Modelling Time Series Count Data: An Autoregressive Conditional Poisson Model

Heinen, Andreas (2003): Modelling Time Series Count Data: An Autoregressive Conditional Poisson Model.

Abstract

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.

• Item Type: MPRA Paper
• Language: English
• Keywords: Forecast; volatility; transactions data
• Subjects: G - Financial Economics > G1 - General Financial Markets; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C53 - Forecasting and Prediction Methods ; Simulation Methods; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C25 - Discrete Regression and Qualitative Choice Models ; Discrete Regressors ; Proportions ; Probabilities

References

Bollerslev, Tim, 1986, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 52, 5{59. , Robert F. Engle, and Daniel B. Nelson, 1994, Arch models, in Robert F. Engle, and Daniel L. McFadden, ed.: Handbook of Econometrics, Volume 4 (Elsevier Science: Amsterdam, North-Holland). BrÄannÄas, Kurt, and Per Johansson, 1994, Time series count data regression, Communica- tions in Statistics: Theory and Methods 23, 2907{2925. Cameron, A. Colin, and Pravin K. Trivedi, 1998, Regression Analysis of Count Data (Cam- bridge University Press: Cambridge). Cameron, Colin A., and Pravin K. Trivedi, 1996, Count data models for ¯nancial data, in Maddala G.S., and C.R. Rao, ed.: Handbook of Statistics, Volume 14, Statistical Methods in Finance (Elsevier Science: Amsterdam, North-Holland). Campbell, M.J., 1994, Time series regression for counts: an investigation into the relation- ship between sudden infant death syndrome and environmental temperature, Journal of the Royal Statistical Society A 157, 191{208. Chang, Tiao J., M.L. Kavvas, and J.W. Delleur, 1984, Daily precipitation modeling by discrete autoregressive moving average processes, Water Resources Research 20, 565{ 580. Davis, Richard A., William Dunsmuir, and YinWang, 2000, On autocorrelation in a poisson regression model, Biometrika 87, 491{505. Diebold, Francis X., Todd A. Gunther, and Anthony S. Tay, 1998, Evaluating density fore- casts with applications to ¯nancial risk management, International Economic Review 39, 863{883. Efron, Bradley, 1986, Double exponential families and their use in generalized linear regres- sion, Journal of the American Statistical Association 81, 709{721. Engle, Robert, 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. in°ation, Econometrica 50, 987{1008. Engle, Robert F., and Je®rey Russell, 1998, Autoregressive conditional duration: A new model for irregularly spaced transaction data, Econometrica 66, 1127,1162. Fahrmeir, Ludwig, and Gerhard Tutz, 1994, Multivariate Statistical Modeling Based on Generalized Linear Models (Springer-Verlag: New York). Gurmu, Shiferaw, and Pravin K. Trivedi, 1993, Variable augmentation speci¯cation tests in the exponential family, Econometric Theory 9, 94{113. Harvey, A.C., and C. Fernandes, 1989, Time series models for count or qualitative obser- vations, Journal of Business and Economic Statistics 7, 407{417. Johansson, Per, 1996, Speed limitation and motorway casualties: a time series count data regression approach, Accident Analysis and Prevention 28, 73{87. Jorgensen, Bent, Soren Lundbye-Christensen, Peter Xue-Kun Song, and Li Sun, 1999, A state space model for multivariate longitudinal count data, Biometrika 96, 169{181. Lee, Charles M., and Mark J. Ready, 1991, Inferring trade direction from intraday data, Journal of Finance 66, 733{746. MacDonald, Iain L., and Walter Zucchini, 1997, Hidden Markov and Other Models for Discrete-valued Time Series (Chapman and Hall: London). McKenzie, Ed., 1985, Some simple models for discrete variate time series, Water Resources Bulletin 21, 645{650. Rydberg, Tina H., and Neil Shephard, 1998, A modeling framework for the prices and times of trades on the nyse, To appear in Nonlinear and nonstationary signal processing edited by W.J. Fitzgerald, R.L. Smith, A.T. Walden and P.C. Young. Cambridge University Press, 2000. , 1999a, Dynamics of trade-by-trade movements decomposition and models,Working paper, Nu±eld College, Oxford. , 1999b, Modelling trade-by-trade price movements of multiple assets using multi- variate compound poisson processes, Working paper, Nu±eld College, Oxford. Zeger, Scott L., 1988, A regression model for time series of counts, Biometrika 75, 621{629. , and Bahjat Qaqish, 1988, Markov regression models for time series: A quasi- likelihood approach, Biometrics 44, 1019{1031.