Bernoulli

A Bayesian nonparametric approach to modeling market share dynamics

Igor Prünster and Matteo Ruggiero

Full-text: Open access

Abstract

We propose a flexible stochastic framework for modeling the market share dynamics over time in a multiple markets setting, where firms interact within and between markets. Firms undergo stochastic idiosyncratic shocks, which contract their shares, and compete to consolidate their position by acquiring new ones in both the market where they operate and in new markets. The model parameters can meaningfully account for phenomena such as barriers to entry and exit, fixed and sunk costs, costs of expanding to new sectors with different technologies and competitive advantage among firms. The construction is obtained in a Bayesian framework by means of a collection of nonparametric hierarchical mixtures, which induce the dependence between markets and provide a generalization of the Blackwell–MacQueen Pólya urn scheme, which in turn is used to generate a partially exchangeable dynamical particle system. A Markov Chain Monte Carlo algorithm is provided for simulating trajectories of the system, by means of which we perform a simulation study for transitions to different economic regimes. Moreover, it is shown that the infinite-dimensional properties of the system, when appropriately transformed and rescaled, are those of a collection of interacting Fleming–Viot diffusions.

Article information

Source
Bernoulli, Volume 19, Number 1 (2013), 64-92.

Dates
First available in Project Euclid: 18 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.bj/1358531741

Digital Object Identifier
doi:10.3150/11-BEJ392

Mathematical Reviews number (MathSciNet)
MR3019486

Zentralblatt MATH identifier
1288.62042

Keywords
Bayesian nonparametrics Gibbs sampler interacting Fleming–Viot processes interacting Pòlya urns market dynamics particle system species sampling models

Citation

Prünster, Igor; Ruggiero, Matteo. A Bayesian nonparametric approach to modeling market share dynamics. Bernoulli 19 (2013), no. 1, 64--92. doi:10.3150/11-BEJ392. https://projecteuclid.org/euclid.bj/1358531741


Export citation

References

  • [1] Billingsley, P. (1968). Convergence of Probability Measures. New York: Wiley.
  • [2] Blackwell, D. and MacQueen, J.B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 353–355.
  • [3] Burda, M., Harding, M. and Hausman, J. (2008). A Bayesian mixed logit-probit model for multinomial choice. J. Econometrics 147 232–246.
  • [4] Cifarelli, D.M. and Regazzini, E. (1996). De Finetti’s contribution to probability and statistics. Statist. Sci. 11 253–282.
  • [5] Dai Pra, P., Runggaldier, W.J., Sartori, E. and Tolotti, M. (2009). Large portfolio losses: A dynamic contagion model. Ann. Appl. Probab. 19 347–394.
  • [6] Dawson, D.A. and Greven, A. (1999). Hierarchically interacting Fleming–Viot processes with selection and mutation: Multiple space time scale analysis and quasi-equilibria. Electron. J. Probab. 4 no. 4, 81 pp. (electronic).
  • [7] Dawson, D.A., Greven, A. and Vaillancourt, J. (1995). Equilibria and quasiequilibria for infinite collections of interacting Fleming–Viot processes. Trans. Amer. Math. Soc. 347 2277–2360.
  • [8] De Blasi, P., James, L.F. and Lau, J.W. (2010). Bayesian nonparametric estimation and consistency of mixed multinomial logit choice models. Bernoulli 16 679–704.
  • [9] De Iorio, M., Müller, P., Rosner, G.L. and MacEachern, S.N. (2004). An ANOVA model for dependent random measures. J. Amer. Statist. Assoc. 99 205–215.
  • [10] Duan, J.A., Guindani, M. and Gelfand, A.E. (2007). Generalized spatial Dirichlet process models. Biometrika 94 809–825.
  • [11] Dunson, D.B. and Park, J.H. (2008). Kernel stick-breaking processes. Biometrika 95 307–323.
  • [12] Ericson, R. and Pakes, A. (1985). Markov-perfect industry dynamics: A framework for empirical work. Rev. Econ. Stud. 62 53–82.
  • [13] Ethier, S.N. (1981). A class of infinite-dimensional diffusions occurring in population genetics. Indiana Univ. Math. J. 30 925–935.
  • [14] Ethier, S.N. and Kurtz, T.G. (1986). Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics. New York: Wiley.
  • [15] Ethier, S.N. and Kurtz, T.G. (1993). Fleming–Viot processes in population genetics. SIAM J. Control Optim. 31 345–386.
  • [16] Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209–230.
  • [17] Fleming, W.H. and Viot, M. (1979). Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J. 28 817–843.
  • [18] Gelfand, A.E. and Smith, A.F.M. (1990). Sampling-based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85 398–409.
  • [19] Griffin, J.E. (2011). The Ornstein–Uhlenbeck Dirichlet process and other time-varying processes for Bayesian nonparametric inference. J. Statist. Plann. Inference 141 3648–3664.
  • [20] Griffin, J.E. and Steel, M.F.J. (2004). Semiparametric Bayesian inference for stochastic frontier models. J. Econometrics 123 121–152.
  • [21] Griffin, J.E. and Steel, M.F.J. (2006). Order-based dependent Dirichlet processes. J. Amer. Statist. Assoc. 101 179–194.
  • [22] Griffin, J.E. and Steel, M.F.J. (2011). Stick-breaking autoregressive processes. J. Econometrics 162 383–396.
  • [23] Hjort, N.L., Holmes, C.C., Müller, P. and Walker, S.G., eds. (2010). Bayesian Nonparametrics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge Univ. Press.
  • [24] Hopenhayn, H.A. (1992). Entry, exit, and firm dynamics in long run equilibrium. Econometrica 60 1127–1150.
  • [25] Ishwaran, H. and James, L.F. (2001). Gibbs sampling methods for stick-breaking priors. J. Amer. Statist. Assoc. 96 161–173.
  • [26] Jovanovic, B. (1982). Selection and the evolution of industry. Econometrica 50 649–670.
  • [27] Lau, J.W. and Siu, T.K. (2008). Modelling long-term investment returns via Bayesian infinite mixture time series models. Scand. Actuar. J. 4 243–282.
  • [28] Lau, J.W. and Siu, T.K. (2008). On option pricing under a completely random measure via a generalized Esscher transform. Insurance Math. Econom. 43 99–107.
  • [29] Lo, A.Y. (1984). On a class of Bayesian nonparametric estimates. I. Density estimates. Ann. Statist. 12 351–357.
  • [30] MacEachern, S.N. (1999). Dependent nonparametric Processes. In ASA Proc. of the Section on Bayesian Statistical Science. Alexandria, VA: Amer. Statist. Assoc.
  • [31] MacEachern, S.N. (2000). Dependent Dirichlet processes. Technical Report, Ohio State Univ.
  • [32] Martin, A., Prünster, I., Ruggiero, M. and Taddei, F. (2012). Inefficient credit cycles via generalized Pólya urn schemes. Working paper.
  • [33] Mena, R.H. and Walker, S.G. (2005). Stationary autoregressive models via a Bayesian nonparametric approach. J. Time Ser. Anal. 26 789–805.
  • [34] Park, J.H. and Dunson, D.B. (2010). Bayesian generalized product partition model. Statist. Sinica 20 1203–1226.
  • [35] Petrone, S., Guindani, M. and Gelfand, A.E. (2009). Hybrid Dirichlet mixture models for functional data. J. R. Stat. Soc. Ser. B Stat. Methodol. 71 755–782.
  • [36] Remenik, D. (2009). Limit theorems for individual-based models in economics and finance. Stochastic Process. Appl. 119 2401–2435.
  • [37] Ruggiero, M. and Walker, S.G. (2009). Bayesian nonparametric construction of the Fleming–Viot process with fertility selection. Statist. Sinica 19 707–720.
  • [38] Sutton, J. (2007). Market share dynamics and the “persistence of leadership” debate. Amer. Econ. Rev. 97 222–241.
  • [39] Trippa, L., Müller, P. and Johnson, W. (2011). The multivariate beta process and an extension of the Pólya tree model. Biometrika 98 17–34.
  • [40] Vaillancourt, J. (1990). Interacting Fleming–Viot processes. Stochastic Process. Appl. 36 45–57.
  • [41] Walker, S. and Muliere, P. (2003). A bivariate Dirichlet process. Statist. Probab. Lett. 64 1–7.