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Time-series Modelling, Stationarity and Bayesian Nonparametric Methods

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  • Juan Carlos Martínez-Ovando
  • Stephen G. Walker

Abstract

In this paper we introduce two general non-parametric first-order stationary time-series models for which marginal (invariant) and transition distributions are expressed as infinite-dimensional mixtures. That feature makes them the first Bayesian stationary fully non-parametric models developed so far. We draw on the discussion of using stationary models in practice, as a motivation, and advocate the view that flexible (non-parametric) stationary models might be a source for reliable inferences and predictions. It will be noticed that our models adequately fit in the Bayesian inference framework due to a suitable representation theorem. A stationary scale-mixture model is developed as a particular case along with a computational strategy for posterior inference and predictions. The usefulness of that model is illustrated with the analysis of Euro/USD exchange rate log-returns.

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File URL: http://www.banxico.org.mx/publicaciones-y-discursos/publicaciones/documentos-de-investigacion/banxico/%7BAAD21931-987D-E2DF-E498-4B23679CD8D4%7D.pdf
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Bibliographic Info

Paper provided by Banco de México in its series Working Papers with number 2011-08.

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Date of creation: Sep 2011
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Handle: RePEc:bdm:wpaper:2011-08

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Web page: http://www.banxico.org.mx
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Related research

Keywords: Stationarity; Markov processes; Dynamic mixture models; Random probability measures; Conditional random probability measures; Latent processes.;

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  1. Ramsés Mena & Stephen Walker, 2007. "On the Stationary Version of the Generalized Hyperbolic ARCH Model," Annals of the Institute of Statistical Mathematics, Springer, vol. 59(2), pages 325-348, June.
  2. Nadjib Bouzar & K. Jayakumar, 2008. "Time series with discrete semistable marginals," Statistical Papers, Springer, vol. 49(4), pages 619-635, October.
  3. Pitt, Michael K. & Walker, Stephen G., 2005. "Constructing Stationary Time Series Models Using Auxiliary Variables With Applications," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 554-564, June.
  4. Markus Haas, 2004. "A New Approach to Markov-Switching GARCH Models," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 2(4), pages 493-530.
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  6. Ramsés H. Mena & Stephen G. Walker, 2005. "Stationary Autoregressive Models via a Bayesian Nonparametric Approach," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(6), pages 789-805, November.
  7. Hamilton, James D, 1989. "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle," Econometrica, Econometric Society, vol. 57(2), pages 357-84, March.
  8. Michael K. Pitt, 2002. "Constructing First Order Stationary Autoregressive Models via Latent Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 29(4), pages 657-663.
  9. Chu, Chia-Shang James & Stinchcombe, Maxwell & White, Halbert, 1996. "Monitoring Structural Change," Econometrica, Econometric Society, vol. 64(5), pages 1045-65, September.
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  14. Barry, Christopher B. & Winkler, Robert L., 1976. "Nonstationarity and Portfolio Choice," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 11(02), pages 217-235, June.
  15. Rong Zhu & Harry Joe, 2006. "Modelling Count Data Time Series with Markov Processes Based on Binomial Thinning," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(5), pages 725-738, 09.
  16. Rodríguez, Abel & Dunson, David B & Gelfand, Alan E, 2008. "The Nested Dirichlet Process," Journal of the American Statistical Association, American Statistical Association, vol. 103(483), pages 1131-1154.
  17. Pitt, Michael K. & Walker, Stephen G., 2006. "Extended constructions of stationary autoregressive processes," Statistics & Probability Letters, Elsevier, vol. 76(12), pages 1219-1224, July.
  18. Clements Michael P. & Hendry David F., 2008. "Economic Forecasting in a Changing World," Capitalism and Society, De Gruyter, vol. 3(2), pages 1-20, October.
  19. Lacour, Claire, 2008. "Nonparametric estimation of the stationary density and the transition density of a Markov chain," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 232-260, February.
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