Penalized Estimation for Integer Autoregressive Models



The integer autoregressive model of order p can be employed for the analysis of discrete–valued time series data. It can be shown, under some conditions, that its correlation structure is identical to that of the usual autoregressive process. The model is usually fitted by the method of least squares. However, consider an alternative estimation scheme, which is based on minimizing the least squares criterion subject to some constraints on the parameters of interest. The ridge type of constraints are used in this article and it is shown that under some reasonable conditions on the penalty parameter, the resulting estimates have lessmean square error than that of the ordinary least squares. A real data set and some limited simulations support further the results.


Time Series Analysis Regularization Parameter Penalty Parameter Ridge Regression Asymptotic Covariance Matrix 
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  1. Akaike, H. (1974). A new look at the statistical model identification, IEEE Transactions on Automatic Control AC–19: 716–723.CrossRefMathSciNetGoogle Scholar
  2. Al-Osh,M.A.&Alzaid, A. A. (1987). First-order integer-valued autoregressive (INAR(1)) process, Journal of Time Series Analysis 8: 261–275.MATHCrossRefMathSciNetGoogle Scholar
  3. Alzaid, A. A.&Al-Osh,M. (1990). An integer-valued pth-order autoregressive structure (INAR(p)) process, Journal of Applied Probability 27: 314–324.MATHCrossRefMathSciNetGoogle Scholar
  4. Brockwell, P. J. & Davis, R. A. (1991). Time Series: Data Analysis and Theory, 2nd edn, Springer, New York.MATHGoogle Scholar
  5. Du, J. G.&Li,Y. (1991). The integer-valued autoregressive INAR(p) model, Journal of Time Series Analysis 12: 129–142.MATHCrossRefMathSciNetGoogle Scholar
  6. Fahrmeir, L. & Kaufmann, H. (1985). Consistency and asymptotic Normality of the maximum likelihood estimates in generalized linear models, Annals of Statistics 13: 342–368.MATHCrossRefMathSciNetGoogle Scholar
  7. Fahrmeir, L. & Kaufmann, H. (1987). Regression Models for Nonstationary Categorical Time Series, Journal of Time Series Analysis 8: 147–160.MATHCrossRefMathSciNetGoogle Scholar
  8. Fahrmeir, L. & Tutz, G. (2001). Multivariate Statistical Modelling Based on Generalized Linear Models, 2nd edn, Springer, New York.MATHGoogle Scholar
  9. Ferland, R., Latour, A. & Oraichi, D. (2006). Integer–valued GARCH processes, Journal of Time Series Analysis Analysis 27: 923–942.MATHCrossRefMathSciNetGoogle Scholar
  10. Frank, I. E. & Friedman, J. H. (1993). A statistical view of some chemometrics regression tools, Technometrics 35: 109–148. (with discussion).MATHCrossRefGoogle Scholar
  11. Grunwald, G. K., Hyndman, R. J., Tedesco, L. & Tweedie, R. L. (2000). Non-Gaussian conditional linear AR(1) models, Australian & New Zealand Journal of Statistics 42: 479–495.MATHCrossRefMathSciNetGoogle Scholar
  12. Guttorp, P. (1991). Statistical Inference for Branching Processes, Wiley, New York.MATHGoogle Scholar
  13. Hall, P. & Heyde, C. C. (1980). Martingale Limit Theory and its Applications, Academic Press, New York.Google Scholar
  14. Hoerl, A. E.&Kennard, R.W. (1970a). Ridge regression:Applications to non–orthogonal problems, Technometrics 12: 69–82.MATHCrossRefGoogle Scholar
  15. Hoerl, A. E. & Kennard, R. W. (1970b). Ridge regression:Biased estimation for non–orthogonal problems, Technometrics 12: 55–67.MATHCrossRefGoogle Scholar
  16. Jung, R. C. & Tremayne, A. R. (2006). Coherent forecasting in integer time series models, International Journal of Forecasting 22: 223–238.CrossRefGoogle Scholar
  17. Kedem, B. & Fokianos, K. (2002). Regression Models for Time Series Analysis, Wiley, Hoboken, NJ.MATHCrossRefGoogle Scholar
  18. Klimko, L. A. & Nelson, P. I. (1978). On conditional least squares estimation for stochastic processes, The Annals of Statistics 6: 629–642.MATHCrossRefMathSciNetGoogle Scholar
  19. Knight, K. & Fu,W. (2000). Asymptotics for lasso–type estimators, Annals of Statistics 28: 1356– 1378.MATHCrossRefMathSciNetGoogle Scholar
  20. Latour,A. (1998). Existence and stochastic structure of a non-negative integer-valued autoregressive process, Journal of Time Series Analysis 19: 439–455.MATHCrossRefMathSciNetGoogle Scholar
  21. McKenzie, E. (1985). Some simplemodels for discrete variate time series,Water Resources Bulletin 21: 645–650.Google Scholar
  22. McKenzie, E. (1986). Autoregressive moving-average processes with negative-binomial and geometric marginal distributions, Advances in Applied Probability 18: 679–705.MATHCrossRefMathSciNetGoogle Scholar
  23. McKenzie, E. (1988). Some ARMA models for dependent sequences of Poisson counts, Advances in Applied Probability 20: 822–835.MATHCrossRefMathSciNetGoogle Scholar
  24. Neal, P. & Subba Rao, T. (2007). MCMC for integer–valued ARMA processes, Journal of Time Series Analysis 28: 92–100.MATHCrossRefMathSciNetGoogle Scholar
  25. Taniguchi, M. & Hirukawa, J. (2005). The Stein-James estimator for short- and long-memory Gaussian processes, Biometrika 92: 737–746.MATHCrossRefMathSciNetGoogle Scholar
  26. Tibshirani, R. (1996). Regression shrinkage and selection via the lasso, Journal of the Royal Statistical Society, Series B 58: 267–288.MATHMathSciNetGoogle Scholar
  27. Zheng, H., Basawa, I. V.&Datta, S. (2006). Inference for the pth–order random coefficient integer– valued process, Journal of Time Series Analysis 27: 411–440.MATHCrossRefMathSciNetGoogle Scholar
  28. Zhu, R. & Joe, H. (2006). Modelling count data time series with Markov processes based on binomial thinning, Journal of Time Series Analysis 27: 725–738.MATHCrossRefMathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Department of Mathematics & StatisticsUniversity of CyprusNicosiaCyprus

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