Aknouche, Abdelhakim (2015): Unified quasimaximum likelihood estimation theory for stable and unstable Markov bilinear processes. Published in: Nova Publisher Science (2015)

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Abstract
A unified quasimaximum likelihood (QML) estimation theory for stationary and nonstationary simple Markov bilinear (SMBL) models is proposed. Such models may be seen as generalized random coefficient autoregressions (GRCA) in which the innovation and the random coefficient processes are fully correlated. It is shown that the QML estimate (QMLE) for the SMBL model is always asymptotically Gaussian without assuming strict stationarity, meaning that there is no knife edge effect. The asymptotic variance of the QMLE is different in the stationary and nonstationary cases but is consistently estimated using the same estimator. A perhaps surprising result is that in the nonstationary domain, all SMBL parameters are consistently estimated in contrast with unstable GARCH and GRCA models where the QMLE of the conditional variance intercept is inconsistent. As a result, strict stationarity testing for the SMBL is studied. Simulation experiments and a real application to strict stationarity testing for some financial stock returns illustrate the theory in finite samples.
Item Type:  MPRA Paper 

Original Title:  Unified quasimaximum likelihood estimation theory for stable and unstable Markov bilinear processes 
English Title:  Unified quasimaximum likelihood estimation theory for stable and unstable Markov bilinear processes 
Language:  English 
Keywords:  Markov bilinear process, random coefficient process, stability, instability, Quasimaximum likelihood, knife edge effect, strict stationarity testing. 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C10  General C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C13  Estimation: General C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C18  Methodological Issues: General C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C19  Other 
Item ID:  69572 
Depositing User:  Prof. Abdelhakim Aknouche 
Date Deposited:  18 Feb 2016 12:04 
Last Modified:  18 Feb 2016 12:22 
References:  Aknouche, A. (2012a). Multistage weighted least squares estimation of ARCH processes in the stable and unstable cases. Statistical Inference for Stochastic Processes, 15, 241256. Aknouche, A. (2012b). Implication of instability on econometric and financial time series modeling. In Econometrics: New research, editors: Mendez, S.A. and Vega, A.M., Nova Publishers, New York, pp. 149186. Aknouche, A. (2013). Twostage weighted least squares estimation of nonstationary random coefficient autoregressions. Journal of Time Series Econometrics, 5, 2547. Aknouche, A. (2014). Estimation and strict stationarity testing of ARCH processes based on weighted least squares. Mathematical Methods of Statistics, 23, 81102. Aknouche, A. (2015a). Quadratic random coefficient autoregression with linearinparameters volatility. Statistical Inference for Stochastic Processes, forthcoming, DOI 10.1007/s1120301491083. Aknouche, A. (2015b). Explosive strong periodic autoregression with multiplicity one. Journal of Statistical Planning and Inference, 161, 5072. Aknouche, A. and Touche, N. (2015). Weighted least squaresbased inference for stable and unstable threshold power ARCH processes. Statistics & Probability Letters, 97, 108115. Aknouche, A., AlEid, E.M. and Hmeid, A.M. (2011). Offline and online weighted least squares estimation of nonstationary power ARCH processes. Statistics & Probability Letters, 81, 15351540. Amendola, A. and Francq, C. (2009). Concepts and tools for nonlinear time series modelling. Handbook of Computational Econometrics, Edts: D. Belsley and E. Kontoghiorghes, Wiley. Aue, A. and Horváth, L. (2011). Quasilikelihood estimation in stationary and nonstationary autoregressive models with random coefficients. Statistica Sinica, 21, 973999. Aue, A., Horváth, L. and Steinebach, J. (2006). Estimation in random coefficient autoregressive models. Journal of Time Series Analysis, 27, 6176. Babillot, M., Bougerol, P. and Elie, L. (1997). The random difference equation X_{n}=A_{n}X_{n1}+B_{n} in the critical case. Annals of Probability, 25, 478493. Berkes, I. and Horváth, L. (2004). The efficiency of the estimators of the parameters in GARCH processes. Annals of Statistics, 32, 633655. Berkes, I., Horváth, L. and Ling, S. (2009). Estimation in nonstationary random coefficient autoregressive models. Journal of Time Series Analysis, 30, 395416. Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, 307327. Bougerol, P. and Picard, N. (1992). Strict stationarity of generalized autoregressive processes. Annals of Probability, 20, 17141730. Brandt, A. (1986). The stochastic equation Y_{n+1}=A_{n}Y_{n}+B_{n} with stationary coefficients. Advances in Applied Probability, 18, 211220. Chen, M., Li, D. and Ling, S. (2014). Nonstationarity and quasimaximum likelihood estimation on a double autoregressive model. Journal of Time Series Analysis, 35, 189202. Cline, D.B.H. (2007). Stability of nonlinear stochastic recursions with application to nonlinear ARGARCH models. Advances in Applied Probability, 39, 462491. Cline, D.B.H. and Pu, H.M.H. (2002). A note on a simple Markov bilinear stochastic process. Statistics & Probability Letters, 56, 283288. Engle, R.F. (1982). Autoregressive Conditional Heteroskedasticity with estimates of variance of U.K. inflation. Econometrica, 50, 9871008. Fan, J., Qi, L. and Xiu, D. (2014). Quasi maximum likelihood estimation of GARCH models with heavytailed likelihoods. Journal of Business and Economic Statistics, 32, 178191. Feigin, P.D. and Tweedie, R.L. (1985). Random coefficient autoregressive processes: a Markov chain analysis of stationarity and finiteness of moments. Journal of Time Series Analysis, 6, 114. Ferrante, M., Fonseca, G. and Vidoni, P. (2003). Geometric ergodicity, regularity of the invariant distribution and inference for a threshold bilinear Markov process. Statistica Sinica, 13, 367384. Francq, C. and Zakoïan, J.M. (2010). GARCH Models: Structure, statistical inference and financial applications. Wiley. Francq, C. and Zakoïan, J.M. (2012). Strict stationarity testing and estimation of stationary and explosive GARCH models. Econometrica, 80, 821861. Francq, C. and Zakoïan, J.M. (2013a). Inference in nonstationary asymmetric GARCH models. Annals of Statistics, 41, 16932262. Francq, C. and Zakoïan, J.M. (2013b). Optimal predictions of powers of conditionally heteroskedastic processes. Journal of the Royal Statistical Society, B75, 345367. Goldie, C. and Maller, R. (2000). Stability of perpetuities. Annals of Probability, 28, 11951218. Holan, S.H., Lund, R. and Davis, G. (2010). The ARMA alphabet soup: A tour of ARMA model variants. Statistics Survey, 4, 232274. Hwang, S.Y. and Basawa, I.V. (1998). Parameter estimation for generalized random coefficient autoregressive processes. Journal of Statistical Planning and Inference, 68, 323337. Hwang, S.Y. and Basawa, I.V. (2005). Explosive randomcoefficient AR(1) processes and related asymptotics for least squares estimation. Journal of Time Series Analysis, 26, 807824. Jensen, S.T. and Rahbek, A. (2004). Asymptotic normality of the QML estimator of ARCH in the nonstationary case. Econometrica, 72, 641646. Ling, S. and Li, D. (2008). Asymptotic inference for a nonstationary double AR(1) model. Biometrika, 95, 257263. Lumsdaine, R.L. (1996). Consistency and asymptotic normality of the quasimaximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models. Econometrica, 64, 575596. Meyn, S. and Tweedie, R. (2009). Markov chains and stochastic stability. 2nd edition, Springer Verlag, New York. Nicholls, D.F. and Quinn, B.G. (1982). Random coefficient autoregressive model: An introduction. Springer Verlag, New York. Peng, L. and Yao, Q. (2003). Least absolute deviations estimation for ARCH and GARCH models. Biometrika, 90, 967975. Schick, A. (1996). √nconsistent estimation in a random coefficient autoregressive model. Australian Journal of Statistics, 38, 15560. Taylor, S. (1986). Financial time series analysis. Wiley. Tong, H. (1981). A note on a Markov bilinear stochastic process in discrete time. Journal of Time Series Analysis, 2, 279284. Truquet, L. and Yao, J. (2012). On the quasilikelihood estimation for random coefficient autoregressions. Statistics, 46, 505521. Tsay R.S. (1987). Conditional heteroskedastic time series models. Journal of the American Statistical Association, 7, 590604. Tsay, R.S. (2002). Analysis of financial time series: Financial econometrics. Wiley. Vervaat, W. (1979). On a stochastic difference equation and a representation of non negative infinitely divisible random variables. Advances in Applied Probability, 11, 750783. Weiss, A.A. (1984). ARMA models with ARCH errors. Journal of Time Series Analysis, 5, 12943. Zhao, Z.W. and Wang, D.H. (2012). Statistical inference for generalized random coefficient autoregressive model. Mathematical and Computer Modelling, 56, 152166. Zhao, Z.W., Wang, D.H. and Peng, C.X. (2013). Coefficient constancy test in generalized random coefficient autoregressive m 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/69572 