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

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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.

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References

  1. J. Andĕl, “Autoregressive Series with Random Parameters”, Math. Operationsforsch. Statist. 7 (5), 735–741 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Aue and L. Horváth, “Quasi-Likelihood Estimation in Stationary and Nonstationary Autoregressive Models with Random Coefficients”, Statist. Sinica. 21, 973–999 (2011).

    MathSciNet  MATH  Google Scholar 

  3. A. Aue L. Horváth, and J. Steinebach, “Estimation in Random Coefficient AutoregressiveModels”, J. Time Ser. Anal. 27 (1), 61–76 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  4. I. Berkes L. Horváth, and S. Ling, “Estimation in Nonstationary Random Coefficient Autoregressive Models”, J. Time Ser. Anal. 30 (4), 395–416 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Billingsley, “The Lindeberg–Lévy TheoremforMartingales”, Proc. Amer. Math. Soc. 12, 788–792 (1961).

    MathSciNet  MATH  Google Scholar 

  6. A. Brandt, “The Stochastic Equation Y N+1 = A NYN + B N with Stationary Coefficients”, Adv. Appl. Probab. 18, 211–220 (1986).

    Article  Google Scholar 

  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. F. Chaabane and F. Maaouia, “Théorèmes limites avec poids pour les martingales vectorielles”, ESAIM Probab. Statist. 4, 137–189 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Duflo, Random IterativeModels, in Applications of Mathematics (Springer-Verlag, New York–Berlin, 1997), Vol. 34.

  10. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge–New York, Cambridge Univ, Press, 1985).

    Book  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  13. U. Jürgens, “The Estimation of a Random Coefficient AR(1) Process Under Moment Conditions”, Statist. Hefte 26, 237–249 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Koubkovà, “First-Order Autoregressive Processes with Time-Dependent Random Parameters”, Kybernetika 18 (5), 408–414 (1982).

    MathSciNet  MATH  Google Scholar 

  15. D. F. Nicholls and B. G. Quinn, “The Estimation of Multivariate Random Coefficient Autoregressive Models”, J.Multivar.Anal. 11, 544–555 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  16. D. F. Nicholls and B. G. Quinn, “Multiple Autoregressive Models with Random Coefficients”, J.Multivar. Anal. 11, 185–198 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  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.

  18. P. M. Robinson, “Statistical Inference for a Random Coefficient Autoregressive Model”, Scand. J. Statist. 5 (3), 163–168 (1978).

    MathSciNet  Google Scholar 

  19. A. Schick, “n-Consistent Estimation in a Random Coefficient Autoregressive Model”, Austral. J. Statist. 38 (2), 155–160 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  20. W. F. Stout, “The Hartman–Wintner Law of the Iterated Logarithm forMartingales”, Ann.Math. Statist. 41 (6), 2158–2160 (1970).

    Article  MATH  Google Scholar 

  21. W. F. Stout, Almost Sure Convergence, in Probability and Mathematical Statistics (Academic Press, New York–London, 1974), Vol. 24.

  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 

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Authors and Affiliations

  1. Laboratoire Angevin de Recherche en Math. (LAREMA), CNRS, Univ. d’Angers, Univ. Bretagne Loire, Angers, France

    F. Proïa

  2. Laboratoire Manceau de Math., Le Mans Univ., Le Mans, France

    M. Soltane

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  1. F. Proïa
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  2. M. Soltane
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Correspondence to F. Proïa.

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Proïa, F., Soltane, M. A Test of Correlation in the Random Coefficients of an Autoregressive Process. Math. Meth. Stat. 27, 119–144 (2018). https://doi.org/10.3103/S1066530718020035

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  • DOI: https://doi.org/10.3103/S1066530718020035

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