Hostname: page-component-7479d7b7d-k7p5g Total loading time: 0 Render date: 2024-07-10T15:55:06.287Z Has data issue: false hasContentIssue false
  • English
  • Français

The Multivariate Ginar(p) Process

Part of: Inference from stochastic processes Multivariate analysis

Published online by Cambridge University Press:  01 July 2016

Alain Latour
Alain Latour*
Affiliation:
Université du Québec à Montréal
*
∗ Postal address: Université du Québec à Montréal, Département de mathématiques, Case postale 8888, succursale Centre-Ville, Montréal, Québec, H3C 3P8, Canada.
Get access Rights & Permissions [Opens in a new window]

Abstract

A criterion is given for the existence of a stationary and causal multivariate integer-valued autoregressive process, MGINAR(p). The autocovariance function of this process being identical to the autocovariance function of a standard Gaussian MAR(p), we deduce that the MGINAR(p) process is nothing but a MAR(p) process. Consequently, the spectral density is directly found and gives good insight into the stochastic structure of a MGINAR(p). The estimation of parameters of the model, as well as the forecasting of the series, is discussed.

Keywords

MULTIVARIATE INTEGER-VALUED TIME SERIESAUTOREGRESSIVE MODELMULTIVARIATE MARTINGALEASYMPTOTIC DISTRIBUTION

MSC classification

Primary: 62M10: Time series, auto-correlation, regression, etc.
Secondary: 62H12: Estimation
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 1 , March 1997 , pp. 228 - 248
Copyright
Copyright © Applied Probability Trust 1997 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported by the Natural Sciences and Engineering Research Council of Canada (NSERCC).

References

Al-Osh, M. A. and Aly, E. E. (1992) First-order autoregressive time series with negative binomial and geometric marginals. Commun. Statist. Theory Meth. 21, 2483–2492.Google Scholar
Al-Osh, M. A. and Alzaid, A. A. (1987) First-order integer-valued autoregressive (INAR(l)) process. J. Time Series Anal. 8, 261–275.Google Scholar
Al-Osh, M. A. and Alzaid, A. A. (1988) Integer-valued moving average (INMA) process. Stat. Hefte 29, 281–300.Google Scholar
Al-Osh, M. A. and Alzaid, A. A. (1991). Binomial autoregressive moving average models. Commun. Statist. Stoch. Models 7, 261–282.Google Scholar
Alzaid, A. A. and Al-Osh, M. A. (1990) An integer-valued pth order autoregressive structure (INAR(p)) process. J. Appl. Prob. 27, 314–324.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Brockwell, P. J. and Davis, R. A. (1991) Time Series: Theory and Methods. 2nd edn. Springer, New York.Google Scholar
Dion, J.-P., Gauthier, G. and Latour, A. (1995) Branching processes with immigration and integer-valued time series. Serdica Math. J. 21, 123–136.Google Scholar
Du, J.-G. and Li, Y. (1991) The integer-valued autoregressive (INAR(p)) model. J. Time Series Anal. 12, 129–142.Google Scholar
Gauthier, G. (1991) Modèles de type autorégressif pour les séries chronologiques à valeurs entières non négatives. Mémoire de maîtrise. Univ. du Québec à Montréal.Google Scholar
Gauthier, G. and Latour, A. (1994) Convergence forte des estimateurs des paramètres d'un processus GENAR(p). Ann. Sc. Math. Québec 18, 37–59.Google Scholar
Graybill, F. A. (1983) Matrices with Applications in Statistics. Wadsworth, Belmont, CA.Google Scholar
Hannan, E. J. (1970) Multiple Time Series. Wiley, New York.Google Scholar
Heyde, C. C. and Seneta, E. (1972) Estimation theory for growth and immigration rates in a multiplicative process. J. Appl. Prob. 9, 235–256.Google Scholar
Jacobs, P. A. and Lewis, P. A. W. (1978a) Discrete time series generated by mixture I: correlational and runs properties J. R. Statist. Soc. B 40, 94–105.Google Scholar
Jacobs, P. A. and Lewis, P. A. W. (1978b) Discrete time series generated by mixture II: asymptotic properties. J. R. Statist. Soc. B 40, 222–228.Google Scholar
Jacobs, P. A. and Lewis, P. A. W. (1978c) Discrete time series generated by mixture III: autoregressive processes (DAR(p)). Tech. report no NPS55-78-022. Naval Postgraduate School, Monterey, CA.Google Scholar
Jacobs, P. A. and Lewis, P. A. W. (1983) Stationary discrete autoregressive-moving average time series generated by mixtures. J. Time Series Anal. 4, 19–36.Google Scholar
Klimko, L. A. and Nelson, P. I. (1978) On conditional least squares estimation for stochastic processes. Ann. Statist. 6, 629–642.Google Scholar
Latour, A. (1994) Existence and stochastic structure of a non-negative integer-valued autoregressive process. Tech. report no 212. Université du Québec à Montréal.Google Scholar
Lewis, P. A. W. (1980) Simple models for positive-valued and discrete-valued time series with ARMA correlation structure. In Multivariate Analysis. ed. Krishnaiah, V. P. R.. North Holland, Amsterdam. pp. 151–166.Google Scholar
Lütkepohl, H. (1991) Introduction to Multiple Time Series Analysis. Springer, New York.Google Scholar
Mckenzie, E. (1981) Extending the correlation structure of exponential autoregressive-moving-average processes. J. Appl. Prob. 18, 181–189.Google Scholar
Mckenzie, E. (1985) Some simple models for discrete variate time series. Water Res. Bullet. 21, 645–650.CrossRefGoogle Scholar
Mckenzie, E. (1986) Autoregressive moving average processes with negative binomial and geometric marginal distributions. Adv. Appl. Prob. 18, 679–705.Google Scholar
Mckenzie, E. (1988a) Some ARMA models for dependent sequences of Poisson counts. Adv. Appl. Prob. 20, 822–835.Google Scholar
Mckenzie, E. (1988b) The distributional structure of finite moving-average processes. J. Appl. Prob. 25, 313–321.Google Scholar
Priestley, M. B. (1981) Spectral Analysis and Time Series. Academic Press, New York.Google Scholar
Reinsel, G. C. (1993) Elements of Multivariate Time Series Analysis. Springer, New York.Google Scholar
Seneta, E. (1981) Non-Negative Matrices and Markov Chains. Springer, New York.Google Scholar
Steutel, F. W. and Van Harn, K. (1979) Discrete analogues of self-decomposability and stability. Ann. Prob. 7, 893–899.CrossRefGoogle Scholar
Wang, Z.-K. (1982) Stochastic Processes. Scientific Press, Beijing.Google Scholar