Brazilian Journal of Probability and Statistics

Time series of count data: A review, empirical comparisons and data analysis

Glaura C. Franco, Helio S. Migon, and Marcos O. Prates

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Abstract

Observation and parameter driven models are commonly used in the literature to analyse time series of counts. In this paper, we study the characteristics of a variety of models and point out the main differences and similarities among these procedures, concerning parameter estimation, model fitting and forecasting. Alternatively to the literature, all inference was performed under the Bayesian paradigm. The models are fitted with a latent AR($p$) process in the mean, which accounts for autocorrelation in the data. An extensive simulation study shows that the estimates for the covariate parameters are remarkably similar across the different models. However, estimates for autoregressive coefficients and forecasts of future values depend heavily on the underlying process which generates the data. A real data set of bankruptcy in the United States is also analysed.

Article information

Source
Braz. J. Probab. Stat., Volume 33, Number 4 (2019), 756-781.

Dates
Received: May 2018
Accepted: March 2019
First available in Project Euclid: 26 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1566806432

Digital Object Identifier
doi:10.1214/19-BJPS437

Mathematical Reviews number (MathSciNet)
MR3996316

Zentralblatt MATH identifier
07120733

Keywords
Observation driven model parameter driven model autoregressive processes Bayesian inference

Citation

Franco, Glaura C.; Migon, Helio S.; Prates, Marcos O. Time series of count data: A review, empirical comparisons and data analysis. Braz. J. Probab. Stat. 33 (2019), no. 4, 756--781. doi:10.1214/19-BJPS437. https://projecteuclid.org/euclid.bjps/1566806432


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References

  • Andrieu, C., Doucet, A. and Holenstein, R. (2010). Particle Markov chain Monte Carlo methods (with discussion). Journal of the Royal Statistical Society, Series B, Statistical Methodology 72, 1–33.
  • Benjamin, M. A., Rigby, R. A. and Stasinopoulos, D. M. (2003). Generalized autoregressive moving average models. Journal of the American Statistical Association 98, 214–223.
  • Box, G. E. P. and Jenkins, G. M. (1976). Time Series Analysis: Forecasting and Control. San Francisco: Holden-Day.
  • Chan, K. and Ledolter, J. (1995). Monte Carlo EM estimation for time series models involving counts. Journal of the American Statistical Association 90, 242–251.
  • Cox, D. R. (1981). Statistical analysis of time series: Some recent developments. Scandinavian Journal of Statistics 8, 93–115.
  • Creal, D., Koopman, S. J. and Lucas, A. (2013). Generalized autoregressive score models with applications. Journal of Applied Econometrics 28, 777–795.
  • Creal, D. D. (2012). A survey of sequential Monte Carlo methods for economics and finance. Econometric Reviews 31, 245–296.
  • Czado, C. and Kolbe, A. (2004). Empirical study of intraday option price changes using extended count regression models. Germany. Collaborative Research Center 386, Discussion Paper 403. Ludwig-Maximilians-Universitat.
  • Davis, R. A., Dunsmuir, W. T. M. and Streett, S. B. (2003). Observation-driven models for Poisson counts. Biometrika 90, 777–790.
  • Davis, R. A., Dunsmuir, W. T. M. and Streett, S. B. (2005). Maximum likelihood estimation for an observation driven model for Poisson counts. Methodology and Computing in Applied Probability 7, 149–159.
  • Davis, R. A., Dunsmuir, W. T. M. and Wang, Y. (1999). Modelling time series of counts. In Asymptotics, Nonparametrics, and Time Series: A Tribute to Madan Lal Puri (S. Ghosh, ed.) 63–113. NewYork: Marcel Dekker.
  • Diebold, F. X. and Schuermann, T. (1996). Exact maximum likelihood estimation of observation-driven econometric models. NBER Technical Working Paper No. 194.
  • Durbin, J. and Koopman, S. J. (2002). A simple and efficient simulation smoother for state space time series analysis. Biometrika 89, 603–615.
  • Gamerman, D. and Lopes, H. F. (2006). Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference. London: Chapman and Hall.
  • Gamerman, D., Santos, T. R. and Franco, G. C. (2013). A non-Gaussian family of state-space models with exact marginal likelihood. Journal of Time Series Analysis 34, 625–645.
  • Gelman, A. (1996). Inference and monitoring convergence. In Markov Chain Monte Carlo in Practice (W. R. Gilks, S. Richardson and D. J. Spiegelhalter, eds.) 131–143. London: Chapman and Hall.
  • Goudie, R. J. B., Turner, R. M., De Angelis, D., Thomas, A. and MultiBUGS (2017). A parallel implementation of the BUGS modelling framework for faster Bayesian inference. Available at https://arxiv.org/abs/1704.03216.
  • Harvey, A. C. and Fernandez, C. (1989). Time series models for count or qualitative observations. Journal of Business and Economic Statistics 7, 407–417.
  • Jung, R. C., Kukuk, M. and Liesenfeld, R. (2006). Time series of count data: Modeling, estimation and diagnostics. Computational Statistics & Data Analysis 51, 2350–2364.
  • Jung, R. C. and Tremayne, A. R. (2011). Useful models for time series of counts or simply wrong ones?. AStA Advances in Statistical Analysis 95, 59–91.
  • Koopman, S. J., Lucas, A. and Scharth, M. (2016). Predicting time-varying parameters with parameter-driven and observation-driven models. Review of Economics and Statistics 98, 97–110.
  • Li, W. K. (1994). Time series models based on generalized linear models: Some further results. Biometrics 50, 506–511.
  • Lunn, D. J., Thomas, A., Best, N. and Spiegelhalter, D. (2000). WinBUGS—a Bayesian modelling framework: Concepts, structure, and extensibility. Statistics and Computing 10, 325–337.
  • Meyer, R. and Yu, J. (2000). BUGS for a Bayesian analisys of stochastic volatility models. Econometrics Journal 3, 198–215.
  • Migon, H. S., Gamerman, D., Lopes, H. F. and Ferreira, M. A. R. (2005). Dynamic Models. Handbook of Statistics 25. Amsterdam: Elsevier.
  • R Core Team (2018). R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing.
  • Rue, H., Martino, S. and Chopin, N. (2009). Approximate Bayesian inference for latent Gaussian models using integrated nested Laplace approximations (with discussion). Journal of the Royal Statistical Society, Series B, Statistical Methodology 71, 319–392.
  • Shephard, N. (1995). Generalized linear autoregressions. Oxford, Nuffield College. Working Paper.
  • Shephard, N. and Pitt, M. K. (1995). Likelihood analysis of non-Gaussian measurement time series. Biometrika 84, 653–667.
  • Smith, R. L. and Miller, J. E. (1986). A non-Gaussian state space model and application to prediction of records. Journal of the Royal Statistical Society, Series B, Statistical Methodology 48, 79–88.
  • Souza, M. A. O. and Migon, H. S. (2018). Extended dynamic generalized linear models: The two-parameter exponential family. Computational Statistics & Data Analysis 121, 164–179.
  • Tierney, L. and Kadane, J. B. (1986). Accurate approximations for posterior moments and marginal densities. Journal of the American Statistical Association 81, 82–86.
  • West, M., Harrison, J. and Migon, H. S. (1985). Dynamic generalized linear models and Bayesian forecasting. Journal of the American Statistical Association 80, 73–83.
  • Zeger, S. L. (1988). A regression model for time series of counts. Biometrika 75, 621–629.
  • Zeger, S. L. and Qaqish, B. (1988). Markov regression models for time series: A quasi-likelihood approach. Biometrics 44, 1019–1031.