On an integer-valued stochastic intensity model for time series of counts

Aknouche, Abdelhakim and Dimitrakopoulos, Stefanos (2020): On an integer-valued stochastic intensity model for time series of counts.

Abstract

We propose a broad class of count time series models, the mixed Poisson integer-valued stochastic intensity models. The proposed specification encompasses a wide range of conditional distributions of counts. We study its probabilistic structure and design Markov chain Monte Carlo algorithms for two cases; the Poisson and the negative binomial distributions. The methodology is applied to simulated data as well as to various data sets. Model comparison using marginal likelihoods and forecast evaluation using point and density forecasts are also considered.

• Item Type: MPRA Paper
• Language: English
• Keywords: Markov chain Monte Carlo, mixed Poisson process, parameter-driven models, count time series models.
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C11 - Bayesian Analysis: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C15 - Statistical Simulation Methods: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C18 - Methodological Issues: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C25 - Discrete Regression and Qualitative Choice Models ; Discrete Regressors ; Proportions ; Probabilities; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C51 - Model Construction and Estimation; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C53 - Forecasting and Prediction Methods ; Simulation Methods; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63 - Computational Techniques ; Simulation Modeling

References

Ahmad, A., Francq, C. (2016). Poisson QMLE of count time series models. Journal of Time Series Analysis 37: 291-314.

Aknouche, A. (2017). Periodic autoregressive stochastic volatility. Statistical Inference for Stochastic Processes 20: 139-177.

Aknouche, A., Bendjeddou, S., Touche, N. (2018). Negative binomial quasi-likelihood inference for general integer-valued time series models. Journal of Time Series Analysis 39: 192-211.

Aknouche, A., Demmouche, N. (2019). Ergodicity conditions for a double mixed Poisson autoregression. Statistics and Probability Letters: 147: 6-11.

Aknouche, A., Francq, C. (2021). Count and duration time series with equal conditional stochastic and mean orders. Econometric Theory, forthcoming.

Atchad�e, Y.F. (2006). An adaptive version for the Metropolis adjusted Langevin algorithm with a truncated drift. Methodology and Computing in Applied Probability 8: 235-254.

Barra, I., Borowska, A., Koopman, S.J. (2018). Bayesian dynamic modeling of high-frequency integer price changes. Journal of Financial Econometrics 16: 384-424.

Benjamin, M.A., Rigby, R.A., Stasinopoulos, D.M. (2003). Generalized autoregressive moving average models. Journal of the American Statistical Association 98: 214-23.

Berg, A., Meyer, R., Yu, J. (2004). Deviance information criterion for comparing stochastic volatility models. Journal of Business & Economic Statistics 22: 107-120.

Bradlow, E.T., Hardie, B.G.S, Fader, P.S. (2002). Bayesian inference for the negative binomial distribution via polynomial expansions. Journal of Computational & Graphical Statistics 11: 189-201.

Carrasco, M., Chen, X. (2002). Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18: 17-39.

Chan, J.C.C., Eisenstat, E. (2015). Marginal likelihood estimation with the Cross-Entropy method. Econometric Reviews, 34: 256-285.

Chan, K.S., Ledholter, J. (1995). Monte Carlo EM Estimation for Time Series Models Involving Counts. Journal of the American Statistical Association, 90, 242-252.

Chen, C.W.S., So, M., Li, J.C., Sriboonchitta, S. (2016). Autoregressive conditional negative binomial model applied to over-dispersed time series of counts. Statistical Methodology 31: 73-90.

Chib, S. (2001). Markov chain Monte Carlo methods: computation and inference. In: Handbook of Econometrics, Volume 5. North Holland, Amsterdam, pp. 3569-3649.

Christou, V., Fokianos, K. (2014). Quasi-likelihood inference for negative binomial time series models. Journal of Time Series Analysis 35: 55-78.

Christou, V., Fokianos, K. (2015). Estimation and testing linearity for non-linear mixed Poisson autoregressions. Electronic Journal of Statistics 9: 1357-1377.

Cox, D.R. (1981). Statistical analysis of time series: Some recent developments. Scandinavian Journal of Statistics 8: 93-115.

Davis, R.A., Dunsmuir, W.T.M., Wang, Y. (1999). Modelling time series of count data. In Asymptotics, Nonparametrics, and Time Series. CRC Press, New York, pp. 63-114.

Davis, R.A., Dunsmuir, W.T.M., Wang, Y. (2000). On autocorrelation in a Poisson regression model. Biometrika 87: 491-505.

Davis, R.A., Dunsmuir, W.T.M. (2016). State space models for count time series. In: Handbook of Discrete-Valued Time Series. Chapman and Hall: CRC Press, pp. 121-144.

Davis, R.A., Holan, S.H., Lund, R., Ravishanker, N. (2016). Handbook of Discrete-Valued Time Series. Chapman and Hall: CRC Press.

Davis, R.A., Liu, H. (2016). Theory and inference for a class of observation-driven models with application to time series of counts. Statistica Sinica 26: 1673-1707.

Davis, R.A., Rodriguez-Yam, G. (2005). Estimation for state-space models based on likelihood approximation. Statistica Sinica 15: 381-406.

Davis, R.A., Wu, R. (2009). A negative binomial model for time series of counts. Biometrika 96: 735-749.

Dean, C., Lawless, J., Willmot, G. (1989). A mixed Poisson-Inverse-Gaussian regression model. The Canadian Journal of Statistics 17: 171-181.

Doukhan, P., Fokianos, K., Tj�stheim, D. (2012). On weak dependence conditions for Poisson autoregressions. Statistics and Probability Letters 82: 942-948.

Doukhan, P., Latour, A., Oraichi, D. (2006). A simple integer-valued bilinear time series model. Advances in Applied Probability 32: 559-578.

Dunn, P.K., Smyth, G.K. (1996). Randomized quantile residuals. Journal of Computational and Graphical Statistics 5: 236-244.

Efron, B. (1986). Double exponential families and their use in generalized linear Regression. Journal of the American Statistical Association 81: 709-721.

Engle, R. (2002). New frontiers for ARCH models. Journal of Applied Econometrics 17: 425-446.

Feller, W. (1943). On a general class of contagious distributions. Annals of Mathematical Statistics 14: 389-400.

Ferland, R., Latour, A., Oraichi, D. (2006). Integer-valued GARCH process. Journal of Time Series Analysis 27: 923-942.

Fokianos, K., Rahbek, A., Tj�stheim, D. (2009). Poisson autoregression. Journal of the American Statistical Association 140: 1430-1439.

Fokianos, K. (2016). Statistical analysis of count time Series models: A GLM perspective. In: Handbook of Discrete-Valued Time Series. Chapman and Hall: CRC Press.

Francq, C., Zakoian, J.M. (2019). GARCH Models: Structure, Statistical Inference and Applications. Wiley.

Freeland, R.K., McCabe, B.P.M. (2004). Forecasting Discrete Valued Low Count Time Series. International Journal of Forecasting, 20: 427-434.

Frühwirth-Schnatter, S., Frühwirth, R., Held, L., Rue, H. (2009). Improved auxiliary mixture sampling for hierarchical models of non-Gaussian data. Statistics and Computing, 19: 479.

Frühwirth-Schnatter, S., Wagner, H. (2006). Auxiliary mixture sampling for parameter-driven models of time series of counts with applications to state space modelling. Biometrika 93: 827-841.

Geweke, J. (1992). Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In: Bayesian Statistics 4, Oxford: Clarendon Press, pp. 641-649.

Geweke, J., Amisano, G. (2011a). Optimal prediction pools. Journal of Econometrics 164: 130-141.

Geweke, J., Amisano, G. (2011b). Hierarchical Markov normal mixture models with applications to financial asset returns. Journal of Applied Econometrics 26:1-29.

Graham, R.L., Knuth, D.E., Patashnik, O. (1989). Concrete Mathematics. A Foundation for Computer Science. Reading, MA: Addison Wesley.

Grunwald, G., Hyndman, R.J.S., Tedesco, L., Tweedie, R.L. (2000). Theory and methods: Non-Gaussian conditional linear AR(1) models. Australian & New Zealand Journal of Statistics 42: 479-495.

Hay, J., Pettitt, A. (2001). Bayesian analysis of a time series of counts with covariates: an application to the control of an infectious disease. Biostatistics 2: 433-444.

Hinde, J.P. (1982). Compound Poisson regression models. In: Proceedings of the International Conference on Generalized Linear Models. Lecture Notes in Statistics, Vol 14. Springer, pp. 109-121.

Jorgensen, B. (1997). The Theory of Dispersion Models. Chapman & Hall, London.

Jung, R.C., Kukuk, M., Liesenfeld, R. (2006). Time series of count data: modeling, estimation and diagnostics. Computational Statistics & Data Analysis 51: 2350-2364.

Jung, R.C., Liesenfeld, R., Richard, J.-F. (2011). Dynamic Factor Models for Multivariate Count Data: An Application to Stock-Market Trading Activity. Journal of Business and Economic Statistics, 29: 73-86.

Kim, S., Shephard, N., Chib, S. (1998). Stochastic volatility: Likelihood inference and comparison with ARCH models. Review of Economic Studies 65: 361-393.

Kokonendji, C.C., Dossou-Gbété, S., Demétrio, C.G.B. (2004). Some discrete exponential dispersion models: Poisson-Tweedie and Hinde-Demétrio classes. Statistics and Operations Research Transactions 28: 201-213.

Koop, G., Poirier, D.J., Tobias, J.L. (2012). Bayesian Econometric methods (1st edition) Cambridge University Press, New York.

Liesenfeld, R., Nolte, I., Pohlmeier, W. (2006). Modelling �nancial transaction price movements: a dynamic integer count data model. Empirical Economics 30: 795-825.

Leroux, B.G. (1992). Maximum likelihood estimation for hidden Markov models. Stochastic Processes and their Applications 40: 127-143.

McCabe, B.P.M., Martin, G.M. (2005). Bayesian predictions of low count time series. International Journal of Forecasting 21: 315-330.

Martino, L., Yang, H., Luengo, D., Kanniainen, J., Corander, J. (2015). A Fast Universal Self-Tuned Sampler Within Gibbs Sampling. Digital Signal Processing 47: 68-83.

Meyn, S.P., Tweedie, R.L. (2009). Markov Chains and Stochastic Stability. Springer Verlag, New York.

Mikosch, T. (2009). Non-life Insurance Mathematics, An Introduction with the Poisson Process. Springer-Verlag, Berlin.

Neumann, M.H. (2011). Absolute regularity and ergodicity of Poisson count processes. Bernoulli 17: 1268-1284.

Priestley, M.B. (1981). Spectral analysis and time series. New York: Academic Press.

Roberts, G.O., Rosenthal, J.S. (1998). Optimal scaling of discrete approximations to Langevin difusions. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 60: 255-268.

Roberts, G.O., Tweedie, R.L. (1996). Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli 2: 341-363.

Rydberg, T.H., Shephard, N. (2000). BIN models for trade-by-trade data. Modelling the number of trades in a �fixed interval of time. Technical Report 0740, Econometric Society.

Silva, R.B., Barreto-Souza, W. (2019). Flexible and robust mixed Poisson INGARCH models. Journal of Time Series Analysis 40: 788-814.

Sorensen, H. (2019). Independence, successive and conditional likelihood for time series of counts. Journal of Statistical Planning and Inference 200: 20-31.

Weiss, C.H (2009). Modelling time series of counts with overdispersion. Statistical Methods and Applications, 18: 507-519. Weiss, C.H. (2017). An Introduction to Discrete-Valued Time Series. Wiley.

Zeger, S.L. (1988). A regression model for time series of counts. Biometrika 75: 621-629.

Zeger, S., Qaqish, B. (1988). Markov regression models for time series: A quasi-likelihood approach. Biometrics 44: 1019-1031.

Zhu, F. (2011). A negative binomial integer-valued GARCH model. Journal of Time Series Analysis 32: 54-67.