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Approximate Bayesian inference for agent-based models in economics: a case study

  • Thomas Lux EMAIL logo
Published/Copyright: June 27, 2022
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Studies in Nonlinear Dynamics & Econometrics
From the journal Studies in Nonlinear Dynamics & Econometrics Volume 27 Issue 4

Abstract

Estimation of agent-based models in economics and finance confronts researchers with a number of challenges. Typically, the complex structures of such models do not allow to derive closed-form likelihood functions so that either numerical approximations to the likelihood or moment-based estimators have to be used for parameter inference. However, all these approaches suffer from extremely high computational demands as they typically work with simulations (of the agent-based model) embedded in (Monte Carlo) simulations conducted for the purpose of parameter identification. One approach that is very generally applicable and that has the potential of alleviating the computational burden is Approximate Bayesian Computation (ABC). While popular in other areas of agent-based modelling, it seems not to have been used so far in economics and finance. This paper provides an introduction to this methodology and demonstrates its potential with the example of a well-studied model of speculative dynamics. As it turns out, ABC appears to make more efficient use of moment-based information than frequentist SMM (Simulated Method of Moments), and it can be used for sample sizes of an order far beyond the reach of numerical likelihood methods.

Keywords: agent-based models; Bayesian estimation; sequential Monte Carlo
JEL Classification: G12; C15; C58
Corresponding author: Thomas Lux, Department of Economics, University of Kiel, Olshausenstr. 40, Kiel 24118, Germany, E-mail:

Acknowledgments

The very detailed and helpful comments of an anonymous reviewer are gratefully acknowledged.

  1. Author contribution: The author has accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The author declares no conflicts of interest regarding this article.

Appendix A: Details of Alfarano, Lux, and Wagner (2008) model

The model assumes that two groups of traders exist in a financial market: chartists and fundamentalists. The number of chartists is N c with each one of them commanding a trading volume T c per integer time unit (which might be a day). Chartists are subject to fluctuations of sentiment which either leads to an optimistic or pessimistic attitude. Sentiment formation is the agent-based part of the model, and its formalization via pairwise interaction is formalized by the transition rates of Eq. (9) in the main text, which are used to determine the switches of each member of this group from optimistic to pessimistic and vice versa as a Poisson process with time-varying intensity depending on the overall numbers of optimists or pessimists among their peers. Equation (9) thus, captures in a very direct way, how sentiment becomes contagious through herd behavior of investors. In the simulations of this model, the number of chartists is set equal to N c = 100 and the behavior of each one of them is simulated with the Gillespie (1977) algorithm for discrete events in continuous time. In contrast to chartists who are considered as individuals, the group of fundamentalists is aggregated in a traditional way summing over the excess demand functions of its members. With N f members of this group and trading volume T f measured per unit of deviation between the assumed fundamental value (p f ) and the current (log) market price (p t ), their excess demand EDf,t is:

E D f , t = N f T f ( p t − p f , t ) .

Excess demand of chartists is simply assumed to be positive if they are optimistic, and negative if pessimistic, so that the second component of market excess demand amounts to:

E D c , t = T c N c x t

with x t the sentiment indicator x t ≡ n + , t − n − , t N c computed very much along the lines of such indicators as they are used in practice. Instantaneous market clearing leads to an equilibrium price

p t = p t , f + T c N c T f N f x t

which shows the potential for mispricing generated by the non-fundamental traders. We, finally, obtain the simple expression of Eq. (10) in the main text for returns over unit time intervals by assuming that the fundamental value follows Brownian motion with a variance σ f 2 and setting T c N c T f N f = 1 . The later assumption is justified by the observation that the two parameters of the agent-based process depicted in eq. (9), a and b, are already capable of generating a wide variety of outcomes for the conditional and unconditional distribution of asset returns. Due to proximity to collinearity of the prefactor and the behavioral parameters, including the composite expression T c N c T f N f as another parameter for the estimation, would be cumbersome. The model above is attractive as a test example for an agent-based framework as its interesting properties are intrinsically linked to the interaction of the chartist agents, and cannot be replicated easily by any aggregation device.

Appendix B: Results of exemplary estimations of Table 2 with moment sets M = 4, 7 and 11

References

Alfarano, S., and T. Lux. 2007. “A Noise Trader Model as a Generator of Apparent Financial Power Laws and Long Memory.” Macroeconomic Dynamics 11 (1): 80–101. https://doi.org/10.1017/s1365100506060299.Search in Google Scholar

Alfarano, S., T. Lux, and F. Wagner. 2008. “Time Variation of Higher Moments in a Financial Market with Heterogeneous Agents: An Analytical Approach.” Journal of Economic Dynamics and Control 32 (1): 101–36. https://doi.org/10.1016/j.jedc.2006.12.014.Search in Google Scholar

Beaumont, M. A., J.-M. Cornuet, J.-M. Marin, and C. P. Robert. 2009. “Adaptive Approximate Bayesian Computation.” Biometrika 96 (4): 983–90. https://doi.org/10.1093/biomet/asp052.Search in Google Scholar

Beaumont, M. A., W. Zhang, and D. J. Balding. 2002. “Approximate Bayesian Computation in Population Genetics.” Genetics 162 (4): 2025–35. https://doi.org/10.1093/genetics/162.4.2025.Search in Google Scholar PubMed PubMed Central

Blum, M. G., and O. François. 2010. “Non-Linear Regression Models for Approximate Bayesian Computation.” Statistics and Computing 20 (1): 63–73. https://doi.org/10.1007/s11222-009-9116-0.Search in Google Scholar

Blum, M. G., M. A. Nunes, D. Prangle, S. A. Sisson. 2013. “A Comparative Review of Dimension Reduction Methods in Approximate Bayesian Computation.” Statistical Science 28 (2): 189–208. https://doi.org/10.1214/12-sts406.Search in Google Scholar

Chen, Z., and T. Lux. 2018. “Estimation of Sentiment Effects in Financial Markets: A Simulated Method of Moments Approach.” Computational Economics 52: 711–44. https://doi.org/10.1007/s10614-016-9638-4.Search in Google Scholar

Csilléry, K., M. Blum, O. Gaggiotti, and O. François. 2010. “Approximate Bayesian Computation (ABC) in Practice.” Trends in Ecology & Evolution 25: 410–8. https://doi.org/10.1016/j.tree.2010.04.001.Search in Google Scholar PubMed

Del Moral, P., A. Doucet, and A. Jasra. 2012. “An Adaptive Sequential Monte Carlo Method for Approximate Bayesian Computation.” Statistics and Computing 22 (5): 1009–20. https://doi.org/10.1007/s11222-011-9271-y.Search in Google Scholar

Drovandi, C. C., and A. N. Pettitt. 2011. “Estimation of Parameters for Macroparasite Population Evolution Using Approximate Bayesian Computation.” Biometrics 67: 225–33. https://doi.org/10.1111/j.1541-0420.2010.01410.x.Search in Google Scholar PubMed

Fearnhead, P. 2019. “Asymptotics of ABC.” In Handbook of Approximate Bayesian Computation, edited by Y. F. Sisson, and S. M. Beaumont. London: Chapman & Hall. chapter 10.10.1201/9781315117195-10Search in Google Scholar

Fearnhead, P., and D. Prangle. 2012. “Constructing Summary Statistics for Approximate Bayesian Computation: Semi-automatic Approximate Bayesian Computation.” Journal of the Royal Statistical Society: Series B 74 (3): 419–74. https://doi.org/10.1111/j.1467-9868.2011.01010.x.Search in Google Scholar

Franke, R., and F. Westerhoff. 2012. “Structural Stochastic Volatility in Asset Pricing Dynamics: Estimation and Model Contest.” Journal of Economic Dynamics and Control 36 (8): 1193–211. https://doi.org/10.1016/j.jedc.2011.10.004.Search in Google Scholar

Frazier, D., W. Maneesoonthorn, G. Martin, and B. McCabe. 2019. “Approximate Bayesian Forecasting.” International Journal of Forecasting 35: 521–39. https://doi.org/10.1016/j.ijforecast.2018.08.003.Search in Google Scholar

Frazier, D. T., G. M. Martin, C. P. Robert, and J. Rousseau. 2018. “Asymptotic Properties of Approximate Bayesian Computation.” Biometrika 105 (3): 593–607. https://doi.org/10.1093/biomet/asy027.Search in Google Scholar

Frazier, D. T., C. P. Robert, and J. Rousseau. 2020. “Model Misspecification in Approximate Bayesian Computation: Consequences and Diagnostics.” Journal of the Royal Statistical Society: Series B 82 (2): 421–44. https://doi.org/10.1111/rssb.12356.Search in Google Scholar

Ghonghadze, J., and T. Lux. 2016. “Bringing an Elementary Agent-Based Model to the Data: Estimation via GMM and an Application to Forecasting of Asset Price Volatility.” Journal of Empirical Finance 37: 1–19. https://doi.org/10.1016/j.jempfin.2016.02.002.Search in Google Scholar

Gillespie, D. T. 1977. “Exact Stochastic Simulation of Coupled Chemical Reactions.” Journal of Physical Chemistry 81 (25): 2340–61. https://doi.org/10.1021/j100540a008.Search in Google Scholar

Gilli, M., and P. Winker. 2001. “Indirect Estimation of the Parameters of Agent Based Models of Financial Markets.” In Working Paper. University of Geneva.10.2139/ssrn.300220Search in Google Scholar

Grazzini, J., M. G. Richiardi, and M. Tsionas. 2017. “Bayesian Estimation of Agent-Based Models.” Journal of Economic Dynamics and Control 77: 26–47. https://doi.org/10.1016/j.jedc.2017.01.014.Search in Google Scholar

Hansen, L. P. 1982. “Large Sample Properties of Generalized Method of Moments Estimators.” Econometrica 50 (4): 1029–54. https://doi.org/10.2307/1912775.Search in Google Scholar

Jang, T.-S., and S. Sacht. 2016. “Animal Spirits and the Business Cycle: Empirical Evidence from Moment Matching.” Metroeconomica 67 (1): 76–113. https://doi.org/10.1111/meca.12091.Search in Google Scholar

Jasra, A., S. Singh, J. S. Martin, and E. McCoy. 2012. “Filtering via Approximate Bayesian Computation.” Statistics and Computing 22 (6): 1223–37. https://doi.org/10.1007/s11222-010-9185-0.Search in Google Scholar

Lenormand, M., F. Jabot, and G. Deffuant. 2013. “Adaptive Approximate Bayesian Computation for Complex Models.” Computational Statistics 28 (6): 2777–96. https://doi.org/10.1007/s00180-013-0428-3.Search in Google Scholar

Li, W., and P. Fearnhead. 2018. “On the Asymptotic Efficiency of Approximate Bayesian Computation Estimators.” Biometrika 105 (2): 285–99. https://doi.org/10.1093/biomet/asx078.Search in Google Scholar

Lux, T. 2018. “Estimation of Agent-Based Models Using Sequential Monte Carlo Methods.” Journal of Economic Dynamics and Control 91: 391–408. https://doi.org/10.1016/j.jedc.2018.01.021.Search in Google Scholar

Lux, T. 2022. ““Bayesian Estimation of Agent-Based Models via Adaptive Particle Markov Chain Monte Carlo”, Computational Economics, in press.” In Working Paper. University of Kiel.10.1007/s10614-021-10155-0Search in Google Scholar

Lux, T., and R. C. Zwinkels. 2018. “Empirical Validation of Agent-Based Models.” In Handbook of Computational Economics, 4, edited by C. Hommes, and B. Lebaron, 437–88. Amsterdam: Elsevier.10.1016/bs.hescom.2018.02.003Search in Google Scholar

Prangle, D. 2017. “Adapting the ABC Distance Function.” Bayesian Analysis 12: 289–309. https://doi.org/10.1214/16-ba1002.Search in Google Scholar

Prangle, D., M. G. Blum, G. Popovic, and S. Sisson. 2014. “Diagnostic Tools for Approximate Bayesian Computation Using the Coverage Property.” Australian & New Zealand Journal of Statistics 56 (4): 309–29. https://doi.org/10.1111/anzs.12087.Search in Google Scholar

Rubin, D. B. 1984. “Bayesianly Justifiable and Relevant Frequency Calculations for the Applied Statistician.” Annals of Statistics 12 (1): 1151–72 https://doi.org/10.1214/aos/1176346785 .Search in Google Scholar

Sisson, S. A., Y. Fan, and M. M. Tanaka. 2007. “Sequential Monte Carlo without Likelihoods.” Proceedings of the National Academy of Sciences of the United States of America 104 (6): 1760–5. https://doi.org/10.1073/pnas.0607208104.Search in Google Scholar PubMed PubMed Central

Tavaré, S., D. J. Balding, R. C. Griffiths, and P. Donnelly. 1997. “Inferring Coalescence Times from DNA Sequence Data.” Genetics 145 (2): 505–18. https://doi.org/10.1093/genetics/145.2.505.Search in Google Scholar PubMed PubMed Central

Toni, T., D. Welch, N. Strelkowa, A. Ipsen, and M. P. Stumpf. 2009. “Approximate Bayesian Computation Scheme for Parameter Inference and Model Selection in Dynamical Systems.” Journal of The Royal Society Interface 6 (31): 187–202. https://doi.org/10.1098/rsif.2008.0172.Search in Google Scholar PubMed PubMed Central

Tubbenhauer, T., C. Fieberg, and T. Poddig. 2021. “Multi-Agent-Based VaR Forecasting.” Journal of Economic Dynamics and Control 131. in press. https://doi.org/10.1016/j.jedc.2021.104231.Search in Google Scholar
Received: 2021-05-26
Revised: 2022-06-08
Accepted: 2022-06-08
Published Online: 2022-06-27

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Approximate Bayesian inference for agent-based models in economics: a case study
  4. Anticipating extreme losses using score-driven shape filters
  5. Does real interest rate parity really work? Historical evidence from a discrete wavelet perspective
  6. The impact of forward guidance and large-scale asset purchase programs on commodity markets
  7. Middle-income traps and complexity in economic development
  8. Bayesian inference for order determination of double threshold variables autoregressive models
  9. Score-driven multi-regime Markov-switching EGARCH: empirical evidence using the Meixner distribution
https://doi.org/10.1515/snde-2021-0052
Keywords for this article
agent-based models; Bayesian estimation; sequential Monte Carlo

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Approximate Bayesian inference for agent-based models in economics: a case study
  4. Anticipating extreme losses using score-driven shape filters
  5. Does real interest rate parity really work? Historical evidence from a discrete wavelet perspective
  6. The impact of forward guidance and large-scale asset purchase programs on commodity markets
  7. Middle-income traps and complexity in economic development
  8. Bayesian inference for order determination of double threshold variables autoregressive models
  9. Score-driven multi-regime Markov-switching EGARCH: empirical evidence using the Meixner distribution
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