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LADE-based inference for ARMA models with unspecified and heavy-tailed heteroscedastic noises

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  • Zhu, Ke
  • Ling, Shiqing
Abstract
This paper develops a systematic procedure of statistical inference for the ARMA model with unspecified and heavy-tailed heteroscedastic noises. We first investigate the least absolute deviation estimator (LADE) and the self-weighted LADE for the model. Both estimators are shown to be strongly consistent and asymptotically normal when the noise has a finite variance and infinite variance, respectively. The rates of convergence of the LADE and the self-weighted LADE are $n^{-1/2}$ which is faster than those of LSE for the AR model when the tail index of GARCH noises is in (0,4], and thus they are more efficient in this case. Since their asymptotic covariance matrices can not be estimated directly from the sample, we develop the random weighting approach for statistical inference under this nonstandard case. We further propose a novel sign-based portmanteau test for model adequacy. Simulation study is carried out to assess the performance of our procedure and one real illustrating example is given.

Suggested Citation

  • Zhu, Ke & Ling, Shiqing, 2014. "LADE-based inference for ARMA models with unspecified and heavy-tailed heteroscedastic noises," MPRA Paper 59099, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:59099
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    1. Yang, Minxian & Bewley, Ronald, 1995. "Moving average conditional heteroskedastic processes," Economics Letters, Elsevier, vol. 49(4), pages 367-372, October.
    2. Ling, Shiqing & McAleer, Michael, 2003. "Asymptotic Theory For A Vector Arma-Garch Model," Econometric Theory, Cambridge University Press, vol. 19(2), pages 280-310, April.
    3. Zhang, Rong-Mao & Sin, Chor-yiu (CY) & Ling, Shiqing, 2015. "On functional limits of short- and long-memory linear processes with GARCH(1,1) noises," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 482-512.
    4. Schwert, G William & Seguin, Paul J, 1990. "Heteroskedasticity in Stock Returns," Journal of Finance, American Finance Association, vol. 45(4), pages 1129-1155, September.
    5. Kani Chen & Zhiliang Ying & Hong Zhang & Lincheng Zhao, 2008. "Analysis of least absolute deviation," Biometrika, Biometrika Trust, vol. 95(1), pages 107-122.
    6. Robinson, P. M., 1977. "The estimation of a nonlinear moving average model," Stochastic Processes and their Applications, Elsevier, vol. 5(1), pages 81-90, February.
    7. Ling, Shiqing, 2007. "Self-weighted and local quasi-maximum likelihood estimators for ARMA-GARCH/IGARCH models," Journal of Econometrics, Elsevier, vol. 140(2), pages 849-873, October.
    8. 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.
    9. repec:bla:jfinan:v:44:y:1989:i:5:p:1115-53 is not listed on IDEAS
    10. Zhu, Ke & Li, Wai Keung, 2015. "A bootstrapped spectral test for adequacy in weak ARMA models," Journal of Econometrics, Elsevier, vol. 187(1), pages 113-130.
    11. Glosten, Lawrence R & Jagannathan, Ravi & Runkle, David E, 1993. "On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks," Journal of Finance, American Finance Association, vol. 48(5), pages 1779-1801, December.
    12. Shiqing Ling & Michael McAleer, 2010. "A general asymptotic theory for time‐series models," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(1), pages 97-111, February.
    13. He, Xuming, et al, 1990. "Tail Behavior of Regression Estimators and Their Breakdown Points," Econometrica, Econometric Society, vol. 58(5), pages 1195-1214, September.
    14. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    15. 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.
    16. Chen, Kani & Guo, Shaojun & Lin, Yuanyuan & Ying, Zhiliang, 2010. "Least Absolute Relative Error Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 105(491), pages 1104-1112.
    17. Zhu, Ke & Ling, Shiqing, 2012. "THE GLOBAL WEIGHTED LAD ESTIMATORS FOR FINITE/INFINITE VARIANCE ARMA(p,q) MODELS," Econometric Theory, Cambridge University Press, vol. 28(5), pages 1065-1086, October.
    18. Koenker, Roger & Bassett, Gilbert, Jr, 1982. "Robust Tests for Heteroscedasticity Based on Regression Quantiles," Econometrica, Econometric Society, vol. 50(1), pages 43-61, January.
    19. Fornari, Fabio & Mele, Antonio, 1997. "Sign- and Volatility-Switching ARCH Models: Theory and Applications to International Stock Markets," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 12(1), pages 49-65, Jan.-Feb..
    20. He, Changli & Terasvirta, Timo, 1999. "Properties of moments of a family of GARCH processes," Journal of Econometrics, Elsevier, vol. 92(1), pages 173-192, September.
    21. Yao, Qiwei & Brockwell, Peter J, 2006. "Gaussian maximum likelihood estimation for ARMA models II: spatial processes," LSE Research Online Documents on Economics 5416, London School of Economics and Political Science, LSE Library.
    22. 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.
    23. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
    24. 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.
    25. Yao, Qiwei & Brockwell, Peter J, 2006. "Gaussian maximum likelihood estimation for ARMA models. I. Time series," LSE Research Online Documents on Economics 57580, London School of Economics and Political Science, LSE Library.
    26. Guodong Li & Wai Keung Li, 2008. "Least absolute deviation estimation for fractionally integrated autoregressive moving average time series models with conditional heteroscedasticity," Biometrika, Biometrika Trust, vol. 95(2), pages 399-414.
    27. Lucas, André, 1995. "Unit Root Tests Based on M Estimators," Econometric Theory, Cambridge University Press, vol. 11(2), pages 331-346, February.
    28. Shiqing Ling, 2005. "Self‐weighted least absolute deviation estimation for infinite variance autoregressive models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(3), pages 381-393, June.
    29. Qiwei Yao & Peter J. Brockwell, 2006. "Gaussian Maximum Likelihood Estimation For ARMA Models. I. Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(6), pages 857-875, November.
    30. Zakoian, Jean-Michel, 1994. "Threshold heteroskedastic models," Journal of Economic Dynamics and Control, Elsevier, vol. 18(5), pages 931-955, September.
    31. Keith Knight, 1987. "Rate Of Convergence Of Centred Estimates Of Autoregressive Parameters For Infinite Variance Autoregressions," Journal of Time Series Analysis, Wiley Blackwell, vol. 8(1), pages 51-60, January.
    32. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
    33. Ngai Hang Chan & Liang Peng, 2005. "Weighted least absolute deviations estimation for an AR(1) process with ARCH(1) errors," Biometrika, Biometrika Trust, vol. 92(2), pages 477-484, June.
    34. 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.
    35. Yao, Qiwei & Brockwell, Peter J., 2006. "Gaussian maximum likelihood estimation for ARMA models I: time series," LSE Research Online Documents on Economics 5825, London School of Economics and Political Science, LSE Library.
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    Citations

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    Cited by:

    1. Ke Zhu, 2016. "Bootstrapping the portmanteau tests in weak auto-regressive moving average models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(2), pages 463-485, March.
    2. Ding, Jing & Jiang, Lei & Liu, Xiaohui & Peng, Liang, 2023. "Nonparametric tests for market timing ability using daily mutual fund returns," Journal of Economic Dynamics and Control, Elsevier, vol. 150(C).
    3. Feiyu Jiang & Dong Li & Ke Zhu, 2019. "Non-standard inference for augmented double autoregressive models with null volatility coefficients," Papers 1905.01798, arXiv.org.
    4. Ke Zhu, 2018. "Statistical inference for autoregressive models under heteroscedasticity of unknown form," Papers 1804.02348, arXiv.org, revised Aug 2018.
    5. Jiang, Feiyu & Li, Dong & Zhu, Ke, 2020. "Non-standard inference for augmented double autoregressive models with null volatility coefficients," Journal of Econometrics, Elsevier, vol. 215(1), pages 165-183.
    6. Zhang, Xingfa & Zhang, Rongmao & Li, Yuan & Ling, Shiqing, 2022. "LADE-based inferences for autoregressive models with heavy-tailed G-GARCH(1, 1) noise," Journal of Econometrics, Elsevier, vol. 227(1), pages 228-240.
    7. Yang, Yaxing & Ling, Shiqing & Wang, Qiying, 2022. "Consistency of global LSE for MA(1) models," Statistics & Probability Letters, Elsevier, vol. 182(C).
    8. Aboagye, Ernest & Ko, Stanley Iat-Meng & Lo, Chia Chun & Hsiao, Cody Yu-Ling & Peng, Liang, 2024. "A contagion test with unspecified heteroscedastic errors," Journal of Economic Dynamics and Control, Elsevier, vol. 159(C).
    9. Li, Dong & Ling, Shiqing & Zhu, Ke, 2016. "ZD-GARCH model: a new way to study heteroscedasticity," MPRA Paper 68621, University Library of Munich, Germany.

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    More about this item

    Keywords

    ARMA(p; q) models; Asymptotic normality; Heavy-tailed noises; G/ARCH noises; LADE; Random weighting approach; Self-weighted LADE; Sign-based portmanteau test; Strong consistency.;
    All these keywords.

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General

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