Summary
The paper deals with the identification of a stationary autoregressive model for a time series and the contemporary detection of a change in its mean. We adopt the Bayesian approach with weak prior information on the parameters of the models under comparison and an exact form of the likelihood function. When necessary, we resort to fractional Bayes factors to choose between models, and to importance sampling to solve computational issues.
Similar content being viewed by others
References
Albert, J. H. andChib, S (1993), Bayes inference via Gibbs sampling of autoregressive time series subject to Markov mean and variance shift.J. Business and Econ. Statist., 11, 1–15.
Atkinson, A. C., Koopman, S. J. andShephard, N. (1997), Detecting shocks: outliers and breaks in time series.J. Econometrics, 80, 387–422.
Barbieri, M. M. (1996), Fattori di Bayes per l’identificazione di dati anomali in serie temporali. InAtti della XXXVIII Riunione Scientifica della Società Italiana di Statistica, vol. 2, 345–352. Rimini: Maggioli Editore.
Barbieri, M. M. andBattaglia, F. (1995),Outlier detection in time series: a decision theoretic approach. Pubbl. del Dip. di Statistica, Probabilità and Stat. Appl., Università di Roma «La Sapienza», no. 19.
Barndorff-Nielsen, O. andSchou, G. (1973), On the parametrization of autoregressive models by partial autocorrelation.J. Multivar. Anal., 3, 408–419.
Berger, J. O. andPericchi, L. R. (1996), The intrinsic Bayes factor for model selection and prediction.J. Amer. Statist. Assoc., 91, 109–122.
Berger, J. O. andPericchi, L. R. (1997), On criticism and comparisons of default Bayes factors for model selection and hypothesis testing. InProceedings of the Workshop on Model Selection (ed. W. Racugno), 1–50. Bologna: Pitagora.
Berger, J. O. andPericchi, L. R. (1998), Accurate and stable Bayesian model selection: the median intrinsic Bayes factor.Sankhyã B, 60, 1–18.
Booth, N. B. andSmith, A. F. M. (1982), A Bayesian approach to retrospective identification of change-points.J. Econometrics, 19, 7–22.
Box, G. E. P., Jenkins, G. M. andReinsel, G. C. (1994),Time Series Analysis: Forecasting and Control, 3rd ed. San Francisco: Holden Day.
Chen, C. andTiao, G. C. (1990), Random level-shift time series models, ARIMA approximations, and level-shift detection.J. Bus. Econ. Statist., 8, 83–97.
Cobb, G. W. (1978), The problem of the Nile: conditional solution to a change point problem.Biometrika, 65, 243–251.
Conigliani, C. andO’Hagan, A. (2000), Sensitivity of the fractional Bayes factor to prior distributions.Canadian Journal of Statistics, 28, 343–352.
Freeman, J. M. (1986), An unknown change point and goodness of fit.Statistician, 35, 335–344.
Galbraith, R. F. andGalbraith, J. I. (1974), On the inverses of some patterned matrices arising in the theory of stationary time series.J. Appl. Prob., 11, 63–71.
Jones, M. C. (1987), Randomly choosing parameters from the stationarity and invertibility region of autoregressive-moving average models.Appl. Statist., 36, 134–138.
Kass, R. andRaftery, A. (1995), Bayes Factors.J. Amer. Statist. Assoc., 90, 773–795.
Le, N. D., Martin, R. D. andRaftery, A. E. (1996), Modeling flat stretches, burnts, and outliers in time series using mixture transition distribution models.J. Amer. Statist. Assoc., 91, 1504–1515.
Maddala, G. S. andKim, I.-M. (1996), Structural change and unit roots.J. Statist. Plann. and Inf., 49, 73–103.
McCulloch, R. E. andTsay, R. S. (1993), Bayesian inference and prediction for mean and variance shifts in autoregressive time series.J. Amer. Statist. Assoc., 88, 968–978.
McCulloch, R. E. andTsay, R. S. (1994), Statistical analysis of economic time series via Markov switching models.J. Time Ser. Anal., 15, 523–539.
O’Hagan, A. (1995), Fractional Bayes factors for model comparison.J. Roy. Statist. Soc., B 57, 99–138.
O’Hagan, A. (1997), Properties of intrinsic and fractional Bayes factors.Test, 6, 101–118.
Phillips, D. B. andSmith, A. F. M. (1996), Bayesian model comparison via jump diffusions. InMarkov Chain Monte Carlo in Practice, eds. W. R. Gilks, S. Richardson and D. J. Spiegelhalter. London: Chapman and Hall, 215–239.
Piccolo, D. (1982), The size of the stationarity and invertibility region of an autoregressive-moving average process.J. Time Ser. Anal., 3, 245–247.
Pole, A., West, M. andHarrison, J. (1994),Applied Bayesian Forecasting and Time Series Analysis. London: Chapman and Hall.
Spiegelhalter, D. J. andSmith, A. F. M. (1982), Bayes factors for linear and log-linear models with vague prior information.J. Roy. Statist. Soc. B 44, 377–387.
Tsay, R. S. (1988), Outliers, level shift, and variance changes in time series.J. Forecasting, 7, 1–20.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Barbieri, M.M., Conigliani, C. Bayesian analysis of autoregressive time series with change points. J. Ital. Statist. Soc. 7, 243 (1998). https://doi.org/10.1007/BF03178933
DOI: https://doi.org/10.1007/BF03178933