422 Handbook of Discrete-Valued Time Series
dened on Z. Models for time series on Z are becoming popular in several disciplines and
some of them can be derived by higher-dimension models. Multivariate time series in Z
d
would also be of interest. For example, in nance one needs to model the number of ticks
that a stock is going up or down during consecutive time points, and this can create a large
number of time series, when managing a portfolio.
Acknowledgment
The author would like to thank Dr. Pedeli for helpful comments during the preparation of
this chapter.
References
Al-Osh, M. and Alzaid, A. (1987). First–order integer–valued autoregressive process. Journal of Time
Series Analysis, 8(3):261–275.
Berkhout, P. and Plug, E. (2004). Abivariate Poisson count data model using conditional probabilities.
Statistica Neerlandica, 58(3):349–364.
Bien, K., Nolte, I., and Pohlmeier, W. (2011). An inated multivariate integer count hurdle model: An
application to bid and ask quote dynamics. Journal of Applied Econometrics, 26(4):669–707.
Billingsley, P. (1961). Statistical Inference for Markov Processes. University of Chicago Press,
Chicago, IL.
Boudreault, M. and Charpentier, A. (2011). Multivariate integer-valued autoregressive models
applied to earthquake counts. Available at http://arxiv.org/abs/1112.0929.
Brandt, P. and Sandler, T. (2012). A Bayesian Poisson vector autoregression model. Political Analysis,
20(3):292–315.
Brännäs, K. and Nordström, J. (2000). A bivariate integer valued allocation model for guest nights in
hotels and cottages. Umeå Economic Studies, 547, Department of Economics, Umea University,
Umea, Sweden.
Bulla, J., Chesneau, C., and Kachour, M. (2011). A bivariate rst-order signed integer-valued
autoregressive process. Available at http://hal.archives-ouvertes.fr/hal-00655102.
Chib, S. and Winkelmann, R. (2001). Markov chain Monte Carlo analysis of correlated count data.
Journal of Business & Economic Statistics, 19(4):428–435.
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. doi:10.5705/ss.2014.145t (to appear).
Davis, R. A. and Yau, C. Y. (2011). Comments on pairwise likelihood in time series models. Statistica
Sinica, 21(1):255–277.
Dewald, L., Lewis, P., and McKenzie, E. (1989). A bivariate first–order autoregressive time series
model in exponential variables BEAR(1). Management Science, 35(10):1236–1246.
Efron, B. (1986). Double exponential families and their use in generalized linear regression. Journal of
the American Statistical Association, 81(395):709–721.
Fokianos, K., Rahbek, A., and Tjøstheim, D. (2009). Poisson autoregression. Journal of the American
Statistical Association, 104:1430–1439.
Frank, M. J. (1979). On the simultaneous associativity of F(x, y) and x + y − F(x, y). Aequationes
Mathematicae, 19:194–226.
Franke, J. and Rao, T. (1995). Multivariate first–order integer–valued autoregressions. Technical
Report, Math. Dep., UMIST.
423 Models for Multivariate Count Time Series
Genest, C. and Nešlehová, J. (2007). A primer on copulas for count data. The ASTIN Bulletin,
37:475–515.
Heinen, A. and Rengifo, E. (2007). Multivariate autoregressive modeling of time series count data
using copulas. Journal of Empirical Finance, 14(4):564–583.
Held, L., Hohle, M., and Hofmann, M. (2005). A statistical framework for the analysis of multivariate
infectious disease surveillance counts. Statistical Modelling, 5(3):187–199.
Joe, H. (1996). Time series models with univariate margins in the convolution-closed innitely
divisible class. Journal of Applied Probability, 33(3):664–677.
Jorgensen, B., Lundbye-Christensen, S., Song, P.-K., and Sun, L. (1999). A state space model for
multivariate longitudinal count data. Biometrika, 86(1):169–181.
Jung, R., Liesenfeld, R., and Richard, J. (2011). Dynamic factor models for multivariate count
data: An application to stock-market trading activity. Journal of Business and Economic Statistics,
29(1):73–85.
Jung, R. and Tremayne, A. (2006). Binomial thinning models for integer time series. Statistical
Modelling, 6(2):81–96.
Karlis, D. and Meligkotsidou, L. (2005). Multivariate Poisson regression with covariance structure.
Statistics and Computing, 15(4):255–265.
Karlis, D. and Meligkotsidou, L. (2007). Finite multivariate Poisson mixtures with applications.
Journal of Statistical Planning and Inference, 137:1942–1960.
Karlis, D. and Pedeli, X. (2013). Flexible bivariate INAR(1) processes using copulas. Communications
in Statistics—Theory and Methods, 42(4):723–740.
Kocherlakota, S. and Kocherlakota, K. (1992). Bivariate Discrete Distributions, Statistics: Textbooks and
Monographs, vol. 132. Markel Dekker, New York.
Latour, A. (1997). The multivariate GINAR(p) process. Advances in Applied Probability, 29(1):
228–248.
Lee, A. H., Wang, K., Yau, K. K., Carrivick, P. J., and Stevenson, M. R. (2005). Modelling bivariate
count series with excess zeros. Mathematical Biosciences, 196(2):226–237.
Liu, H. (2012). Some models for time series of counts. PhD thesis, Columbia University, New York.
McKenzie, E. (1985). Some simple models for discrete variate time series. Water Resources Bulletin,
21(4):645–650.
McKenzie, E. (1988). Some ARMA models for dependent sequences of Poisson counts. Advances in
Applied Probability, 20(4):822–835.
McKenzie, E. (2003). Discrete Variate Time Series, vol. 21. In C. Rap. and D. Shanbhag, (eds.), Stochastic
Processes: Modelling and Simulation, Handbook of Statistics, vol. 21, pp. 573606. Elsevier Science
B.V., North-Holland, Amsterdam, the Netherlands.
Nastic, A., Ristic, M., and Popovic, P. (2014). Estimation in a bivariate integer-valued autoregressive
process. In Communications in Statistics, Theory and Methods, page (forthcoming).
Nelsen, R. B. (2006). An Introduction to Copulas, 2nd edn. Springer–Verlag, New York.
Nikoloulopoulos, A. (2013a). On the estimation of normal copula discrete regression models using
the continuous extension and simulated likelihood. Journal of Statistical Planning and Inference,
143(11):1923–1937.
Nikoloulopoulos, A. K. (2013b). Copula-based models for multivariate discrete response data. In
Copulae in Mathematical and Quantitative Finance, pp. 231–249. Lecture Notes in Statistics,
Springer.
Orfanogiannaki, K. and Karlis, D. (2013). Hidden Markov models in modeling time series of earth-
quakes. In Proceedings of the 18th European Young Statisticians Meeting, Osijek, Croatia, 2013,
pp. 95–100. Available at http://www.mathos.unios.hr/eysm18/Proceedings_web.pdf.
Pedeli, X. and Karlis, D. (2011). A bivariate INAR(1) process with application. Statistical Modelling,
11(4):325–349.
Pedeli, X. and Karlis, D. (2013a). On composite likelihood estimation of a multivariate INAR(1)
model. Journal of Time Series Analysis, 34(2):206–220.
Pedeli, X. and Karlis, D. (2013b). On estimation of the bivariate Poisson INAR process. Communica-
tions in Statistics: Simulation and Computation, 42(3):514–533.
424 Handbook of Discrete-Valued Time Series
Pedeli, X. and Karlis, D. (2013c). Some properties of multivariate INAR(1) processes. Computational
Statistics and Data Analysis, 67:213–225.
Quoreshi, A. (2006). Bivariate time series modeling of nancial count data. Communications in
Statistics—Theory and Methods, 35(7):1343–1358.
Quoreshi, A. (2008). A vector integer-valued moving average model for high frequency nancial
count data. Economics Letters, 101(3):258–261.
Ristic, M., Nastic, A., Jayakumar, K., and Bakouch, H. (2012). A bivariate INAR(1) time series model
with geometric marginals. Applied Mathematics Letters, 25(3):481–485.
Scotto, M. G., Weiß, C. H., Silva, M. E., and Pereira, I. (2014). Bivariate binomial autoregressive
models. Journal of Multivariate Analysis, 125:233–251.
Sklar, M. (1959). Fonctions de répartition à n dimensions et leurs marges. l’Institut de Statistique de
Paris., 8:229–231.
Sofronas, G. (2012). Bayesian estimation for bivariate integer autoregressive model. Unpublished
Master thesis, Department of Statistics, Athens University of Economics, Athens, Greece.
Steutel, F. and van Harn, K. (1979). Discrete analogues of self–decomposability and stability. The
Annals of Probability, 7:893–899.
Trostheim, D. (2012). Some recent theory for autoregressive count time series. Test, 21(3):413–438.
Varin, C. (2008). On composite marginal likelihoods. Advances in Statistical Analysis, 92(1):1–28.
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.
Weiß, C. H. (2008). Thinning operations for modeling time series of counts—a survey. AStA Advances
in Statistical Analysis, 92(3):319–341.
Weiß, C. H. (2009). Modelling time series of counts with overdispersion. Statistical Methods and
Applications, 18(4):507–519.
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.