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Two‐Step Estimation for Time Varying Arch Models

Author

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  • Yuanyuan Zhang
  • Rong Liu
  • Qin Shao
  • Lijian Yang

Abstract

A time varying autoregressive conditional heteroskedasticity (ARCH) model is proposed to describe the changing volatility of a financial return series over long time horizon, along with two‐step least squares and maximum likelihood estimation procedures. After preliminary estimation of the time varying trend in volatility scale, approximations to the latent stationary ARCH series are obtained, which are used to compute the least squares estimator (LSE) and maximum likelihood estimator (MLE) of the ARCH coefficients. Under elementary and mild assumptions, oracle efficiency of the two‐step LSE for ARCH coefficients is established, that is, the two‐step LSE is asymptotically as efficient as the infeasible LSE based on the unobserved ARCH series. As a matter of fact, the two‐step LSE deviates from the infeasible LSE by opn−1/2. The two‐step MLE, however, does not enjoy such efficiency, but n1/2 asymptotic normality is established for both the two‐step MLE as well as its deviation from the infeasible MLE. Simulation studies corroborate the asymptotic theory, and application to the S&P 500 index daily returns from 1950 to 2018 indicates significant change in volatility scale over time.

Suggested Citation

  • Yuanyuan Zhang & Rong Liu & Qin Shao & Lijian Yang, 2020. "Two‐Step Estimation for Time Varying Arch Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 41(4), pages 551-570, July.
    • Handle: RePEc:bla:jtsera:v:41:y:2020:i:4:p:551-570
    DOI: 10.1111/jtsa.12522
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    1. Duan, Jin-Chuan, 1997. "Augmented GARCH (p,q) process and its diffusion limit," Journal of Econometrics, Elsevier, vol. 79(1), pages 97-127, July.
    2. Amado, Cristina & Teräsvirta, Timo, 2013. "Modelling volatility by variance decomposition," Journal of Econometrics, Elsevier, vol. 175(2), pages 142-153.
    3. Hafner, Christian M. & Linton, Oliver, 2010. "Efficient estimation of a multivariate multiplicative volatility model," Journal of Econometrics, Elsevier, vol. 159(1), pages 55-73, November.
    4. Liang Peng, 2003. "Least absolute deviations estimation for ARCH and GARCH models," Biometrika, Biometrika Trust, vol. 90(4), pages 967-975, December.
    5. 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.
    6. Engle, Robert F & Ng, Victor K, 1993. "Measuring and Testing the Impact of News on Volatility," Journal of Finance, American Finance Association, vol. 48(5), pages 1749-1778, December.
    7. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    8. Hall, Peter & Yao, Qiwei, 2003. "Inference in ARCH and GARCH models with heavy-tailed errors," LSE Research Online Documents on Economics 5875, London School of Economics and Political Science, LSE Library.
    9. Engle, Robert F. & White (the late), Halbert (ed.), 1999. "Cointegration, Causality, and Forecasting: Festschrift in Honour of Clive W. J. Granger," OUP Catalogue, Oxford University Press, number 9780198296836.
    10. Yang, Lijian, 2006. "A semiparametric GARCH model for foreign exchange volatility," Journal of Econometrics, Elsevier, vol. 130(2), pages 365-384, February.
    11. Cristina Amado & Timo Teräsvirta, 2014. "Conditional Correlation Models of Autoregressive Conditional Heteroscedasticity With Nonstationary GARCH Equations," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 32(1), pages 69-87, January.
    12. repec:hal:journl:peer-00732539 is not listed on IDEAS
    13. Amado, Cristina & Teräsvirta, Timo, 2014. "Modelling changes in the unconditional variance of long stock return series," Journal of Empirical Finance, Elsevier, vol. 25(C), pages 15-35.
    14. Weiss, Andrew A., 1986. "Asymptotic Theory for ARCH Models: Estimation and Testing," Econometric Theory, Cambridge University Press, vol. 2(1), pages 107-131, April.
    15. Peter Hall & Qiwei Yao, 2003. "Inference in Arch and Garch Models with Heavy--Tailed Errors," Econometrica, Econometric Society, vol. 71(1), pages 285-317, January.
    16. 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.
    17. Hafner, Christian M. & Preminger, Arie, 2009. "Asymptotic Theory For A Factor Garch Model," Econometric Theory, Cambridge University Press, vol. 25(2), pages 336-363, April.
    18. Hafner, Christian M., 2008. "Temporal aggregation of multivariate GARCH processes," Journal of Econometrics, Elsevier, vol. 142(1), pages 467-483, January.
    19. Lionel Truquet, 2017. "Parameter stability and semiparametric inference in time varying auto-regressive conditional heteroscedasticity models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(5), pages 1391-1414, November.
    20. Peng, Liang & Yao, Qiwei, 2003. "Least absolute deviations estimation for ARCH and GARCH models," LSE Research Online Documents on Economics 5828, London School of Economics and Political Science, LSE Library.
    21. Hafner, Christian M. & Herwartz, Helmut, 2006. "Volatility impulse responses for multivariate GARCH models: An exchange rate illustration," Journal of International Money and Finance, Elsevier, vol. 25(5), pages 719-740, August.
    22. Hafner, Christian M. & Preminger, Arie, 2009. "On asymptotic theory for multivariate GARCH models," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 2044-2054, October.
    23. Liu, Rong & Yang, Lijian, 2016. "Spline Estimation Of A Semiparametric Garch Model," Econometric Theory, Cambridge University Press, vol. 32(4), pages 1023-1054, August.
    24. Christian M. Hafner & Arie Preminger, 2017. "On Asymptotic Theory for ARCH (∞) Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 38(6), pages 865-879, November.
    25. Qin Shao & Lijian Yang, 2017. "Oracally efficient estimation and consistent model selection for auto-regressive moving average time series with trend," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(2), pages 507-524, March.
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    Cited by:

    1. Jie Li & Jiangyan Wang & Lijian Yang, 2022. "Kolmogorov–Smirnov simultaneous confidence bands for time series distribution function," Computational Statistics, Springer, vol. 37(3), pages 1015-1039, July.

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