A., Pianese, A.: Modeling stock prices by multifractional Brownian motion: an improved estimation of the pointwise regularity. Quant. Fin. 13(8), 1317–1330 (2013)
Bianchi, S., Pianese, A.: Multifractional properties of stock indices decomposed by filtering their pointwise Hoelder regularity. Int. J. Theor. Appl. Fin. 11(6), 567–595 (2008)
Bischi, G.I., Gallegati, M., Gardini, L., Leombruni, R., Palestrini, A.: Herd behavior and nonfundamental asset price fluctuations in financial markets. Macroecon. Dyn. 10, 502–528 (2006)
Bollerslev, T., Mikkelsen, H.O.: Modelling and pricing long memory in stock market volatility. J. Econom. 73, 151–184 (1996)
Box-Steffenmaier, J.M., Smith, R.M.: The dynamics of aggregate partisanship. Am. Polit. Sci. Rev. 90, 567–580 (1996)
Brianzoni, S., Cerqueti, R., Michetti, E.: A dynamics stochastic model of asset pricing with heterogeneous beliefs. Comput. Econ. 35(2), 165–188 (2010)
Brock, W.A., Hommes, C.H.: Rational route to randomness. Econometrica 65, 1059–1095 (1997)
Brock, W.A., Hommes, C.H.: Heterogeneous beliefs and routes to chaos in a simple asset pricing model. J. Econ. Dyn. Control 22, 1235–1274 (1998)
Byers, D., Davidson, J., Peel, D.: Modelling political popularity: an analysis of long range dependence in opinion poll series. J. R. Stat. Soc. A 160, 471–490 (1997)
Cerqueti, R., Rotundo, G.: Microeconomic modeling of financial time series with long term memory. Communicated to the conference C.I.F.E.r (2003 IEEE International Conference on Computational Intelligence for Financial Engineering), sponsored by IEEE Neural Network Society and organised by the technical committee of Financial Engineering, held in Hong Kong, March 20th–23th 2003, Proceedings, pp. 191–198, IEEE catalog number: 03TH8653, ISBN 0-7803-7654-4
Cerqueti, R., Rotundo, G.: Dynamics of financial time series in an inhomogeneous aggregation framework. In: Perna, C., Sibillo, M. (eds.) Mathematical and Statistical Methods in Insurance and Finance, pp. 67–74. Springer, New York (2007). ISBN:978-88-470-0703-1
Cerqueti, R., Rotundo, G.: Memory property in heterogeneously populated markets. In: Greco, S., Marques Pereira R.A., Squillante, M., Yager, R.R., Kacprzyk, J. (eds.) Preferences and Decisions, vol. 257, pp. 53–67. Springer Series Studies in Fuzziness and Soft Computing (2010). ISBN/ISSN:978-3-642-15975-6
Cerqueti, R., Rotundo, G.: The role of diversity in persistence aggregation. Int. J. Intell. Syst. 27, 176–187 (2012)
Cheung, Y.W., Lai, K.S.: A fractional cointegration analysis of purchasing power parity. J. Bus. Econ. Stat. 11, 103–112 (1993)
Chiarella, C., Dieci, R., Gardini, L.: Asset price and wealth dynamics in a financial market with heterogeneous agents. J. Econ. Dyn. Control 30, 1755–1786 (2006)
Chiarella, C., Gallegati, M., Leombruni, R., Palestrini, A.: Asset price dynamics among heterogeneous interacting agents. Comput. Econ. 22(2), 213–223 (2002)
Chiarella, C., He, X.: Heterogeneous beliefs, risk and learning in a simple asset pricing model. Comput. Econ. 19, 95–132 (2002)
Diebolt, C., Guiraud, V.: A note on long memory time series. Qual. Quant. 39, 827–836 (2005)
Ding, Z., Engle, R.F., Granger, C.W.J.: A long memory property of stock market returns and a new model. J. Empir. Fin. 1, 83–106 (1993)
Ding, Z., Granger, C.W.J.: Modelling volatility persistence of speculative returns: a new approach. J. Econom. 73, 185–215 (1996a)
Ding, Z., Granger, C.W.J.: Varieties of long memory models. J. Econom. 73, 61–77 (1996b)
Dittman, I., Granger, C.W.J.: Properties of nonlinear transformations of fractionally integrated processes. J. Econom. 110, 113–133 (2002)
Foellmer, H., Horst, U., Kirman, A.: Equilibria in financial markets with heterogeneous agents: a probabilistic perspective. J. Math. Econ. 41, 123–155 (2005)
Fung, H.K., Lai, S., Lai, M.: Fractal structure in currency futures price dynamics. J. Futures Mark. 14, 169–181 (1994)
Granger, C.W.J.: Long memory relationships and the aggregation of dynamic models. J. Econom. 14, 227–228 (1980)
Granger, C.W.J., Joyeux, R.: An introduction to long-memory time series and fractional differencing. J. Time Ser. Anal. 1, 15–39 (1980)
Hommes, C.H.: Financial markets as nonlinear adaptive evolutionary systems. Quant. Fin. 1, 149–167 (2001)
Hommes, C.H.: Heterogeneous agent models in economics and finance. In: Tesfatsion, L., Judd, K.L. (eds.) Handbook of Computational Economics, vol. 2, pp. 1109–1186. North Holland, Elsevier, Amsterdam (2006)
Hosking, J.R.M.: Fractional differencing. Biometrica 68(1), 165–176 (1981)
Hurst, H.: Long term storage capacity of reservoirs. Trans. Am. Soc. Civ. Eng. 116, 770–799 (1951)
Hurst, H.E.: A suggested statistical model of some time seris which occur in nature. Nature 180, 494 (1957)
Ivanova, K., Ausloos, M.: Low-order variability diagrams for short-range correlation evidence in financial data: BGL–USD exchange rate, Dow Jones industrial average, gold ounce price. Phys. A 265, 279–291 (1999)
Jonas, A.: Long memory self similar time series models. Harward University Manuscript (1981)
Kirman, A.P., Teyssiére, G.: Microeconomic models for long-memory in the volatility of financial time series. Stud. Nonlinear Dyn. Econom. 5, 281–302 (2002)
Kirman, A.P.: Heterogeneity in economics. J. Econ. Interact. Coord. 1, 89–117 (2006)
Lo, A.W.: Long memory in stock market prices. Econometrica 59, 1279–1313 (1991)
Lux, T., Ausloos, M.: Market fluctuations I: scaling, multi-scaling and their possible origins. In: Bunde, A., Kropp, J., Schellnhuber, H.-J. (eds.) The Science of Disaster: Scaling Laws Governing Weather, Body, Stock-Market Dynamics, pp. 377–413. Springer, Berlin (2002)
Mandelbrot, B.B.: Possible refinements of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence. In: Rosenblatt, M., Van Atta, C. (eds.) Statistical Models and Turbulence. Springer, New York (1972)
Mandelbrot, B.B., Wallis, J.: Noah, Joseph and operational hydrology. Water Resour. Res. 4, 909–918 (1968)
McLeod, A.I., Hipel, A.W.: Preservation of the rescaled adjusted range, 1: a reassessment of the Hurst phenomenon. Water Resour. Res. 14, 491–508 (1978)
Menna, M., Rotundo, G., Tirozzi, B.: Distinguishing between chaotic and stochastic systems in financial time series. Int. J. Mod. Phys. C 13(1), 31–39 (2002)
Osborne, A.R., Provenzale, A.: Finite correlation dimension for stochastic systems with power-law spectra. Phys. D 35, 357–381 (1989)
Peng, C.K., Buldyrev, S.V., Havlin, S., Simons, M., Stanley, H.E., Goldberger, A.L.: Mosaic organization of DNA nucleotides. Phys. Rev. E 49, 1685–1689 (1994)
Rangarajan, G., Ding, M.: Integrated approach to the assessment of long range correlation in time series data. Phys. Rev. E 61(5), 4991–5001 (2000)
Reka, A., Barabasi, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)
Reboredo, J.C., Rivera-Castro, M.A., Miranda, J.G.V., Gara-Rubio, R.: How fast do stock prices adjust to market efficiency? Evidence from a detrended fluctuation analysis. Phys. A 392, 1631–1637 (2013)
Rotundo, G., Ausloos, M.: Microeconomic coevolution model for financial technical analysis signals. Phys. A 373, 569–585 (2007)
Rotundo G, Blasi M, Petroni F (2007) Testing long memory in synthetic time series. Working paper Dipartimento di Studi Aziendali, vol. 1, pp. 1–14. Tecnologici e Quantitativi, Universitá della Tuscia
Simon, H.: Models of Man. Wiley, New York (1957)
Stanley, H.E., Kantelhardt, J.W., Zschiegner, S.A., Koscielny-Bunde, E., Havlin, S., Bunde, A.: Multifractal detrended fluctuation analysis of nonstationary time series. Phys. A 316, 87 (2002)
Tscherning, R.: Long memory in foreign exchange rates revisited. J. Int. Fin. Mark. Inst. Money 5, 53–78 (1995)
Vandewalle, N., Ausloos, M.: Spareness and roughness of foreign exchange rates. Int. J. Mod. Phys. C 9, 711–720 (1998)
Vandervalle, N., D’Hulst, R., Ausloos, M.: Phase segregation in binary sandpiles on fractal bases. Phys. Rev. E 59, 631–635 (1999)
Wei, A., Leuthold, R.M.: Agricultural Futures Prices and Long Memory Processes. OFOR Working Paper No. 00.04. Available at SSRN: http://ssrn.com/abstract=229795 or 2000. doi:10.2139/ssrn.229795
Weron, R.: Estimating long-range dependence: finite sample properties and confidence intervals. Phys. A 312, 285–299 (2002)
Willinger, W., Paxson, V., Taqqu, M.S.: Self similarity and heavy tails: structural modeling of network traffic. In: A Practical Guide To Heavy Tails: Statistical Techniques and Applications. Birkhäuser, Boston (1998)
Yager, R.R.: Including a diversity criterion in decision making. Int. J. Intell. Syst. 25, 958–969 (2010)
Zaffaroni, P.: Contemporaneous aggregation of linear dynamic models in large economies. J. Econom. 120, 75–102 (2004)
Zaffaroni, P.: Memory and aggregation for models of changing volatility. J. Econom. 136, 237–249 (2007a)
Zaffaroni, P.: Contemporaneous aggregation of GARCH processes. J. Time Ser. Anal. 28, 521–544 (2007b)
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The authors would like to thank Thomas Lux, Cars Hommes, Jorgen-Vitting Andersen, Doyne Farmer and Alan Kirman for helpful suggestions.
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Cerqueti, R., Rotundo, G. A review of aggregation techniques for agent-based models: understanding the presence of long-term memory. Qual Quant 49, 1693–1717 (2015). https://doi.org/10.1007/s11135-014-9995-9
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DOI: https://doi.org/10.1007/s11135-014-9995-9