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A review of aggregation techniques for agent-based models: understanding the presence of long-term memory

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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

  1. For the sake of simplicity, we will denote hereafter the entire time series or process as \(x_t\) instead of \(\{x_t\}\).

  2. 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.

  3. see the Appendix for the definition of Beta distribution, generalized beta distribution and the related normalization coefficient \(\beta (p,q)\).

  4. This hypothesis is not strictly necessary, but it simplifies the calculus.

  5. \(\delta _{i,j}\) is the usual Kronecker symbol, e.g. \(\delta _{i,j}=1\) for \(i=j;\, \delta _{i,j}=0\) for \(i \not =j\).

  6. 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Ausloos, M., Ivanova, K.: Introducing False EUR and False EUR exchange rates. Phys. A 286, 353–366 (2000)

    Article  Google Scholar 

  • 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)

    Google Scholar 

  • Avery, C., Zemsky, P.: Multidimensional uncertainty and herd behavior in financial markets. Am. Econ. Rev. 88(4), 724–748 (1988)

    Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Bollerslev, T., Mikkelsen, H.O.: Modelling and pricing long memory in stock market volatility. J. Econom. 73, 151–184 (1996)

    Article  Google Scholar 

  • Box-Steffenmaier, J.M., Smith, R.M.: The dynamics of aggregate partisanship. Am. Polit. Sci. Rev. 90, 567–580 (1996)

    Google Scholar 

  • Brianzoni, S., Cerqueti, R., Michetti, E.: A dynamics stochastic model of asset pricing with heterogeneous beliefs. Comput. Econ. 35(2), 165–188 (2010)

    Article  Google Scholar 

  • Brock, W.A., Hommes, C.H.: Rational route to randomness. Econometrica 65, 1059–1095 (1997)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Cheung, Y.W., Lai, K.S.: A fractional cointegration analysis of purchasing power parity. J. Bus. Econ. Stat. 11, 103–112 (1993)

    Google Scholar 

  • 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)

    Article  Google Scholar 

  • Chiarella, C., Gallegati, M., Leombruni, R., Palestrini, A.: Asset price dynamics among heterogeneous interacting agents. Comput. Econ. 22(2), 213–223 (2002)

    Google Scholar 

  • Chiarella, C., He, X.: Heterogeneous beliefs, risk and learning in a simple asset pricing model. Comput. Econ. 19, 95–132 (2002)

    Article  Google Scholar 

  • Diebolt, C., Guiraud, V.: A note on long memory time series. Qual. Quant. 39, 827–836 (2005)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Ding, Z., Granger, C.W.J.: Modelling volatility persistence of speculative returns: a new approach. J. Econom. 73, 185–215 (1996a)

    Google Scholar 

  • Ding, Z., Granger, C.W.J.: Varieties of long memory models. J. Econom. 73, 61–77 (1996b)

    Google Scholar 

  • Dittman, I., Granger, C.W.J.: Properties of nonlinear transformations of fractionally integrated processes. J. Econom. 110, 113–133 (2002)

    Article  Google Scholar 

  • Foellmer, H., Horst, U., Kirman, A.: Equilibria in financial markets with heterogeneous agents: a probabilistic perspective. J. Math. Econ. 41, 123–155 (2005)

    Article  Google Scholar 

  • Fung, H.K., Lai, S., Lai, M.: Fractal structure in currency futures price dynamics. J. Futures Mark. 14, 169–181 (1994)

    Article  Google Scholar 

  • Granger, C.W.J.: Long memory relationships and the aggregation of dynamic models. J. Econom. 14, 227–228 (1980)

    Article  Google Scholar 

  • Granger, C.W.J., Joyeux, R.: An introduction to long-memory time series and fractional differencing. J. Time Ser. Anal. 1, 15–39 (1980)

    Article  Google Scholar 

  • Hommes, C.H.: Financial markets as nonlinear adaptive evolutionary systems. Quant. Fin. 1, 149–167 (2001)

    Article  Google Scholar 

  • 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)

    Google Scholar 

  • Hosking, J.R.M.: Fractional differencing. Biometrica 68(1), 165–176 (1981)

    Article  Google Scholar 

  • Hurst, H.: Long term storage capacity of reservoirs. Trans. Am. Soc. Civ. Eng. 116, 770–799 (1951)

    Google Scholar 

  • Hurst, H.E.: A suggested statistical model of some time seris which occur in nature. Nature 180, 494 (1957)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Kirman, A.P.: Heterogeneity in economics. J. Econ. Interact. Coord. 1, 89–117 (2006)

    Article  Google Scholar 

  • Lo, A.W.: Long memory in stock market prices. Econometrica 59, 1279–1313 (1991)

    Article  Google Scholar 

  • 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)

    Google Scholar 

  • 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)

    Google Scholar 

  • Mandelbrot, B.B., Wallis, J.: Noah, Joseph and operational hydrology. Water Resour. Res. 4, 909–918 (1968)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Osborne, A.R., Provenzale, A.: Finite correlation dimension for stochastic systems with power-law spectra. Phys. D 35, 357–381 (1989)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Reka, A., Barabasi, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Rotundo, G., Ausloos, M.: Microeconomic coevolution model for financial technical analysis signals. Phys. A 373, 569–585 (2007)

    Article  Google Scholar 

  • 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)

    Google Scholar 

  • 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)

    Article  Google Scholar 

  • Tscherning, R.: Long memory in foreign exchange rates revisited. J. Int. Fin. Mark. Inst. Money 5, 53–78 (1995)

    Google Scholar 

  • Vandewalle, N., Ausloos, M.: Spareness and roughness of foreign exchange rates. Int. J. Mod. Phys. C 9, 711–720 (1998)

    Article  Google Scholar 

  • Vandervalle, N., D’Hulst, R., Ausloos, M.: Phase segregation in binary sandpiles on fractal bases. Phys. Rev. E 59, 631–635 (1999)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Google Scholar 

  • Zaffaroni, P.: Contemporaneous aggregation of linear dynamic models in large economies. J. Econom. 120, 75–102 (2004)

    Article  Google Scholar 

  • Zaffaroni, P.: Memory and aggregation for models of changing volatility. J. Econom. 136, 237–249 (2007a)

    Google Scholar 

  • Zaffaroni, P.: Contemporaneous aggregation of GARCH processes. J. Time Ser. Anal. 28, 521–544 (2007b)

    Google Scholar 

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Acknowledgments

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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Authors and Affiliations

  1. Department of Economics and Law, University of Macerata, via Crescimbeni, 20, 62100 , Macerata, Italy

    Roy Cerqueti

  2. Department of Methods and Models for Economics, Territory and Finance, Sapienza University of Rome, via del Castro Laurenziano 9, 00161 , Rome, Italy

    Giulia Rotundo

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Correspondence to Roy Cerqueti.

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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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