Abstract
We consider estimation in the class of first order conditional linear autoregressive models with discrete support that are routinely used to model time series of counts. Various groups of estimators proposed in the literature are discussed: moment-based estimators; regression-based estimators; and likelihood-based estimators. Some of these have been used previously and others not. In particular, we address the performance of new types of generalized method of moments estimators and propose an exact maximum likelihood procedure valid for a Poisson marginal model using backcasting. The small sample properties of all estimators are comprehensively analyzed using simulation. Three situations are considered using data generated with: a fixed autoregressive parameter and equidispersed Poisson innovations; negative binomial innovations; and, additionally, a random autoregressive coefficient. The first set of experiments indicates that bias correction methods, not hitherto used in this context to our knowledge, are some-times needed and that likelihood-based estimators, as might be expected, perform well. The second two scenarios are representative of overdispersion. Methods designed specifically for the Poisson context now perform uniformly badly, but simple, bias-corrected, Yule-Walker and least squares estimators perform well in all cases.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Al-Osh, M.A. and Aly, E.-E.A.A. (1992). First order autoregressive time series with negative binomial and geometric marginals.Comm. Statist. Theory Methods 21, 2483–2492.
Al-Osh, M.A. and Alzaid, A.A. (1987). First-order integer-valued autoregressive (INAR(1)) process.J. Time Ser. Anal. 8, 261–275.
Alzaid, A.A. and Al-Osh, M.A. (1988). First-order integer-valued autore-gressive (INAR(1)) process: distributional and regression properties.Statist. Neerlandica 42, 53–61.
Andrews, D.W.K. (1993). Exactly median-unbiased estimation of first-order autore gressive/unit-root models.Econometrica 61, 139–165.
Box, G.E.P.; Jenkins, G.M. and Reinsel, G.C. (1994).Time series analysis: forecasting and control (3rd. ed.). Prentice Hall, Englewood Cliffs.
Brännäs, K. (1994). Estimating and testing in integer-valued AR (1) models.Working Paper No. 335, Department of Economics, University of Umeå, Sweden.
Brännäs, K. and Hall, A. (2001). Estimation in integer-valued moving average models.Appl. Stoch. Models Bus. Ind. 17, 277–291.
Brännäs, K. and Hellström, J. (2001). Generalized integer-valued autoregression.Econometric Rev. 20, 425–433.
Cameron, A.C. and Trivedi, P.K. (1998).Regression analysis of count data. Cambridge University Press, Cambridge.
Freeland, K. (1998).Statistical analysis of discrete time series with application to the analysis of worker's compensation claims data. Ph.D. thesis, University of British Columbia.
Freeland, K. and McCabe, B.P.M. (2003a). Forecasting discrete valued low count time series.Internat. J. Forecasting, forthcoming.
Freeland, K. and McCabe, B.P.M. (2003b). Analysis of count data by means of the Poisson autoregressive model.J. Time Ser. Anal., forthcoming.
Godambe, V.P. (1960). An optimum property of regular maximum-likelihood estimation.Ann. of Math. Stat. 31, 1208–1211.
Greene, W.H. (2000).Econometric analysis (4th ed.). Prentice Hall, Upper Saddle River.
Grunwald, G.K., Hyndman, R., Tedesco, L. and Tweedie, R.L. (2000). Non-Gaussian conditional linear AR(1) models.Aust. N. Z. J. Stat. 42, 479–495.
Heyde, C.C. (1997).Quasi-likelihood and its applications. Springer, New York.
Heyde, C.C. and Lin, Y.-X. (1992). On quasi-likelihood methods and estimation for branching processes and heteroscedastic regression models.Aust. J. Stat. 34, 199–206.
Jung, R. (1999).Zeitreihenanalyse für Zähldaten. Eine Untersuchung ganzzahliger Autoregressiver-Moving-Average-Prozesse. Josef Eul Verlag, Lohmar.
Jung, R.C. and Tremayne, A.R. (2003). Testing for serial dependence in time series models of counts.J. Time Ser. Anal. 24, 65–84.
Kendall, M.G. (1954). Note on the bias in the estimation of autocorrelation.Biometrika 41, 403–404.
Kotz, S. and Johnson, N.L. (1982).Encyclopedia of statistical sciences. Volume 2. Wiley, New York.
MacKinnon, J.G. and Smith, A.A., Jr. (1998). Approximate bias correction in econometrics.J. Econometrics 85, 205–230.
Marriott, F.H.C. and Pope, J.A. (1954). Bias in the estimation of autocorrelations.Biometrika 41, 390–402.
McKenzie, E. (1986). Autoregressive moving-average processes with negative-binomial and geometric marginal distributions.Adv. in Appl. Probab. 18, 679–705.
McKenzie, E. (1988). Some ARMA models for dependent sequences of Poisson counts.Adv. in Appl. Probab. 20, 822–835.
Orcutt, G.H. and Winokur, H.S., Jr. (1969). First order autoregression: inference, estimation, and prediction.Econometrica 37, 1–14.
Park, Y. and Oh, C.W. (1997). Some asymptotic properties in INAR(1) processes with Poisson marginals.Statist. Papers 38, 287–302.
Shaman, P. and Stine, R.A. (1988). The bias of autoregressive coefficient estimators.J. Amer. Statist. Assoc. 83, 842–848.
Steutel, F.W. and Van Harn, K. (1979). Discrete analogues of self-decomposability and stability.Ann. Probab. 7, 893–899.
Venkataraman, K.N. (1982). A time series approach to the study of the simple subcritical Galton-Watson process with immigration.Adv. Appl. Probab. 14, 1–20.
Wei, C.Z. and Winnicki, J. (1989). Some asymptotic results for the branching process with immigration.Stochastic Process. Appl. 31, 261–282.
Winkelmann, R. (2000).Econometric analysis of count data (3rd ed.) Springer, Berlin.
Winnicki, J. (1988). Estimation theory for the branching process with immigration.Contemp. Math. 80, 301–322.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jung, R.C., Ronning, G. & Tremayne, A.R. Estimation in conditional first order autoregression with discrete support. Statistical Papers 46, 195–224 (2005). https://doi.org/10.1007/BF02762968
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02762968