Advertisement

Statistical Methods & Applications

, Volume 27, Issue 3, pp 479–490 | Cite as

Asymptotics of the weighted least squares estimation for AR(1) processes with applications to confidence intervals

  • Ruidong Han
  • Xinghui Wang
  • Shuhe Hu
Original Paper
  • 141 Downloads

Abstract

For the first-order autoregressive model, we establish the asymptotic theory of the weighted least squares estimations whether the underlying autoregressive process is stationary, unit root, near integrated or even explosive under a weaker moment condition of innovations. The asymptotic limit of this estimator is always normal. It is shown that the empirical log-likelihood ratio at the true parameter converges to the standard chi-square distribution. An empirical likelihood confidence interval is proposed for interval estimations of the autoregressive coefficient. The results improve the corresponding ones of Chan et al. (Econ Theory 28:705–717, 2012). Some simulations are conducted to illustrate the proposed method.

Keywords

Weighted least squares estimation Empirical likelihood Interval estimation Autoregressive models 

Mathematics Subject Classification

62F12 60G10 62G20 

Notes

Acknowledgements

The authors thank the Editor-in-Chief Tommaso Proietti and two anonymous referees for their helpful comments and valuable suggestions that greatly improved the paper. This work is supported by the National Natural Science Foundation of China (11701005, 11671012, 11501004, 11501005), the Natural Science Foundation of Anhui Province (1608085QA02), the Science Fund for Distinguished Young Scholars of Anhui Province (1508085J06) and Introduction Projects of Anhui University Academic and Technology Leaders.

References

  1. Chan NH (1990) Inference for near-integrated time series with infinite variance. J Am Stat Assoc 85:1069–1074MathSciNetCrossRefzbMATHGoogle Scholar
  2. Chan NH (2009) Time series with roots on or near the unit circle. In: Andersen TG, Davis RA, Kreiss J, Mikosch T (eds) Springer handbook of financial time series. Springer, Berlin, pp 696–707Google Scholar
  3. Chan NH, Wei CZ (1987) Asymptotic inference for nearly nonstationary autoregressive time series with infinite variance. Stat Sin 16:15–28Google Scholar
  4. Chan NH, Peng L (2005) Weighted least absolute deviations estimation for an AR(1) process with ARCH(1) errors. Biometrika 92:477–484MathSciNetCrossRefzbMATHGoogle Scholar
  5. Chan NH, Li DY, Peng L (2012) Toward a unified interval estimation of autoregressions. Econ Theory 28:705–717Google Scholar
  6. Chen X, Cui HJ (2008) Empirical likelihood inference for partial linear models under martingale difference sequence. Stat Probab Lett 78:2895–3382Google Scholar
  7. Chuang C-S, Chan NH (2002) Empirical likelihood for autoregressive models, with applications to unstable series. Stat Sin 12:387–407MathSciNetzbMATHGoogle Scholar
  8. Datta S (1996) On asymptotic properties of bootrap for AR(1) processes. J Stat Plan Inference 53:361–374CrossRefzbMATHGoogle Scholar
  9. Hall P, Heyde C (1980) Martingale limit theory and its applications. Academic Press, New YorkzbMATHGoogle Scholar
  10. Han C, Phillips PCB, Sul D (2010) Uniform asymptotic normality in stationary and unit root autoregression. Econ Theory 27:1117–1151MathSciNetCrossRefzbMATHGoogle Scholar
  11. Li JY, Liang W, He SY (2011) Empirical likelihood for LAD estimators in infinite variance ARMA models. Stat Probab Lett 81:212–219MathSciNetCrossRefzbMATHGoogle Scholar
  12. Ling S (2005) Self-weighted least absolute deviation estimation for infinite variance autoregressive models. J R Stat Soc Ser B 67:381–393MathSciNetCrossRefzbMATHGoogle Scholar
  13. Mikusheva A (2007) Uniform inference in autoregressive models. Econometrica 74:535–574MathSciNetzbMATHGoogle Scholar
  14. Monti AC (1997) Empirical likelihood confidence regions in time series models. Biometrika 84:395–405MathSciNetCrossRefzbMATHGoogle Scholar
  15. Owen A (1988) Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75:237–249MathSciNetCrossRefzbMATHGoogle Scholar
  16. Owen A (1990) Empirical likelihood ratio confidence regions. Ann Stat 18:90–120MathSciNetCrossRefzbMATHGoogle Scholar
  17. Owen A (1991) Empirical likelihood for linear models. Ann Stat 19:1725–1747MathSciNetCrossRefzbMATHGoogle Scholar
  18. Owen A (2001) Empirical likelihood. Chapman and Hall, New YorkCrossRefzbMATHGoogle Scholar
  19. Phillips PCB (1987) Toward a unified asymptotic theory of autoregression. Biometrika 74:535–574MathSciNetCrossRefzbMATHGoogle Scholar
  20. Qin J, Lawless J (1994) Empirical likelihood and general estimating equations. Ann Stat 22:300–325MathSciNetCrossRefzbMATHGoogle Scholar
  21. Shi J, Lau TS (2000) Empirical likelihood for partially linear models. J Multivar Anal 72:132–148MathSciNetCrossRefzbMATHGoogle Scholar
  22. So BS, Shin DW (1999) Cauchy estimators for autoregressive processes with applications to unit root tests and confidence intervals. Econ Theory 15:165–176MathSciNetCrossRefzbMATHGoogle Scholar
  23. Wang QH, Jing BY (1999) Empirical likelihood for partial linear models with fixed designs. Stat Probab Lett 41:425–433MathSciNetCrossRefzbMATHGoogle Scholar
  24. Wang QH, Rao JNK (2002) Empirical likelihood-based inference in linear errors-in covariables models with validation data. Biometrika 89:345–358MathSciNetCrossRefzbMATHGoogle Scholar
  25. Zhu LX, Xue LG (2006) Empirical likelihood confidence regions in partially linear single-index model. J R Stat Soc Ser B 68:549–570MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of EconomicsAnhui UniversityHefeiPeople’s Republic of China
  2. 2.School of Mathematical ScienceAnhui UniversityHefeiPeople’s Republic of China

Personalised recommendations