Modelling Time Series Count Data: An Autoregressive Conditional Poisson Model

This paper introduces and evaluates new models for time series count data. The Autoregressive Conditional Poisson model (ACP) makes it possible to deal with issues of discreteness, overdispersion (variance greater than the mean) and serial correlation. A fully parametric approach is taken and a marginal distribution for the counts is specified, where conditional on past observations the mean is autoregressive. This enables to attain improved inference on coefficients of exogenous regressors relative to static Poisson regression, which is the main concern of the existing literature, while modelling the serial correlation in a flexible way. A variety of models, based on the double Poisson distribution of Efron (1986) is introduced, which in a first step introduce an additional dispersion parameter and in a second step make this dispersion parameter time-varying. All models are estimated using maximum likelihood which makes the usual tests available. In this framework autocorrelation can be tested with a straightforward likelihood ratio test, whose simplicity is in sharp contrast with test procedures in the latent variable time series count model of Zeger (1988). The models are applied to the time series of monthly polio cases in the U.S between 1970 and 1983 as well as to the daily number of price change durations of .75$ on the IBM stock. A .75$ price-change duration is defined as the time it takes the stock price to move by at least .75$. The variable of interest is the daily number of such durations, which is a measure of intradaily volatility, since the more volatile the stock price is within a day, the larger the counts will be. The ACP models provide good density forecasts of this measure of volatility.

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

[2]  Anthony S. Tay,et al.  Evaluating Density Forecasts , 1997 .

[3]  Charles M. C. Lee,et al.  Inferring Trade Direction from Intraday Data , 1991 .

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

[5]  T. J. Chang,et al.  Daily Precipitation Modeling by Discrete Autoregressive Moving Average Processes , 1984 .

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

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

[8]  Tina Hviid Rydberg,et al.  Modelling Trade-by-Trade Price Movements of Multiple Assets Using Multivariate Compound Poisson Processes , 1999 .

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

[10]  Michael J. Campbell,et al.  Time Series Regression for Counts: An Investigation into the Relationship between Sudden Infant Death Syndrome and Environmental Temperature , 1994 .

[11]  Anthony S. Tay,et al.  Evaluating Density Forecasts with Applications to Financial Risk Management , 1998 .

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

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

[14]  P Johansson,et al.  Speed limitation and motorway casualties: a time series count data regression approach. , 1996, Accident; analysis and prevention.

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

[16]  B. Jørgensen,et al.  A state-space model for multivariate longitudinal count data , 1999 .

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

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

[19]  Calyampudi R. Rao,et al.  Count Data Models for Financial Data , 1996 .

[20]  P. Trivedi,et al.  Variable Augmentation Specification Tests in the Exponential Family , 1993, Econometric Theory.

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

[22]  R. Engle.  Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation , 1982 .