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The Strong Consistency of Quasi-Maximum Likelihood Estimators for p-order Random Coefficient Autoregressive (RCA) Models

Abstract

In this paper, we investigate the strong consistency of the quasi-maximum likelihood estimators derived through the Kalman filter for stationary random coefficient autoregressive (RCA) models. The estimators in question are the subject of Benmoumen et al. (2019) work. The suggested proof exploits both the ergodic theorem and Kalman filter asymptotic proprieties.

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References

  1. Allal, J. and Benmoumen, M. (2013). Parameter estimation for first-order Random Coefficient Autoregressive (RCA) Models based on Kalman Filter. Communications in Statistics-Simulation and Computation 42, 1750–1762.

    MathSciNet  MATH  Google Scholar 

  2. Anderson, B.D. and Moore, J.B. (2012). Optimal filtering. Courier Corporation.

  3. Aue, A. and Horváth, L. (2011). Quasi-likelihood estimation in stationary and nonstationary autoregressive models with random coefficients. Stat. Sin. 21, 973–999.

    MathSciNet  MATH  Google Scholar 

  4. Aue, A., Horvath, L. and Steinebach, J. (2006). Estimation in random coefficient autoregressive models. J. Time Ser. Anal. 27, 61–76.

    MathSciNet  Article  Google Scholar 

  5. Benmoumen, M., Allal, J. and Salhi, I. (2019). Parameter Estimation for p-Order Random Coefficient Autoregressive (RCA) Models Based on Kalman Filter. Journal of Applied Mathematics.

  6. Berkes, I., Horváth, L. and Ling, S. (2009). Estimation in nonstationary random coefficient autoregressive models. J. Time Ser. Anal. 30, 395–416.

    MathSciNet  Article  Google Scholar 

  7. Caines, P.E. and Mayne, D.Q. (1970). On the discrete time matrix Riccati equation of optimal control. Int. J. Control. 12, 785–794.

    MathSciNet  Article  Google Scholar 

  8. Ghosh, D. (1989). Maximum likelihood estimation of the dynamic shock-error model. J. Econ. 41, 121–143.

    MathSciNet  Article  Google Scholar 

  9. Hamilton, J.D. (1994). Time Series Analysis. Princeton University Press, Princeton.

    Book  Google Scholar 

  10. Hautus, M.L.J. (1970). Stabilization controllability and observability of linear autonomous systems, 73, North-Holland, p. 448–455.

  11. Harvey, A.C. (1990). Forecasting, structural time series models and the Kalman filter. Cambridge university press.

  12. Hewer, G.A. (1973). Analysis of a discrete matrix Riccati equation of linear control and Kalman filtering. Journal of Mathematical Analysis and applications42, 226–236.

    MathSciNet  Article  Google Scholar 

  13. Jazwinski, A.H. (1970). Stochastic processes and filtering theory.

  14. Kalman, R. and Bertram, J. (1959). Control system analysis and design via the second method of Lyapunov:(I) continuous-time systems (II) discrete time systems. IRE Trans. Autom. Control. 4, 112–112.

    Article  Google Scholar 

  15. Kalman, R.E. (1960). A new approach to linear filtering and prediction problems.

  16. Ljung, L. (1978). Convergence analysis of parametric identification methods. IEEE Transactions on Automatic Control 23, 770–783.

    MathSciNet  Article  Google Scholar 

  17. Nicholls, D.F. and Quinn, B.G. (1982). Random Coefficient Autoregressive Models: An Introduction. Springer, New York.

    Book  Google Scholar 

  18. Pagan, A. (1980). Some identification and estimation results for regression models with stochastically varying coefficients. J. Econ. 13, 341–363.

    MathSciNet  Article  Google Scholar 

  19. Rissanen, J. and Caines, P. E. (1979). The strong consistency of maximum likelihood estimators for ARMA processes. Ann. Stat. 7, 297–315.

    MathSciNet  Article  Google Scholar 

  20. Tjøstheim, D. (1986). Estimation in nonlinear time series models. Stoch. Process. Appl. 21, 251–273.

    MathSciNet  Article  Google Scholar 

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Correspondence to Mohammed Benmoumen.

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Benmoumen, M., Salhi, I. The Strong Consistency of Quasi-Maximum Likelihood Estimators for p-order Random Coefficient Autoregressive (RCA) Models. Sankhya A (2021). https://doi.org/10.1007/s13171-021-00269-w

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Keywords

  • RCA models
  • Maximum likelihood
  • Kalman filter
  • Strong consistency.

AMS (2000) subject classification

  • Primary 62M10
  • Secondary 62F12