A Bayesian nonparametric approach to modeling market share dynamics. (English) Zbl 1288.62042

Summary: 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.


62F15 Bayesian inference
60G57 Random measures
60K35 Interacting random processes; statistical mechanics type models; percolation theory
62G05 Nonparametric estimation
62P20 Applications of statistics to economics
91G70 Statistical methods; risk measures
Full Text: DOI arXiv Euclid


[1] Billingsley, P. (1968). Convergence of Probability Measures . New York: Wiley. · Zbl 0172.21201
[2] Blackwell, D. and MacQueen, J.B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 353-355. · Zbl 0276.62010
[3] Burda, M., Harding, M. and Hausman, J. (2008). A Bayesian mixed logit-probit model for multinomial choice. J. Econometrics 147 232-246. · Zbl 1429.62363
[4] Cifarelli, D.M. and Regazzini, E. (1996). De Finetti’s contribution to probability and statistics. Statist. Sci. 11 253-282. · Zbl 0955.01552
[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. · Zbl 1159.60353
[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). · Zbl 0920.92016
[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. · Zbl 0831.60102
[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. · Zbl 1220.62036
[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. · Zbl 1089.62513
[10] Duan, J.A., Guindani, M. and Gelfand, A.E. (2007). Generalized spatial Dirichlet process models. Biometrika 94 809-825. · Zbl 1156.62064
[11] Dunson, D.B. and Park, J.H. (2008). Kernel stick-breaking processes. Biometrika 95 307-323. · Zbl 1437.62448
[12] Ericson, R. and Pakes, A. (1985). Markov-perfect industry dynamics: A framework for empirical work. Rev. Econ. Stud. 62 53-82. · Zbl 0828.90026
[13] Ethier, S.N. (1981). A class of infinite-dimensional diffusions occurring in population genetics. Indiana Univ. Math. J. 30 925-935. · Zbl 0478.60084
[14] Ethier, S.N. and Kurtz, T.G. (1986). Markov Processes : Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics . New York: Wiley. · Zbl 0592.60049
[15] Ethier, S.N. and Kurtz, T.G. (1993). Fleming-Viot processes in population genetics. SIAM J. Control Optim. 31 345-386. · Zbl 0774.60045
[16] Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209-230. · Zbl 0255.62037
[17] Fleming, W.H. and Viot, M. (1979). Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J. 28 817-843. · Zbl 0444.60064
[18] Gelfand, A.E. and Smith, A.F.M. (1990). Sampling-based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85 398-409. · Zbl 0702.62020
[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. · Zbl 1219.62084
[20] Griffin, J.E. and Steel, M.F.J. (2004). Semiparametric Bayesian inference for stochastic frontier models. J. Econometrics 123 121-152. · Zbl 1328.62208
[21] Griffin, J.E. and Steel, M.F.J. (2006). Order-based dependent Dirichlet processes. J. Amer. Statist. Assoc. 101 179-194. · Zbl 1118.62360
[22] Griffin, J.E. and Steel, M.F.J. (2011). Stick-breaking autoregressive processes. J. Econometrics 162 383-396. · Zbl 1441.62709
[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. · Zbl 1192.62080
[24] Hopenhayn, H.A. (1992). Entry, exit, and firm dynamics in long run equilibrium. Econometrica 60 1127-1150. · Zbl 0766.90021
[25] Ishwaran, H. and James, L.F. (2001). Gibbs sampling methods for stick-breaking priors. J. Amer. Statist. Assoc. 96 161-173. · Zbl 1014.62006
[26] Jovanovic, B. (1982). Selection and the evolution of industry. Econometrica 50 649-670. · Zbl 0478.90043
[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. · Zbl 1224.91068
[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. · Zbl 1140.91400
[29] Lo, A.Y. (1984). On a class of Bayesian nonparametric estimates. I. Density estimates. Ann. Statist. 12 351-357. · Zbl 0557.62036
[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. · Zbl 1281.62070
[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. · Zbl 1097.62084
[34] Park, J.H. and Dunson, D.B. (2010). Bayesian generalized product partition model. Statist. Sinica 20 1203-1226. · Zbl 05769963
[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. · Zbl 1248.62079
[36] Remenik, D. (2009). Limit theorems for individual-based models in economics and finance. Stochastic Process. Appl. 119 2401-2435. · Zbl 1168.60369
[37] Ruggiero, M. and Walker, S.G. (2009). Bayesian nonparametric construction of the Fleming-Viot process with fertility selection. Statist. Sinica 19 707-720. · Zbl 05586084
[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. · Zbl 1214.62101
[40] Vaillancourt, J. (1990). Interacting Fleming-Viot processes. Stochastic Process. Appl. 36 45-57. · Zbl 0729.92017
[41] Walker, S. and Muliere, P. (2003). A bivariate Dirichlet process. Statist. Probab. Lett. 64 1-7. · Zbl 1113.62313
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