Advertisement

Mathematical Methods of Statistics

, Volume 27, Issue 2, pp 119–144 | Cite as

A Test of Correlation in the Random Coefficients of an Autoregressive Process

Article

Abstract

A random coefficient autoregressive process in which the coefficients are correlated is investigated. First we look at the existence of a strictly stationary causal solution, we give the second-order stationarity conditions and the autocorrelation function of the process. Then we study some asymptotic properties of the empirical mean and the usual estimators of the process, such as convergence, asymptotic normality and rates of convergence, supplied with appropriate assumptions on the driving perturbations. Our objective is to get an overview of the influence of correlated coefficients in the estimation step through a simple model. In particular, the lack of consistency is shown for the estimation of the autoregressive parameter when the independence hypothesis in the random coefficients is violated. Finally, a consistent estimation is given together with a testing procedure for the existence of correlation in the coefficients. While convergence properties rely on ergodicity, we use a martingale approach to reach most of the results.

Keywords

RCAR process,MAprocess random coefficients least squares estimation stationarity ergodicity symptotic normality autocorrelation 

2010 Mathematics Subject Classification

62M10 62F03 62F12 60G42 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Andĕl, “Autoregressive Series with Random Parameters”, Math. Operationsforsch. Statist. 7 (5), 735–741 (1976).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    A. Aue and L. Horváth, “Quasi-Likelihood Estimation in Stationary and Nonstationary Autoregressive Models with Random Coefficients”, Statist. Sinica. 21, 973–999 (2011).MathSciNetMATHGoogle Scholar
  3. 3.
    A. Aue L. Horváth, and J. Steinebach, “Estimation in Random Coefficient AutoregressiveModels”, J. Time Ser. Anal. 27 (1), 61–76 (2006).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    I. Berkes L. Horváth, and S. Ling, “Estimation in Nonstationary Random Coefficient Autoregressive Models”, J. Time Ser. Anal. 30 (4), 395–416 (2009).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    P. Billingsley, “The Lindeberg–Lévy TheoremforMartingales”, Proc. Amer. Math. Soc. 12, 788–792 (1961).MathSciNetMATHGoogle Scholar
  6. 6.
    A. Brandt, “The Stochastic Equation Y N+1 = A NYN + B N with Stationary Coefficients”, Adv. Appl. Probab. 18, 211–220 (1986).CrossRefGoogle Scholar
  7. 7.
    P. J. Brockwell and R. A. Davis, Time Series: Theory andMethods, 2nd ed. in Springer Series in Statistics (Springer-Verlag, New York, 1991).Google Scholar
  8. 8.
    F. Chaabane and F. Maaouia, “Théorèmes limites avec poids pour les martingales vectorielles”, ESAIM Probab. Statist. 4, 137–189 (2000).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    M. Duflo, Random IterativeModels, in Applications of Mathematics (Springer-Verlag, New York–Berlin, 1997), Vol. 34.Google Scholar
  10. 10.
    R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge–New York, Cambridge Univ, Press, 1985).CrossRefMATHGoogle Scholar
  11. 11.
    S. Y. Hwang and I. V. Basawa, “Explosive Random-Coefficient AR(1) Processes and Related Asymptotics for Least-Squares Estimation”, J. Time Ser. Anal. 26 (6), 807–824 (2005).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    S. Y. Hwang I. V. Basawa and T. Y. Kim, “Least Squares Estimation for Critical Random Coefficient First- Order Autoregressive Processes”, Statist. Probab. Lett. 76, 310–317 (2006).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    U. Jürgens, “The Estimation of a Random Coefficient AR(1) Process Under Moment Conditions”, Statist. Hefte 26, 237–249 (1985).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    A. Koubkovà, “First-Order Autoregressive Processes with Time-Dependent Random Parameters”, Kybernetika 18 (5), 408–414 (1982).MathSciNetMATHGoogle Scholar
  15. 15.
    D. F. Nicholls and B. G. Quinn, “The Estimation of Multivariate Random Coefficient Autoregressive Models”, J.Multivar.Anal. 11, 544–555 (1981).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    D. F. Nicholls and B. G. Quinn, “Multiple Autoregressive Models with Random Coefficients”, J.Multivar. Anal. 11, 185–198 (1981).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    D. F. Nicholls and B. G. Quinn, Random Coefficient Autoregressive Models: An Introduction, in Lecture Notes in Statistics (Springer-Verlag, New York, 1982), Vol. 11.Google Scholar
  18. 18.
    P. M. Robinson, “Statistical Inference for a Random Coefficient Autoregressive Model”, Scand. J. Statist. 5 (3), 163–168 (1978).MathSciNetGoogle Scholar
  19. 19.
    A. Schick, “n-Consistent Estimation in a Random Coefficient Autoregressive Model”, Austral. J. Statist. 38 (2), 155–160 (1996).MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    W. F. Stout, “The Hartman–Wintner Law of the Iterated Logarithm forMartingales”, Ann.Math. Statist. 41 (6), 2158–2160 (1970).CrossRefMATHGoogle Scholar
  21. 21.
    W. F. Stout, Almost Sure Convergence, in Probability and Mathematical Statistics (Academic Press, New York–London, 1974), Vol. 24.Google Scholar
  22. 22.
    M. Taniguchi and Y. Kakizawa, Asymptotic Theory of Statistical Inference for Time Series, in Springer Series in Statistics (Springer, New York, 2000).Google Scholar

Copyright information

© Allerton Press, Inc. 2018

Authors and Affiliations

  1. 1.Laboratoire Angevin de Recherche en Math. (LAREMA), CNRSUniv. d’Angers, Univ. Bretagne LoireAngersFrance
  2. 2.Laboratoire Manceau de Math.Le Mans Univ.Le MansFrance

Personalised recommendations