Kybernetika 54 no. 2, 375-399, 2018

QMLE of periodic bilinear models and of PARMA models with periodic bilinear innovations

Abdelouahab Bibi and Ahmed GhezalDOI: 10.14736/kyb-2018-2-0375

Abstract:

This paper develops an asymptotic inference theory for bilinear $\left( BL\right) $ time series models with periodic coefficients $\left( PBL\text{ for short}\right) $. For this purpose, we establish firstly a necessary and sufficient conditions for such models to have a unique stationary and ergodic solutions (in periodic sense). Secondly, we examine the consistency and the asymptotic normality of the quasi-maximum likelihood estimator $\left( QMLE\right) $ under very mild moment condition for the innovation errors. As a result, it is shown that whenever the model is strictly stationary, the moment of some positive order of $PBL$ model exists and is finite, under which the strong consistency and asymptotic normality of $QMLE$ for $PBL$ are proved. Moreover, we consider also the periodic $ARMA$ $\left( PARMA\right) $ models with $PBL$ innovations and we prove the consistency and the asymptotic normality of its $QMLE.$

Keywords:

asymptotic normality, periodic bilinear model, periodic $ARMA$ model, strict and second-order periodic stationarity, strong consistency

Classification:

2M10, 62M15

paper.pdf

References:

  1. A. Aknouche and A. Bibi: Quasi-maximum likelihood estimation of periodic $GARCH$ and periodic $ARMA-GARCH$ processes. J. Time Ser. Anal. 30 (2008), 1, 19-46.   DOI:10.1111/j.1467-9892.2008.00598.x
  2. A. Aknouche and H. Guerbyenne: Periodic stationarity of random coefficient periodic autoregressions. Statist. Probab. Lett. 79 (2009), 7, 990-996.   DOI:10.1016/j.spl.2008.12.012
  3. A. Aknouche and A. Bibi: Quasi-maximum likelihood estimation of periodic $GARCH$ and periodic $ARMA-GARCH$ processes. J. Time Ser. Anal. 30 (2008), 1, 19-46.   DOI:10.1111/j.1467-9892.2008.00598.x
  4. I. V. Basawa and R. Lund: Large sample properties of parameters estimates for periodic $ARMA$ models. J. Time Series Anal. 22 (2001), 6, 651-663.   DOI:10.1111/1467-9892.00246
  5. A. Bibi: On the covariance structure of time-varying bilinear models. Stochastic Anal. App. 21 (2003), 1, 25-60.   DOI:10.1081/sap-120017531
  6. A. Bibi and C. Francq: Consistent and asymptotically normal estimators for cyclically time-dependent linear models. Ann. Inst. Statist. Math. 55 (2003), 1, 41-68.   DOI:10.1007/bf02530484
  7. A. Bibi and A. J. Oyet: Estimation of some bilinear time series models with time-varying coefficients. Stochastic Anal. Appl. 22 (2004), 2, 355-376.   DOI:10.1081/sap-120028595
  8. A. Bibi and A. Aknouche: Yule-Walker type estimators in periodic bilinear models: strong consistency and asymptotic normality. Statist. Methods Appl. 19 (2010), 1, 1-30.   DOI:10.1007/s10260-008-0110-z
  9. A. Bibi and R. Lessak: On stationarity and $\beta$-mixing of periodic bilinear processes. Statist. Probab. Lett. 79 (2009), 1, 79-87.   DOI:10.1016/j.spl.2008.07.024
  10. A. Bibi and I. Lescheb: On general periodic time-varying bilinear processes. Econom. Lett. 114 (2012), 3, 353-357.   DOI:10.1016/j.econlet.2011.11.013
  11. A. Bibi and A. Ghezal: On periodic time-varying bilinear processes: Structure and asymptotic inference. Stat. Methods Appl. 25 (2016), 3, 395-420.   DOI:10.1007/s10260-015-0344-5
  12. P. Billingsley: Probability and Measure. Third edition. Wiley - Interscience 1995.   CrossRef
  13. P. Bougerol and N. Picard: Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 (1992), 4, 1714-1730.   DOI:10.1214/aop/1176989526
  14. R. A. Boyles and W. A. Gardner: Cycloergodic properties of discrete-parameter nonstationary stochastic processes. IEEE, Trans. Inform. Theory 29 (1983), 105-114.   DOI:10.1109/tit.1983.1056613
  15. A. Brandt: The stochastic equation $Y_{n+1}=A_{n}Y_{n-1}+B_{n}$ with stationary coefficients. Adv. Appl. Probab. 18 (1986), 1, 211-220.   CrossRef
  16. S. Chatterjee and S. Das: Parameter estimation in conditional heteroscedastic models. Comm. Stat. Theory Methods 32 (2003), 6, 1135-1153.   DOI:10.1081/sta-120021324
  17. Z. Florian: Quasi-maximum likelihood estimation of periodic autoregressive, conditionally heteroscedastic. Time Series. Stochastic Models, Statistics and their Applications 122 (2015), 207-214.   DOI:10.1007/978-3-319-13881-7_23
  18. C. Francq, R. Roy and A. Saidi: Asymptotic properties of weighted least squares estimation in weak PARMA models. J. of Time Ser. Anal. 32 (2011), 699-723.   DOI:10.1111/j.1467-9892.2011.00728.x
  19. C. Francq: ARMA models with bilinear innovations. Comm. Statist. Stochastic Models 15 (1999), 1, 29-52.   DOI:10.1080/15326349908807524
  20. C. Francq and J. M. Zako\^ıan: Maximum likelihood estimation of pure $GARCH$ and $ARMA-GARCH$ processes. Bernoulli 10 (2004), 4, 605-637.   DOI:10.3150/bj/1093265632
  21. W. A. Gardner, A. Nopolitano and L. Paura: Cyclostationarity: Half a century of research. Signal Process. 86 (2006), 4, 639-697.   DOI:10.1016/j.sigpro.2005.06.016
  22. E. G. Gladyshev: Periodically correlated random sequences. Soviet Math. 2 (1961), 385-388.   CrossRef
  23. T. Grahn: A conditional least squares approach to bilinear time series estimation. J. Time Series Analysis 16 (1995), 509-529.   DOI:10.1111/j.1467-9892.1995.tb00251.x
  24. J. He, S. Yu and J. Cai: Numerical Analysis and improved algorithms for Lyapunov-exponent calculation of discrete-time chaotic systems. Int. J. Bifurcation Chaos 26 (2016), 13, 1-11.   DOI:10.1142/s0218127416502199
  25. H. Kesten and F. Spitzer: Convergence in distribution for products of random matrices. Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 67 (1984), 363-386.   DOI:10.1007/bf00532045
  26. D. Kristensen: On stationarity and ergodicity of the bilinear model with applications to the GARCH models. J. Time Series Anal. 30 (2009), 1, 125-144.   DOI:10.1111/j.1467-9892.2008.00603.x
  27. S. Ling, L. Peng and F. Zho: Inference for special bilinear time series model. arXiv 2014.   CrossRef
  28. J. Liu: Estimation for some bilinear time series. Stochastic Models 6 (1990), 649-665.   DOI:10.1080/15326349908807167
  29. J. Ngatchou-W: Estimation in a class of nonlinear heteroscedastic time series models. Elec. J. Stat. 2 (2008), 40-62.   DOI:10.1214/07-ejs157
  30. J. Z. Pan, G. D. Li and Z.J. Xie: Stationary solution and parametric estimation for bilinear model driven by $ARCH$ noises. Science in China (Series $A$) 45 (2002), 12, 1523-1537.   CrossRef
  31. D. T. Pham: Bilinear Markovian representation of bilinear models. Stoch. Proc. Appl. 20 (1983), 295-306.   DOI:10.1016/0304-4149(85)90216-9
  32. T. Subba Rao and M. M. Gabr: An introduction to bispectral analysis and bilinear time series models. Lecture Notes In Statistics 24 (1984), Springer Verlag, N.Y.   DOI:10.1007/978-1-4684-6318-7
  33. D. Tjøstheim: Estimation in nonlinear time series models. Stoch. Proc. Appl. 21 (1986), 251-273.   DOI:10.1016/0304-4149(86)90099-2
  34. G. S. Wittwer: Some remarks on bilinear time series models. Statistics 20 (1989), 521-529.   DOI:10.1080/02331888908802201