, Volume 27, Issue 1, pp 52–69 | Cite as

Asymptotic normality and parameter change test for bivariate Poisson INGARCH models

  • Youngmi Lee
  • Sangyeol Lee
  • Dag Tjøstheim
Original Paper


In this paper, we consider the problem of testing for a parameter change in bivariate Poisson integer-valued GARCH(1, 1) models, constructed via a trivariate reduction method of independent Poisson variables. We verify that the conditional maximum-likelihood estimator of the model parameters is asymptotically normal. Then, based on these results, we construct CMLE- and residual-based CUSUM tests and derive that their limiting null distributions are a function of independent Brownian bridges. A simulation study and real data analysis are conducted for illustration.


Time series of counts Bivariate Poisson INGARCH model Test for a parameter change CUSUM test 

Mathematics Subject Classification

62M10 62G20 



We thank the Editor and the two anonymous referees for their careful reading and valuable comments to improve the quality of the paper. Sangyeol Lee’s research is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (No. 2015R1A2A2A010003894).

Supplementary material

11749_2016_510_MOESM1_ESM.pdf (71 kb)
Supplementary material 1 (pdf 71 KB)


  1. Alzaid A, Al-Osh M (1990) An integer-valued pth-order autoregressive structure (INAR(p)) process. J Appl Probab 27:314–324MathSciNetCrossRefMATHGoogle Scholar
  2. Andreassen CM (2013) Models and inference for correlated count data. Ph.D. thesis, Aarhus UniversityGoogle Scholar
  3. Csörgö M, Horváth L (1997) Limit theorems in change-point analysis, 18th edn. Wiley, New YorkMATHGoogle Scholar
  4. Denuit M, Lambert P (2005) Constraints on concordance measures in bivariate discrete data. J Multi Anal 93(1):40–57MathSciNetCrossRefMATHGoogle Scholar
  5. Doukhan P, Kengne W (2015) Inference and testing for structural change in general poisson autoregressive models. Electron J Stat 9(1):1267–1314MathSciNetCrossRefMATHGoogle Scholar
  6. Doukhan P, Fokianos K, Tjøstheim D (2012) On weak dependence conditions for Poisson autoregressions. Stat Probab Lett 82(5):942–948MathSciNetCrossRefMATHGoogle Scholar
  7. Doukhan P, Fokianos K, Tjøstheim D (2013) Correction to “On weak dependence conditions for Poisson autoregressions”. Stat Probab Lett 83(8):1926–1927CrossRefMATHGoogle Scholar
  8. Efron B (1986) Double exponential families and their use in generalized linear regression. J Am Stat Assoc 81(395):709–721MathSciNetCrossRefMATHGoogle Scholar
  9. Ferland R, Latour A, Oraichi D (2006) Integer-valued GARCH process. J Time Series Anal 27(6):923–942MathSciNetCrossRefMATHGoogle Scholar
  10. Fokianos K, Fried R (2010) Interventions in INGARCH processes. J Time Series Anal 31(3):210–225MathSciNetCrossRefMATHGoogle Scholar
  11. Fokianos K, Fried R (2012) Interventions in log-linear Poisson autoregression. Stat Model 12(4):299–322MathSciNetCrossRefGoogle Scholar
  12. Fokianos K, Rahbek A, Tjøstheim D (2009) Poisson autoregression. J Am Stat Assoc 104(488):1430–1439MathSciNetCrossRefMATHGoogle Scholar
  13. Fokianos K, Gombay E, Hussein A (2014) Retrospective change detection for binary time series models. J Stat Plan Inference 145:102–112MathSciNetCrossRefMATHGoogle Scholar
  14. Franke J, Kirch C, Kamgaing JT (2012) Changepoints in times series of counts. J Time Series Anal 33(5):757–770MathSciNetCrossRefMATHGoogle Scholar
  15. Heinen A (2003) Modelling time series count data: an autoregressive conditional Poisson model. CORE Discussion Paper 2003/62, Université catholique de LouvainGoogle Scholar
  16. Heinen A, Rengifo E (2003) Multivariate modeling of time series count data: an AR conditional Poisson model. CORE Discussion Paper 2003/23, Université catholique de LouvainGoogle Scholar
  17. Heinen A, Rengifo E (2007) Multivariate autoregressive modeling of time series count data using copulas. J Empir Fin 14(4):564–583CrossRefGoogle Scholar
  18. Hudecová Š (2013) Structural changes in autoregressive models for binary time series. J Stat Plan Inference 143(10):1744–1752MathSciNetCrossRefMATHGoogle Scholar
  19. Jin-Guan D, Yuan L (1991) The integer-valued autoregressive (INAR(p)) model. J Time Series Anal 12(2):129–142MathSciNetCrossRefMATHGoogle Scholar
  20. Kang J, Lee S (2009) Parameter change test for random coefficient integer-valued autoregressive processes with application to polio data analysis. J Time Series Anal 30(2):239–258MathSciNetCrossRefMATHGoogle Scholar
  21. Kang J, Lee S (2014) Parameter change test for Poisson autoregressive models. Scand J Stat 41(4):1136–1152MathSciNetCrossRefMATHGoogle Scholar
  22. Lee S, Ha J, Na O, Na S (2003) The cusum test for parameter change in time series models. Scand J Stat 40(4):781–796MathSciNetCrossRefMATHGoogle Scholar
  23. Lee S, Lee Y, Chen CW (2016) Parameter change test for zero-inflated generalized Poisson autoregressive models. Statistics 50(3):1–18MathSciNetCrossRefMATHGoogle Scholar
  24. Liu H (2012) Some models for time series of counts. Ph.D. thesis, Columbia UniversityGoogle Scholar
  25. McKenzie E (1985) Some simple models for discrete variate time series. Water Resour Bull 21(4):645–650Google Scholar
  26. McKenzie E (2003) Ch. 16. Discrete variate time series. Handb Stat 21:573–606Google Scholar
  27. Neumann MH (2011) Absolute regularity and ergodicity of Poisson count processes. Bernoulli 17(4):1268–1284MathSciNetCrossRefMATHGoogle Scholar
  28. Pedeli X, Karlis D (2011) A bivariate INAR(1) process with application. Stat Model 11(4):325–349MathSciNetCrossRefGoogle Scholar
  29. Pedeli X, Karlis D (2013a) On composite likelihood estimation of a multivariate INAR(1) model. J Time Ser Anal 34(2):206–220MathSciNetCrossRefMATHGoogle Scholar
  30. Pedeli X, Karlis D (2013b) On estimation of the bivariate Poisson INAR process. Commun Stat Simul Comput 42(3):514–533MathSciNetCrossRefMATHGoogle Scholar
  31. Quoreshi AS (2006) Bivariate time series modeling of financial count data. Commun Stat Theory Methods 35(7):1343–1358MathSciNetCrossRefMATHGoogle Scholar
  32. Wang C, Liu H, Yao JF, Davis RA, Li WK (2014) Self-excited threshold Poisson autoregression. J Am Stat Assoc 109(506):777–787MathSciNetCrossRefMATHGoogle Scholar
  33. Weiß CH (2008) Thinning operations for modeling time series of counts-a survey. AStA Adv Stat Anal 92(3):319–341MathSciNetCrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2016

Authors and Affiliations

  1. 1.Department of StatisticsSeoul National UniversitySeoulKorea
  2. 2.Department of MathematicsUniversity of BergenBergenNorway

Personalised recommendations