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Modeling overdispersed or underdispersed count data with generalized Poisson integer-valued GARCH models. (English) Zbl 1232.62120
Summary: Overdispersion in time series of counts is very common and has been well studied by many authors, but the opposite phenomenon of underdispersion may also be encountered in real applications and receives little attention. Based on popularity of the generalized Poisson distribution in regression count models and of Poisson INGARCH models in time series analysis, we introduce a generalized Poisson INGARCH model, which can account for both overdispersion and underdispersion. Compared with the double Poisson INGARCH model, conditions for the existence and ergodicity of such a process are easily given. We analyze the autocorrelation structure and also derive expressions for moments of order $1$ and $2$. We consider the maximum likelihood estimators for the parameters and establish their consistency and asymptotic normality. We apply the proposed model to one overdispersed real example and one underdispersed real example, respectively, which indicates that the proposed methodology performs better than other conventional model-based methods in the literature.

62M10Time series, auto-correlation, regression, etc. (statistics)
62F12Asymptotic properties of parametric estimators
Full Text: DOI
[1] Amemiya, T.: Advanced econometrics, (1985)
[2] Cameron, A. C.; Trivedi, P. K.: Regression analysis of count data, (1998) · Zbl 0924.62004
[3] Consul, P. C.: Generalized Poisson distributions: properties and applications, (1989) · Zbl 0691.62015
[4] Consul, P. C.; Famoye, F.: Generalized Poisson regression model, Comm. statist. Theory methods 21, 89-109 (1992) · Zbl 0800.62355 · doi:10.1080/03610929208830766
[5] Consul, P. C.; Famoye, F.: Lagrangian probability distributions, (2006) · Zbl 1103.62013
[6] Consul, P. C.; Jain, G. C.: A generalization of the Poisson distribution, Technometrics 15, 791-799 (1973) · Zbl 0271.60020 · doi:10.2307/1267389
[7] Consul, P. C.; Shoukri, M. M.: The negative integer moments of the generalized Poisson distribution, Comm. statist. Simulation comput. 15, 1053-1064 (1986) · Zbl 0606.62014 · doi:10.1080/03610918608812560
[8] Efron, B.: Double exponential families and their use in generalized linear regression, J. amer. Statist. assoc. 81, 709-721 (1986) · Zbl 0611.62072 · doi:10.2307/2289002
[9] Famoye, F.: Restricted generalized Poisson regression model, Comm. statist. Theory methods 22, 1335-1354 (1993) · Zbl 0784.62018 · doi:10.1080/03610929308831089
[10] Famoye, F.; Wulu, J. T.; Singh, K. P.: On the generalized Poisson regression model with an application to accident data, J. data sci. 2, 287-295 (2004)
[11] Ferland, R.; Latour, A.; Oraichi, D.: Integer-valued GARCH process, J. time ser. Anal. 27, 923-942 (2006) · Zbl 1150.62046 · doi:10.1111/j.1467-9892.2006.00496.x
[12] Fokianos, K.: Some recent progress in count time series, Statistics 45, 49-58 (2011) · Zbl 1291.62164
[13] Fokianos, K.; Fried, R.: Interventions in INGARCH processes, J. time ser. Anal. 31, 210-225 (2010) · Zbl 1242.62095
[14] Fokianos, K.; Rahbek, A.; Tjøstheim, D.: Poisson autoregression, J. amer. Statist. assoc. 104, 1430-1439 (2009) · Zbl 1205.62130 · doi:10.1198/jasa.2009.tm08270
[15] Fokianos, K.; Tjøstheim, D.: Log-linear Poisson autoregression, J. multivariate anal. 102, 563-578 (2011) · Zbl 1207.62165 · doi:10.1016/j.jmva.2010.11.002
[16] J. Franke, Weak dependence of functional INGARCH processes, Technical report, University of Kaiserslautern, 2010.
[17] A. Heinen, Modeling time series count data: an autoregressive conditional Poisson model, CORE Discussion paper 2003/62, Université catholique de Louvain, 2003.
[18] Jensen, S. T.; Rahbek, A.: Asymptotic inference for nonstationary GARCH, Econometric theory 20, 1203-1226 (2004) · Zbl 1069.62067 · doi:10.1017/S0266466604206065
[19] Jung, R. C.; Kukuk, M.; Liesenfeld, R.: Time series of count data: modeling, estimation and diagnostics, Comput. statist. Data anal. 51, 2350-2364 (2006) · Zbl 1157.62492 · doi:10.1016/j.csda.2006.08.001
[20] Jung, R. C.; Tremayne, A. R.: Useful models for time series of counts or simply wrong ones?, Adv. stat. Anal. 95, 59-91 (2011)
[21] Kedem, B.; Fokianos, K.: Regression models for time series analysis, (2002) · Zbl 1011.62089
[22] Ling, S.: Estimation and testing stationarity for double autoregressive models, J. roy. Statist. soc. Ser. B 66, 63-78 (2004) · Zbl 1061.62138 · doi:10.1111/j.1467-9868.2004.00432.x
[23] Matteson, D. S.; Mclean, M. W.; Woodard, D. B.; Henderson, S. G.: Forecasting emergency medical service call arrival rates, Ann. appl. Stat. 5, 1379-1406 (2011) · Zbl 1223.62161 · doi:10.1214/10-AOAS442
[24] Mckenzie, E.: Discrete variate time series, Handbook of statistics, vol. 21 21, 573-606 (2003) · Zbl 1064.62560
[25] Neumann, M. H.: Absolute regularity and ergodicity of Poisson count processes, Bernoulli 17, 1268-1284 (2011) · Zbl 1277.60089
[26] Özmen, .I.: Quasi likelihood/moment method for generalized and restricted generalized Poisson regression models and its application, Biom. J. 42, 303-314 (2000) · Zbl 0955.62093 · doi:10.1002/1521-4036(200007)42:3<303::AID-BIMJ303>3.0.CO;2-Q
[27] Ridout, M. S.; Besbeas, P.: An empirical model for underdispersed count data, Stat. model. 4, 77-89 (2004) · Zbl 1111.62017 · doi:10.1191/1471082X04st064oa
[28] Wang, W.; Famoye, F.: Modeling household fertility decisions with generalized Poisson regression, J. population econ. 10, 273-283 (1997)
[29] Weiß, C. H.: Controlling correlated processes of Poisson counts, Qual. reliab. Eng. internat. 23, 741-754 (2007)
[30] Weiß, C. H.: Thinning operations for modeling time series of counts--a survey, Adv. stat. Anal. 92, 319-341 (2008)
[31] Weiß, C. H.: Serial dependence and regression of Poisson INARMA models, J. statist. Plann. inference 138, 2975-2990 (2008) · Zbl 1140.62069 · doi:10.1016/j.jspi.2007.11.009
[32] Weiß, C. H.: Modelling time series of counts with overdispersion, Stat. methods appl. 18, 507-519 (2009)
[33] Weiß, C. H.: The $INARCH(1)$ model for overdispersed time series of counts, Comm. statist. Simulation comput. 39, 1269-1291 (2010) · Zbl 1204.62161 · doi:10.1080/03610918.2010.490317
[34] Weiß, C. H.: $INARCH(1)$ processes: higher-order moments and jumps, Statist. probab. Lett. 80, 1771-1780 (2010) · Zbl 1202.62125 · doi:10.1016/j.spl.2010.08.001
[35] White, H.: Maximum likelihood estimation of misspecified models, Econometrica 50, 1-25 (1982) · Zbl 0478.62088 · doi:10.2307/1912526
[36] Zhu, F.: A negative binomial integer-valued GARCH model, J. time ser. Anal. 32, 54-67 (2011) · Zbl 1290.62092
[37] F. Zhu, Zero-inflated Poisson and negative binomial integer-valued GARCH models, J. Statist. Plann. Inference (2012), doi:10.1016/j.jspi.2011.10.002, in press. · Zbl 1232.62121
[38] Zhu, F.; Li, Q.: Moment and Bayesian estimation of parameters in the INGARCH (1,1) model, J. jilin univ. (Science edition) 47, 899-902 (2009) · Zbl 1208.62145
[39] Zhu, F.; Li, Q.; Wang, D.: A mixture integer-valued ARCH model, J. statist. Plann. inference 140, 2025-2036 (2010) · Zbl 1184.62159 · doi:10.1016/j.jspi.2010.01.037
[40] Zhu, F.; Wang, D.: Diagnostic checking integer-valued $ARCH(p)$ models using conditional residual autocorrelations, Comput. statist. Data anal. 54, 496-508 (2010) · Zbl 05689605
[41] Zhu, F.; Wang, D.: Estimation and testing for a Poisson autoregressive model, Metrika 73, 211-230 (2011) · Zbl 1206.62155 · doi:10.1007/s00184-009-0274-z
[42] Zhu, F.; Wang, D.; Li, F.; Li, H.: Empirical likelihood inference for an integer-valued $ARCH(p)$ model, J. jilin univ. (Science edition) 46, 1042-1048 (2008) · Zbl 1193.62163
[43] Zucchini, W.; Macdonald, I. L.: Hidden Markov models for time series, (2009) · Zbl 1180.62130