# Article

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Keywords:
periodic bilinear model; periodic $ARMA$ model; strict and second-order periodic stationarity; strong consistency; asymptotic normality
Summary:
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 $QMLE$.
References:
[1] Aknouche, A., Bibi, A.: 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 | MR 2488634
[2] Aknouche, A., Guerbyenne, H.: Periodic stationarity of random coefficient periodic autoregressions. Statist. Probab. Lett. 79 (2009), 7, 990-996. DOI 10.1016/j.spl.2008.12.012 | MR 2509491
[3] Aknouche, A., Bibi, A.: 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 | MR 2488634
[4] Basawa, I. V., Lund, R.: 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 | MR 1867391
[5] Bibi, A.: On the covariance structure of time-varying bilinear models. Stochastic Anal. App. 21 (2003), 1, 25-60. DOI 10.1081/sap-120017531 | MR 1954074
[6] Bibi, A., Francq, C.: Consistent and asymptotically normal estimators for cyclically time-dependent linear models. Ann. Inst. Statist. Math. 55 (2003), 1, 41-68. DOI 10.1007/bf02530484 | MR 1965962
[7] Bibi, A., Oyet, A. J.: Estimation of some bilinear time series models with time-varying coefficients. Stochastic Anal. Appl. 22 (2004), 2, 355-376. DOI 10.1081/sap-120028595 | MR 2037377
[8] Bibi, A., Aknouche, A.: 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 | MR 2591755
[9] Bibi, A., Lessak, R.: 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 | MR 2483399
[10] Bibi, A., Lescheb, I.: On general periodic time-varying bilinear processes. Econom. Lett. 114 (2012), 3, 353-357. DOI 10.1016/j.econlet.2011.11.013 | MR 2880617
[11] Bibi, A., Ghezal, A.: 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 | MR 3539499
[12] Billingsley, P.: Probability and Measure. Third edition. Wiley - Interscience 1995. MR 1324786
[13] Bougerol, P., Picard, N.: Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 (1992), 4, 1714-1730. DOI 10.1214/aop/1176989526 | MR 1188039
[14] Boyles, R. A., Gardner, W. A.: Cycloergodic properties of discrete-parameter nonstationary stochastic processes. IEEE, Trans. Inform. Theory 29 (1983), 105-114. DOI 10.1109/tit.1983.1056613 | MR 0711279
[15] Brandt, A.: The stochastic equation $Y_{n+1}=A_{n}Y_{n-1}+B_{n}$ with stationary coefficients. Adv. Appl. Probab. 18 (1986), 1, 211-220. DOI 10.2307/1427243 | MR 0827336
[16] Chatterjee, S., Das, S.: Parameter estimation in conditional heteroscedastic models. Comm. Stat. Theory Methods 32 (2003), 6, 1135-1153. DOI 10.1081/sta-120021324 | MR 1983236
[17] Florian, Z.: 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] Francq, C., Roy, R., Saidi, A.: 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 | MR 2846568
[19] Francq, C.: ARMA models with bilinear innovations. Comm. Statist. Stochastic Models 15 (1999), 1, 29-52. DOI 10.1080/15326349908807524 | MR 1674085
[20] Francq, C., Zakoîan, J. M.: Maximum likelihood estimation of pure $GARCH$ and $ARMA-GARCH$ processes. Bernoulli 10 (2004), 4, 605-637. DOI 10.3150/bj/1093265632 | MR 2076065
[21] Gardner, W. A., Nopolitano, A., Paura, L.: Cyclostationarity: Half a century of research. Signal Process. 86 (2006), 4, 639-697. DOI 10.1016/j.sigpro.2005.06.016
[22] Gladyshev, E. G.: Periodically correlated random sequences. Soviet Math. 2 (1961), 385-388. MR 0126873
[23] Grahn, T.: 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 | MR 1365644
[24] He, J., Yu, S., Cai, J.: 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 | MR 3590146
[25] Kesten, H., Spitzer, F.: Convergence in distribution for products of random matrices. Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 67 (1984), 363-386. DOI 10.1007/bf00532045 | MR 0761563
[26] Kristensen, D.: 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 | MR 2488638
[27] Ling, S., Peng, L., Zho, F.: Inference for special bilinear time series model. arXiv 2014. MR 3300205
[28] Liu, J.: Estimation for some bilinear time series. Stochastic Models 6 (1990), 649-665. DOI 10.1080/15326349908807167 | MR 1080415
[29] Ngatchou-W, J.: Estimation in a class of nonlinear heteroscedastic time series models. Elec. J. Stat. 2 (2008), 40-62. DOI 10.1214/07-ejs157 | MR 2386085
[30] Pan, J. Z., Li, G. D., Xie, Z.J.: Stationary solution and parametric estimation for bilinear model driven by $ARCH$ noises. Science in China (Series $A$) 45 (2002), 12, 1523-1537. MR 1955672
[31] Pham, D. T.: Bilinear Markovian representation of bilinear models. Stoch. Proc. Appl. 20 (1983), 295-306. DOI 10.1016/0304-4149(85)90216-9 | MR 0808163
[32] Rao, T. Subba, Gabr, M. M.: 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 | MR 0757536
[33] Tjøstheim, D.: Estimation in nonlinear time series models. Stoch. Proc. Appl. 21 (1986), 251-273. DOI 10.1016/0304-4149(86)90099-2 | MR 0833954
[34] Wittwer, G. S.: Some remarks on bilinear time series models. Statistics 20 (1989), 521-529. DOI 10.1080/02331888908802201 | MR 1047220

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