Electronic Journal of Statistics

Estimation and testing linearity for non-linear mixed poisson autoregressions

Vasiliki Christou and Konstantinos Fokianos

Full-text: Open access

Abstract

Non-linear mixed Poisson autoregressive models are studied for the analysis of count time series. Given a correct mean specification of the model, we discuss quasi maximum likelihood estimation based on Poisson log-likelihood function. A score testing procedure for checking linearity of the mean process is developed. We consider the cases of identifiable and non identifiable parameters under the null hypothesis. When the parameters are identifiable then a chi-square approximation to the distribution of the score test is obtained. In the case of non identifiable parameters, a supremum score type test statistic is employed for checking linearity of the mean process. The methodology is applied to simulated and real data.

Article information

Source
Electron. J. Statist., Volume 9, Number 1 (2015), 1357-1377.

Dates
Received: January 2015
First available in Project Euclid: 23 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1435064563

Digital Object Identifier
doi:10.1214/15-EJS1044

Mathematical Reviews number (MathSciNet)
MR3360730

Zentralblatt MATH identifier
1327.62456

Subjects
Primary: 62M09: Non-Markovian processes: estimation
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Keywords
Bootstrap chi-square contraction identifiability quasi maximum likelihood score test threshold model

Citation

Christou, Vasiliki; Fokianos, Konstantinos. Estimation and testing linearity for non-linear mixed poisson autoregressions. Electron. J. Statist. 9 (2015), no. 1, 1357--1377. doi:10.1214/15-EJS1044. https://projecteuclid.org/euclid.ejs/1435064563


Export citation

References

  • [1] Andrews, D. W. K. (2001). Testing when a parameter is on the boundary of the maintained hypothesis, Econometrica 69, 683–734.
  • [2] Andrews, D. W. K. and Ploberger, W. (1994). Optimal tests when a nuisance parameter is present only under the alternative, Econometrica 62, 1383–1414.
  • [3] Bardet, J.-M. and Wintenberger, O. (2009). Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes, The Annals of Statistics 37, 2730–2759.
  • [4] Berkes, I., Horváth, L. and Kokoszka, P. (2003). GARCH processes: Structure and estimation, Bernoulli 9, 201–227.
  • [5] Boos, D. D. (1992). On generalized score tests, The American Statistician 46, 327–333.
  • [6] Breslow, N. (1990). Tests of hypotheses in overdispersed Poisson regression and other quasi-likelihood models, Journal of the American Statistical Association 85, 565–571.
  • [7] Cameron, A. C. and Trivedi, P. K. (1998)., Regression Analysis of Count Data, 1st ed, Cambridge University Press, Cambridge.
  • [8] Chan, K. S. (1993). Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model, The Annals of Statistics 21, 520–533.
  • [9] Christou, V. and Fokianos, K. (2014). Quasi–likelihood inference for negative binomial time series models, Journal of Time Series Analysis 35, 55–78.
  • [10] Christou, V. and Fokianos, K. (2015). On count time series prediction, Journal of Statistical Computation and Simulation 85, 357–373.
  • [11] Davies, R. B. (1987). Hypothesis testing when a nuisance parameter is present only under the alternative, Biometrika 74, 33–43.
  • [12] Davis, R. A., Dunsmuir, W. T. M. and Streett, S. B. (2003). Observation-driven models for Poisson counts, Biometrika 90, 777–790.
  • [13] Davis, R. A. and Liu, H. (2015). Theory and inference for a class of observation-driven models with application to time series of counts, Statistica Sinica. to appear.
  • [14] Dean, C. B., Eaves, D. M. and Martinez, C. J. (1995). A comment on the use of empirical covariance matrices in the analysis of count data, Journal of Statistical Planning and Inference 48, 197–205.
  • [15] Dedecker, J., Doukhan, P., Lang, G., León, J. R., Louhichi, S. and Prieur, C. (2007)., Weak Dependence: With Examples and Applications, Vol. 190 of Lecture Notes in Statistics, Springer, New York.
  • [16] Dedecker, J. and Prieur, C. (2007). An empirical central limit theorem for dependent sequences, Stochastic Process. Appl. 117, 121–142.
  • [17] Dehling, H., Durieu, O. and Tusche, M. (2014). Approximating class approach for empirical processes of dependent sequences indexed by functions, Bernoulli 20, 1372–1403.
  • [18] Douc, R., Doukhan, P. and Moulines, E. (2013). Ergodicity of observation-driven time series models and consistency of the maximum likelihood estimator, Stochastic Processes and their Applications 123, 2620–2647.
  • [19] Doukhan, P., Fokianos, K. and Tjøstheim, D. (2012). On weak dependence conditions for Poisson autoregressions, Statistics & Probability Letters 82, 942–948.
  • [20] Doukhan, P. and Wintenberger, O. (2008). Weakly dependent chains with infinite memory, Stochastic Processes and Their Applications 118, 1997–2013.
  • [21] Engle, R. F. (1984)., Wald, Likelihood Ratio, and Lagrange Multiplier Tests in Econometrics, Vol. 2 of Handbook of Econometrics, Elsevier, Amsterdam: North Holland, 775–826.
  • [22] Fokianos, K., Rahbek, A. and Tjøstheim, D. (2009). Poisson autoregression, Journal of the American Statistical Association 104, 1430–1439.
  • [23] Fokianos, K. and Tjøstheim, D. (2012). Nonlinear Poisson autoregression, Annals of the Institute of Statistical Mathematics 64, 1205–1225.
  • [24] Francq, C., Horvath, L. and Zakoïan, J.-M. (2010). Sup-tests for linearity in a general nonlinear AR(1) model, Econometric Theory 26, 965–993.
  • [25] Francq, C. and Zakoïan, J.-M. (2004). Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes, Bernoulli 10, 605–637.
  • [26] Francq, C. and Zakoïan, J.-M. (2010)., GARCH Models: Stracture, Statistical Inference and Financial Applications, John Wiley, United Kingdom.
  • [27] Gallant, A. R. (1987)., Nonlinear statistical models, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley, New York.
  • [28] Godambe, V. P. and Heyde, C. C. (1987). Quasi–likelihood and optimal estimation, International Statistical Review 55, 231–244.
  • [29] Harvey, A. C. (1990)., The Econometric Analysis of Time Series, 2nd ed, MIT Press, Cambridge, MA.
  • [30] Heyde, C. C. (1997)., Quasi-Likelihood and its Applications: A General Approach to Optimal Parameter Estimation, Springer, New York.
  • [31] Joe, H. and Zhu, R. (2005). Generalized Poisson distribution: The property of mixture of Poisson and comparison with negative binomial distribution, Biometrical Journal 47, 219–229.
  • [32] Johnson, N. L., Kotz, S. and Kemp, A. W. (1992)., Univariate Discrete Distributions, 2nd ed, Wiley, New York.
  • [33] Jørgensen, B. (1997)., The Theory of Dispersion Models, Chapman & Hall, London.
  • [34] Jung, R. and Tremayne, A. (2011). Useful models for time series of counts or simply wrong ones?, AStA Advances in Statistical Analysis 95, 59–91.
  • [35] Kedem, B. and Fokianos, K. (2002)., Regression Models for Time Series Analysis, John Wiley, Hoboken, NJ.
  • [36] Kent, J. T. (1982). Robust properties of likelihood ratio tests, Biometrika 69, 19–27.
  • [37] Kokonendji, C. C., Dossou-Gbété, S. and Demétrio, C. G. B. (2004). Some discrete exponential dispersion models: Poisson-Tweedie and Hinde-Demétrio classes, Statistics and Operations Research Transactions 28, 201–213.
  • [38] Lawless, J. F. (1987). Negative binomial and mixed Poisson regression, The Canadian Journal of Statistics 15, 209–225.
  • [39] Li, G. and Li, W. K. (2011). Testing a linear time series model against its threshold extension, Biometrika 98, 243–250.
  • [40] Luukkonen, R., Saikkonen, P. and Teräsvirta, T. (1988). Testing linearity against smooth transition autoregressive models, Biometrika 75, 491–499.
  • [41] Mikosch, T. (2009)., Non-life Insurance Mathematics, An Introduction with the Poisson Process, 2nd ed, Springer-Verlag, Berlin.
  • [42] Mikosch, T. and Straumann, D. (2006). Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach, The Annals of Statistics 34, 2449–2495.
  • [43] Neumann, M. (2011). Absolute regularity and ergodicity of Poisson count processes, Bernoulli 17, 1268–1284.
  • [44] Teräsvirta, T. (1994). Specification, estimation, and evaluation of smooth transition autoregressive models, Journal of the American Statistical Association 89, 208–218.
  • [45] Tong, H. (1990)., Nonlinear Time Series: A Dynamical System Approach, Oxford University Press, New York.
  • [46] Wang, C., Liu, H., Yao, J.-F., Davis, R. A. and Li, W. K. (2014). Self-excited threshold Poisson autoregression, Journal of the American Statistical Association 109, 777–787.
  • [47] Woodard, D. W., Matteson, D. S. and Henderson, S. G. (2011). Stationarity of count-valued and nonlinear time series models, Electronic Journal of Statistics 5, 800–828.
  • [48] Zeger, S. L. and Qaqish, B. (1988). Markov regression models for time series: A quasi-likelihood approach, Biometrics 44, 1019–1031.
  • [49] Zhu, F. (2011). A negative binomial integer-valued GARCH model, Journal of Time Series Analysis 32, 54–67.
  • [50] Zhu, F., (2012a). Modeling overdispersed or underdispersed count data with generalized Poisson integer-valued GARCH models, Journal of Mathematical Analysis and Applications 389, 58–71.
  • [51] Zhu, F., (2012b). Zero-inflated Poisson and negative binomial integer-valued GARCH models, J. Statist. Plann. Inference 142, 826–839.
  • [52] Zivot, E. and Wang, J. (2006)., Modeling Financial Time Series with S-Plus$^\circledR$, 2nd ed, Springer-Verlag, New York.