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Least absolute deviation estimation for general autoregressive moving average time‐series models

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  • Rongning Wu
  • Richard A. Davis

Abstract

We study least absolute deviation (LAD) estimation for general autoregressive moving average time‐series models that may be noncausal, noninvertible or both. For ARMA models with Gaussian noise, causality and invertibility are assumed for the parameterization to be identifiable. The assumptions, however, are not required for models with non‐Gaussian noise, and hence are removed in our study. We derive a functional limit theorem for random processes based on an LAD objective function, and establish the consistency and asymptotic normality of the LAD estimator. The performance of the estimator is evaluated via simulation and compared with the asymptotic theory. Application to real data is also provided.

Suggested Citation

  • Rongning Wu & Richard A. Davis, 2010. "Least absolute deviation estimation for general autoregressive moving average time‐series models," Journal of Time Series Analysis, Wiley Blackwell, vol. 31(2), pages 98-112, March.
    • Handle: RePEc:bla:jtsera:v:31:y:2010:i:2:p:98-112
    DOI: 10.1111/j.1467-9892.2009.00648.x
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    References listed on IDEAS

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    1. Andrews, Beth & Davis, Richard A. & Jay Breidt, F., 2006. "Maximum likelihood estimation for all-pass time series models," Journal of Multivariate Analysis, Elsevier, vol. 97(7), pages 1638-1659, August.
    2. Davis, Richard A. & Knight, Keith & Liu, Jian, 1992. "M-estimation for autoregressions with infinite variance," Stochastic Processes and their Applications, Elsevier, vol. 40(1), pages 145-180, February.
    3. Pan, Jiazhu & Wang, Hui & Yao, Qiwei, 2007. "Weighted Least Absolute Deviations Estimation For Arma Models With Infinite Variance," Econometric Theory, Cambridge University Press, vol. 23(5), pages 852-879, October.
    4. Pan, Jiazhu & Wang, Hui & Yao, Qiwei, 2007. "Weighted least absolute deviations estimation for ARMA models with infinite variance," LSE Research Online Documents on Economics 5405, London School of Economics and Political Science, LSE Library.
    5. Lii, Keh-Shin & Rosenblatt, Murray, 1992. "An approximate maximum likelihood estimation for non-Gaussian non-minimum phase moving average processes," Journal of Multivariate Analysis, Elsevier, vol. 43(2), pages 272-299, November.
    6. Davis, Richard A., 1996. "Gauss-Newton and M-estimation for ARMA processes with infinite variance," Stochastic Processes and their Applications, Elsevier, vol. 63(1), pages 75-95, October.
    7. Jian Huang & Yudi Pawitan, 2000. "Quasi‐likelihood Estimation of Non‐invertible Moving Average Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(4), pages 689-702, December.
    8. Breid, F. Jay & Davis, Richard A. & Lh, Keh-Shin & Rosenblatt, Murray, 1991. "Maximum likelihood estimation for noncausal autoregressive processes," Journal of Multivariate Analysis, Elsevier, vol. 36(2), pages 175-198, February.
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    Cited by:

    1. Rongning Wu, 2013. "M-estimation for general ARMA Processes with Infinite Variance," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(3), pages 571-591, September.
    2. Hecq, A.W. & Lieb, L.M. & Telg, J.M.A., 2015. "Identification of Mixed Causal-Noncausal Models : How Fat Should We Go?," Research Memorandum 035, Maastricht University, Graduate School of Business and Economics (GSBE).
    3. Ke Zhu & Shiqing Ling, 2015. "LADE-Based Inference for ARMA Models With Unspecified and Heavy-Tailed Heteroscedastic Noises," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(510), pages 784-794, June.
    4. Pentti Saikkonen & Rickard Sandberg, 2016. "Testing for a Unit Root in Noncausal Autoregressive Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 37(1), pages 99-125, January.
    5. Meitz, Mika & Saikkonen, Pentti, 2013. "Maximum likelihood estimation of a noninvertible ARMA model with autoregressive conditional heteroskedasticity," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 227-255.
    6. Markku Lanne & Mika Meitz & Pentti Saikkonen, 2012. "Testing for Predictability in a Noninvertible ARMA Model," Koç University-TUSIAD Economic Research Forum Working Papers 1225, Koc University-TUSIAD Economic Research Forum.
    7. Frédérique Bec & Heino Bohn Nielsen & Sarra Saïdi, 2020. "Mixed Causal–Noncausal Autoregressions: Bimodality Issues in Estimation and Unit Root Testing," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 82(6), pages 1413-1428, December.
    8. Jean-Baptiste MICHAU, 2019. "Helicopter Drops of Money under Secular Stagnation," Working Papers 2019-10, Center for Research in Economics and Statistics.
    9. Zhu, Ke & Ling, Shiqing, 2013. "Global self-weighted and local quasi-maximum exponential likelihood estimators for ARMA-GARCH/IGARCH models," MPRA Paper 51509, University Library of Munich, Germany.
    10. Hecq, Alain & Issler, João Victor & Telg, Sean, 2017. "Mixed Causal-Noncausal Autoregressions with Strictly Exogenous Regressors," MPRA Paper 80767, University Library of Munich, Germany.
    11. Lanne Markku & Saikkonen Pentti, 2011. "Noncausal Autoregressions for Economic Time Series," Journal of Time Series Econometrics, De Gruyter, vol. 3(3), pages 1-32, October.
    12. Wu, Rongning, 2014. "Least absolute deviation estimation for general fractionally integrated autoregressive moving average time series models," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 69-76.
    13. Alain Hecq & Daniel Velasquez-Gaviria, 2023. "Spectral identification and estimation of mixed causal-noncausal invertible-noninvertible models," Papers 2310.19543, arXiv.org.
    14. Kramkov, Viacheslav & Maksimov, Andrey, 2020. "Loan market markups and noncausal autoregressions," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 60, pages 48-69.
    15. Nyholm, Juho, 2017. "Residual-based diagnostic tests for noninvertible ARMA models," MPRA Paper 81033, University Library of Munich, Germany.

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