×

Integer-valued GARCH process. (English) Zbl 1150.62046

The authors propose a simple model as an integer-valued analogue of the generalized autoregressive conditional heteroskedastic (GARCH(\(p,q\))) model with Poisson deviates. Putting particular emphasis to the case \(p=1, q=1\), it is shown, from a second-order point of view, that this integer-valued GARCH process is a standard ARMA(1,1) process. The problem of maximum likelihood estimation of the parameters is investigated and the asymptotic distribution of the estimators is derived. A numerical example and an application of this model to real time series are presented.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F10 Point estimation
62E20 Asymptotic distribution theory in statistics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1016/0304-4076(86)90063-1 · Zbl 0616.62119 · doi:10.1016/0304-4076(86)90063-1
[2] Brockwell P., Time Series: Theory and Methods, 2nd edn (1991) · Zbl 0709.62080 · doi:10.1007/978-1-4419-0320-4
[3] Cardinal M., Modelisation temporelle d’incidence de maladies (1995)
[4] DOI: 10.1093/biomet/90.4.777 · Zbl 1436.62418 · doi:10.1093/biomet/90.4.777
[5] Efron B., An Introduction to Bootstrap (1993) · doi:10.1007/978-1-4899-4541-9
[6] Franke J., Developments in Time Series Analysis pp 310– (1993) · doi:10.1007/978-1-4899-4515-0_22
[7] Gauthier G., Annales des Sciences Mathematiques du Quebec 18 pp 49– (1994)
[8] Gourieroux C., ARCH Models and Financial Applications. Springer Series in Statistics (1997) · doi:10.1007/978-1-4612-1860-9
[9] Graham R. L., Concrete Mathematics (1989)
[10] Hamilton J. D., Time Series Analysis (1994) · Zbl 0831.62061
[11] Hardy E., Modelisation de type ARCH pour series chronologiques a valeurs entieres (1996)
[12] Heinen A., Modeling Time Series Count Data: The Autoregressive Conditional Poisson Model (2001)
[13] Kedem B., Regression Models for Time Series Analysis. Wiley series in Probability and Statistics (2002) · Zbl 1011.62089 · doi:10.1002/0471266981
[14] MacDonald I. L., Hidden Markov and Other Models for Discrete-valued Time Series (1997) · Zbl 0868.60036
[15] T. H. Rydberg, and N. Shephard(2000 ) BIN Models for Trade-by-Trade Data. Modelling the Number of Trades in a Fixed Interval of Time . Technical report 0740, Econometric Society. Available at http://ideas.repec.org/p/ecm/wc2000/0740.html (accessed on 25th July 2006).
[16] Streett S. B., Some observaton driven models for time series (2000)
[17] DOI: 10.1016/0304-4076(82)90100-2 · doi:10.1016/0304-4076(82)90100-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.