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  • Antonio Lijoi (a1), Igor Prünster (a2) and Stephen G. Walker (a3)

In this paper, we provide a Doob-style consistency theorem for stationary models. Many applications involving Bayesian inference deal with non independent and identically distributed data, in particular, with stationary data. However, for such models, there is still a theoretical gap to be filled regarding the asymptotic properties of Bayesian procedures. The primary goal to be achieved is establishing consistency of the sequence of posterior distributions. Here we provide an answer to the problem. Bayesian methods have recently gained growing popularity in economic modeling, thus implying the timeliness of the present paper. Indeed, we secure Bayesian procedures against possible inconsistencies. No results of such a generality are known up to now.The authors are grateful for the comments and suggestions of two referees. Antonio Lijoi and Igor Prünster were supported by the Italian Ministry of University and Research, grants 2006134525 and 2006133449, respectively. The research of Stephen G. Walker was funded by an EPSRC Advanced Research Fellowship.

Corresponding author
Address correspondence to Stephen G. Walker, Institute of Mathematics, Statistics and Actuarial Science, University of Kent, Kent CT2 7NZ, UK; e-mail:
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Aït-Sahalia, Y. (2002) Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica 70, 223262.
Aldous, D. (1985) Exchangeability and Related Topics. Lecture Notes in Mathematics 1117. Springer-Verlag.
Berti, P., L. Pratelli, & P. Rigo (2006) Almost sure weak convergence of random probability measures. Stochastics 78, 9197.
Beskos, A., O. Papaspiliopoulos, G.O. Roberts, & P. Fernhead (2006) Exact and efficient likelihood-based estimation for diffusion processes. With discussion. Journal of the Royal Statistical Society, Series B 68, 333382.
Bibby, B.M. & M. Sørensen (1995) Martingale estimation functions for discretely observed diffusion processes. Bernoulli 1, 1739.
Billingsley, P. (1961) Statistical Inference for Markov Processes. Statistical Research Monographs. University of Chicago Press.
Brockwell, P.J. & R.A. Davis (1996) Time Series: Theory and Methods. 3rd ed. Springer-Verlag.
Chib, S. (1996) Calculating posterior distributions and modal estimates in Markov mixture models. Journal of Econometrics 75, 7997.
Chib, S. & E. Greenberg (1996) Markov chain Monte Carlo simulation methods in econometrics. Econometric Theory 12, 409431.
Dacunha-Castelle, D. & D. Florens-Zmirou (1986) Estimation of the coefficients of a diffusion from discrete observations. Stochastics 19, 263284.
Doob, J.L. (1949) Application of the theory of martingales. In Le Calcul des Probabilités et ses Applications, pp. 2327. Colloques Internationaux du Centre National de la Recherche Scientifique, no. 13, Paris.
Elerian, O., S. Chib, & N. Shephard (2001) Likelihood inference for discretely observed non-linear diffusions. Econometrica 69, 959993.
Eraker, B. (2001) MCMC analysis of diffusion models with applications to finance. Journal of Business & Economic Statistics 19, 177191.
Fiorentini, G., E. Sentana, & N. Shephard (2004) Likelihood-based estimation of latent generalised ARCH structures. Econometrica 72, 14811517.
Gourieroux, C., A. Monfort, & E. Renault (1993) Indirect inference. Journal of Applied Econometrics 8, 85118.
Huerta, G. & M. West (1999a) Priors and component structures in autoregressive time series models. Journal of the Royal Statistical Society, Series B 61, 881899.
Huerta, G. & M. West (1999b) Bayesian inference on periodicities and component spectral structure in time series. Journal of Time Series Analysis 20, 401416.
Karatzas, I. & S.E. Shreve (1991) Brownian Motion and Stochastic Calculus. Springer-Verlag.
Lijoi, A., I. Prünster, & S.G. Walker (2004) Extending Doob's consistency theorem to nonparametric densities. Bernoulli 10, 651663.
Maitra, A. (1977) Integral representations of invariant measures. Transactions of the American Mathematical Society 229, 209225.
Mena, R.H. & S.G. Walker (2005) Stationary autoregressive models via a Bayesian nonparametric approach. Journal of Time Series Analysis 26, 789805.
Pedersen, A.R. (1995) A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scandinavian Journal of Statistics 22, 5571.
Pitt, M.K. & N. Shephard (1999) Filtering via simulation: Auxiliary particle filters. Journal of the American Statistical Association 94, 590599.
Roberts, G.O. & O. Stramer (2001) On inference for partially observed non-linear diffusion models using the Metropolis–Hastings algorithm. Biometrika 88, 603621.
West, M. & J. Harrison (1997) Bayesian Forecasting and Dynamic Models. Springer-Verlag.
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Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
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