Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Time Series Econometrics

Editor-in-Chief: Hidalgo, Javier

CiteScore 2018: 0.20

SCImago Journal Rank (SJR) 2018: 0.323
Source Normalized Impact per Paper (SNIP) 2018: 0.291

Mathematical Citation Quotient (MCQ) 2018: 0.03

See all formats and pricing
More options …

Two-Stage Weighted Least Squares Estimation of Nonstationary Random Coefficient Autoregressions

Abdelhakim Aknouche
Published Online: 2013-05-15 | DOI: https://doi.org/10.1515/jtse-2012-0011


This paper proposes a two-stage weighted least squares (2S-WLS) estimate for the autoregressive parameter and the random coefficient variance of a non-(strictly) stationary random coefficient autoregression (RCA). In the first stage, the autoregressive parameter is estimated from the conditional mean equation by a weighted least squares (WLS) method in which the weight is the conditional variance evaluated at any arbitrary known parameter value. In the second stage, based on the estimated conditional variance equation, the random coefficient variance is estimated again using the WLS method, but weighted by the squared conditional variance arbitrarily evaluated. It will be shown that the 2S-WLS estimate is asymptotically Gaussian with the same asymptotic variance as the quasi-maximum likelihood estimate under very mild conditions. Applications to the Gaussian double autoregression and the Markov bilinear model are given.

JEL Classification: AMS Subject Classification (2000) Primary 62M10; Secondary 62M04

Keyword: RCA process; Markov bilinear process; non-strict stationarity; weighted least squares; asymptotic normality.


  • Aknouche, A. 2012. “Multi-Stage Weighted Least Squares Estimation of ARCH Processes in the Stable and Unstable Cases.” Statistical Inference for Stochastic Processes, 15: 241–56.Google Scholar

  • Aknouche, A., and E. Al-Eid, 2012. “Asymptotic Inference of Unstable Periodic ARCHProcesses.” Statistical Inference for Stochastic Processes, 15: 61–79.Google Scholar

  • Andel, J.1976. “Autoregressive Series with Random Parameters.” Mathematische Operations for Schung und Statistik, Series Statistics, 7: 735–41.Google Scholar

  • Anderson, T. W. 1959. “On Asymptotic Distributions of Estimates of Parameters of Stochastic Difference Equations.” Annals of Mathematical Statistics, 30: 676–87.CrossrefGoogle Scholar

  • Aue, A., and L. Horváth. 2011. “Quasi-Likelihood Estimation in Stationary and Nonstationary Autoregressive Models with Random Coefficients.” Statistica Sinica, 21: 973–99.Google Scholar

  • Aue, A., L. Horváth, and J. Steinebach. 2006. “Estimation in Random Coefficient Autoregressive Models.” Journal of Time Series Analysis, 27: 61–76.Web of ScienceCrossrefGoogle Scholar

  • Babillot, M., P. Bougerol, and L. Elie 1997. “The Random Difference Equation Xn = AnXn 1 + Bn in the Critical Case.” Annals of Probability, 25: 478–93.Google Scholar

  • Berkes, I., L. Horváth, and P. Kokoskza. 2003. “GARCH Processes: Structure and Estimation.” Bernoulli, 9: 201–27.CrossrefGoogle Scholar

  • Berkes, I., L. Horváth, and S. Ling. 2009. “Estimation in Nonstationary Random Coefficient Autoregressive Models.” Journal of Time Series Analysis, 30: 395–416.Web of ScienceCrossrefGoogle Scholar

  • Bougerol, P., and N. Picard. 1992. “Strict Stationarity of Generalized Autoregressive Processes.” Annals of Probability, 20: 1714–30.CrossrefGoogle Scholar

  • Brown, B. M. 1971. “Martingale Central Limit Theorems.” Annals of Mathematical Statistics, 42: 59–66.CrossrefGoogle Scholar

  • Cline, D. B. H., and H.M.H. Pu. 2002. “A Note on a Simple Markov Bilinear Stochastic Process.” Statistics & Probability Letters, 56: 283–88.CrossrefGoogle Scholar

  • Feigin, P. D., and R. L. Tweedie. 1985. “Random Coefficient Autoregressive Processes: A Markov Chain Analysis of Stationarity and Finiteness of Moments.” Journal of Time Series Analysis, 6: 1–14.CrossrefGoogle Scholar

  • Francq, C., and J. M. Zakoїan. 2004. “Maximum Likelihood Estimation of Pure GARCHand ARMA-GARCH Processes.” Bernoulli, 10: 605–37.CrossrefGoogle Scholar

  • Francq, C., and J. M. Zakoїan. 2010. GARCH Models: Structure, Statistical Inference and Financial Applications. New York: Wiley.Google Scholar

  • Francq, C., and J. M. Zakoїan. 2012. “Strict Stationarity Testing and Estimation of Stationary and ExplosiveGARCH Models.” Econometrica, 80: 821–61.Google Scholar

  • Goldie, C., and R. Maller. 2000. “Stability of Perpetuities.” Annals of Probability, 28: 1195–218.Google Scholar

  • Hwang, S. Y., and I. V. Basawa. 2005. “Explosive Random-Coefficient AR(1) Processes and Related Asymptotics for Least Squares Estimation.” Journal of Time Series Analysis, 26: 807–24.CrossrefGoogle Scholar

  • Jensen, S. T., and A. Rahbek, 2004a. “Asymptotic Normality of the QML Estimator of ARCH in the Nonstationary Case.” Econometrica, 72: 641–6.CrossrefGoogle Scholar

  • Jensen, S. T., and A.Rahbek, 2004b. “Asymptotic Inference for Nonstationary GARCH.” Econometric Theory, 20: 1203–26.Google Scholar

  • Ling, S. 2007. “A Double AR(p) Model: Structure and Estimation.” Statistica Sinica, 17: 161–75.Google Scholar

  • Ling, S., and D. Li.2008. “Asymptotic Inference for a Nonstationary Double AR(1) Model.” Biometrika, 95: 257–63.CrossrefWeb of ScienceGoogle Scholar

  • Linton, O., J. Pan, and H. Wang. 2010. “Estimation for a Non-Stationary Semi-Strong GARCH(1, 1) Model with Heavy-Tailed Errors.” Econometric Theory, 26: 1–28.CrossrefGoogle Scholar

  • Nicholls, D. F., and B. G. Quinn. 1982. Random Coefficient Autoregressive Model: An Introduction. New York: Springer Verlag.Google Scholar

  • Quinn, B. G. 1982. “Stationarity and Invertibility of Simple Bilinear Models.” Stochastic Processes and Their Applications, 12: 225–30.CrossrefGoogle Scholar

  • Rosenberg, B. 1973. “A Survey of Stochastic Parameter Regression.” Annals of Economic and Social Measurement, 2: 381–98.Google Scholar

  • Schick, A. 1996. “-Consistent Estimation in a Random Coefficient Autoregressive Model.” Australian Journal of Statistics, 38: 155–60.CrossrefGoogle Scholar

  • Tong, H. 1981. “A Note on a Markov Bilinear Stochastic Process in Discrete Time.” Journal of Time Series Analysis, 2: 279–84.CrossrefGoogle Scholar

  • Tsay, R. S.1987. “Conditional Heteroskedastic Time Series Models.” Journal of the American Statistical Association, 7: 590–604.CrossrefGoogle Scholar

About the article

Published Online: 2013-05-15

Citation Information: Journal of Time Series Econometrics, Volume 5, Issue 1, Pages 25–46, ISSN (Online) 1941-1928, ISSN (Print) 2194-6507, DOI: https://doi.org/10.1515/jtse-2012-0011.

Export Citation

©2013 by Walter de Gruyter Berlin / Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in