Convolution‐closed models for count time series with applications

There has recently been an upsurge of interest in time series models for count data. Many papers focus on the model with first-order (Markov) dependence and Poisson innovations. Our paper considers practical models that can capture higher-order dependence based on the work of Joe (1996). In this framework we are able to model both equidispersed and overdispersed marginal distributions of data. The latter is approached using generalized Poisson innovations. Central to the models is the use of the property of closure under convolution of certain families of random variables. The models can be thought of as stationary Markov chains of finite order. Parameter estimation is undertaken by maximum likelihood, inference procedures are considered and means of assessing model adequacy employed. Applications to two new data sets are provided.

[1]  Brendan McCabe,et al.  Analysis of Count Data by means of the Poisson Autoregressive Model. , 2004 .

[2]  Ruijun Bu,et al.  Maximum likelihood estimation of higher‐order integer‐valued autoregressive processes , 2008 .

[3]  Mohamed Alosh,et al.  An integer-valued pth-order autoregressive structure (INAR(p)) process , 1990, Journal of Applied Probability.

[4]  Li Yuan,et al.  THE INTEGER‐VALUED AUTOREGRESSIVE (INAR(p)) MODEL , 1991 .

[5]  Claudia Czado,et al.  Predictive Model Assessment for Count Data , 2009, Biometrics.

[6]  Robert C. Jung,et al.  Useful models for time series of counts or simply wrong ones? , 2011 .

[7]  K. N. Venkataraman.  A time series approach to the study of the simple subcritical Galton–Watson process with immigration , 1982, Advances in Applied Probability.

[8]  Andrew Harvey,et al.  Forecasting, Structural Time Series Models and the Kalman Filter , 1990 .

[9]  C. Gouriéroux,et al.  Likelihood Ratio Test, Wald Test, and Kuhn-Tucker Test in Linear Models with Inequality Constraints on the Regression Parameters , 1982 .

[10]  Gael M. Martin,et al.  Optimal Probabilistic Forecasts for Counts , 2009 .

[11]  Ruey S. Tsay,et al.  Model Checking Via Parametric Bootstraps in Time Series Analysis , 1992 .

[12]  Benjamin Kedem,et al.  Regression Models for Time Series Analysis: Kedem/Time Series Analysis , 2005 .

[13]  Harry Joe.  TIME SERIES MODELS WITH UNIVARIATE MARGINS IN THE CONVOLUTION-CLOSED INFINITELY DIVISIBLE CLASS , 1996 .

[14]  David Harris,et al.  Efficient probabilistic forecasts for counts , 2011 .

[15]  B. McCabe,et al.  Analysis of low count time series data by poisson autoregression , 2004 .

[16]  Gerd Ronning,et al.  Estimation in conditional first order autoregression with discrete support , 2005 .

[17]  J. Franke,et al.  Conditional maximum likelihood estimates for INAR(1) processes and their application to modelling epileptic seizure counts , 1993 .

[18]  Satishs Iyengar,et al.  Multivariate Models and Dependence Concepts , 1998 .

[19]  J. MacKinnon,et al.  Estimation and inference in econometrics , 1994 .

[20]  Lain L. MacDonald,et al.  Hidden Markov and Other Models for Discrete- valued Time Series , 1997 .

[21]  Bootstrapping the autocorrelation coefficient of finite Markov chains , 1992 .

[22]  Ed. McKenzie,et al.  SOME SIMPLE MODELS FOR DISCRETE VARIATE TIME SERIES , 1985 .

[23]  Pravin K. Trivedi,et al.  Regression Analysis of Count Data , 1998 .

[24]  A. W. Kemp,et al.  Generalized Poisson Distributions: Properties and Applications. , 1992 .

[25]  Eddie McKenzie,et al.  Discrete variate time series , 2003 .

[26]  Eugene Seneta,et al.  ESTIMATION THEORY FOR GROWTH AND IMMIGRATION RATES IN A MULTIPLICATIVE PROCESS , 1972 .

[27]  P. Neal,et al.  Efficient order selection algorithms for integer‐valued ARMA processes , 2009 .

[28]  E. Bolthausen.  The Berry-Esseen theorem for functionals of discrete Markov chains , 1980 .

[29]  E. McKenzie,et al.  Some ARMA models for dependent sequences of poisson counts , 1988, Advances in Applied Probability.

[30]  Michael J. LuValle,et al.  Generalized Poisson Distributions: Properties and Applications , 1990 .

[31]  Dimitris Karlis,et al.  Maximum Likelihood Estimation for Integer Valued Time Series Models , 2001 .

[32]  S. Zeger.  A regression model for time series of counts , 1988 .

[33]  Fw Fred Steutel,et al.  Discrete analogues of self-decomposability and stability , 1979 .

[34]  A. Alzaid,et al.  Some autoregressive moving average processes with generalized Poisson marginal distributions , 1993 .

[35]  J. T. Wulu,et al.  Regression analysis of count data , 2002 .

[36]  Jonas Andersson,et al.  Treating Missing Values in INAR(1) Models: An Application to Syndromic Surveillance Data , 2010 .

[37]  P. C. Consul,et al.  A new urn model with predetermined strategy , 1975 .

[38]  Patrick Billingsley,et al.  Statistical inference for Markov processes , 1961 .

[39]  Robert C. Jung,et al.  Coherent forecasting in integer time series models , 2006 .

[40]  Mohamed Alosh,et al.  FIRST‐ORDER INTEGER‐VALUED AUTOREGRESSIVE (INAR(1)) PROCESS , 1987 .

[41]  Stefan Frey,et al.  The Impact of Iceberg Orders in Limit Order Books , 2009 .

[42]  Bonnie K. Ray,et al.  Regression Models for Time Series Analysis , 2003, Technometrics.

[43]  Pravin K. Trivedi,et al.  Regression Analysis of Count Data , 1998 .

[44]  T. N. Sriram.  Invalidity of Bootstrap for Critical Branching Processes with Immigration , 1994 .

[45]  Bas J. M. Werker,et al.  Efficient estimation of auto‐regression parameters and innovation distributions for semiparametric integer‐valued AR(p) models , 2007 .

[46]  Rob J Hyndman,et al.  Theory & Methods: Non‐Gaussian Conditional Linear AR(1) Models , 2000 .

[47]  F. Famoye,et al.  Bivariate generalized Poisson distribution with some applications , 1995 .