Useful models for time series of counts or simply wrong ones?

There has been a considerable and growing interest in low integer-valued time series data leading to a diversification of modelling approaches. In addition to static regression models, both observation-driven and parameter-driven models are considered here. We compare and contrast a variety of time series models for counts using two very different data sets as a testbed. A range of diagnostic devices is employed to help inform model adequacy. Special attention is paid to dynamic structure and underlying distributional assumptions including associated dispersion properties. Competing models show attractive features, but overall no one modelling approach is seen to dominate.

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

[2]  Roman Liesenfeld,et al.  Time series of count data: modeling, estimation and diagnostics , 2006, Comput. Stat. Data Anal..

[3]  Robert C. Jung,et al.  Convolution‐closed models for count time series with applications , 2011 .

[4]  W. Dunsmuir,et al.  On autocorrelation in a Poisson regression model , 2000 .

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

[6]  Benjamin Kedem,et al.  Regression models for time series analysis , 2002 .

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

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

[9]  M. Rosenblatt.  Remarks on a Multivariate Transformation , 1952 .

[10]  Thomas J. Tomberlin,et al.  A Hierarchical Bayes Approach to Estimation and Prediction for Time Series of Counts , 2003 .

[11]  A. Raftery,et al.  Probabilistic forecasts, calibration and sharpness , 2007 .

[12]  Richard A. Davis,et al.  A negative binomial model for time series of counts , 2009 .

[13]  Alain Latour,et al.  Integer‐Valued GARCH Process , 2006 .

[14]  Christian H. Weiß,et al.  Thinning operations for modeling time series of counts—a survey , 2008 .

[15]  Richard A. Davis,et al.  Maximum Likelihood Estimation for an Observation Driven Model for Poisson Counts , 2005 .

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

[17]  Jeffrey M. Wooldridge,et al.  Quasi‐Likelihood Methods for Count Data , 2008 .

[18]  R. Snyder,et al.  Feasible parameter regions for alternative discrete state space models , 2008 .

[19]  A. P. Dawid,et al.  Present position and potential developments: some personal views , 1984 .

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

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

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

[23]  S. Zeger,et al.  Markov regression models for time series: a quasi-likelihood approach. , 1988, Biometrics.

[24]  Kurt Brännäs,et al.  Time series count data regression , 1994 .

[25]  W. Dunsmuir,et al.  Observation-driven models for Poisson counts , 2003 .

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

[27]  Konstantinos Fokianos,et al.  Truncated Poisson Regression for Time Series of Counts , 2001 .

[28]  Ruijun Bu,et al.  Model Selection, Estimation and Forecasting in INAR(p) Models: , 2008 .

[29]  Robert C. Jung,et al.  Binomial thinning models for integer time series , 2006 .

[30]  Andrew Harvey,et al.  Time Series Models for Count or Qualitative Observations , 1989 .

[31]  B. Efron.  Double Exponential Families and Their Use in Generalized Linear Regression , 1986 .

[32]  Siem Jan Koopman,et al.  Time Series Analysis of Non-Gaussian Observations Based on State Space Models from Both Classical and Bayesian Perspectives , 1999 .

[33]  Andréas Heinen,et al.  Modelling Time Series Count Data: An Autoregressive Conditional Poisson Model , 2003 .

[34]  A. Kuk,et al.  The monte carlo newton-raphson algorithm , 1997 .

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

[36]  A. Raftery,et al.  Strictly Proper Scoring Rules, Prediction, and Estimation , 2007 .

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

[38]  Roman Liesenfeld,et al.  Estimating time series models for count data using efficient importance sampling , 2001 .

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

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

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

[42]  Man-Suk Oh,et al.  Bayesian analysis of time series Poisson data , 2001 .

[43]  L. Fahrmeir,et al.  Multivariate statistical modelling based on generalized linear models , 1994 .

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

[45]  Robert C. Jung,et al.  Dynamic Factor Models for Multivariate Count Data: An Application to Stock-Market Trading Activity , 2008 .

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

[47]  Jeffrey R. Russell,et al.  Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data , 1998 .

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

[49]  Christian H. Weiß,et al.  Modelling time series of counts with overdispersion , 2009, Stat. Methods Appl..

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

[51]  K. Chan,et al.  Monte Carlo EM Estimation for Time Series Models Involving Counts , 1995 .

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

[53]  Neil Shephard,et al.  Generalized linear autoregressions , 1995 .

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

[55]  T. Bollerslev,et al.  Generalized autoregressive conditional heteroskedasticity , 1986 .