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Normalized least-squares estimation in time-varying ARCH models

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  • Fryzlewicz, Piotr
  • Sapatinas, Theofanis
  • Subba Rao, Suhasini

Abstract

We investigate the time-varying ARCH (tvARCH) process. It is shown that it can be used to describe the slow decay of the sample autocorrelations of the squared returns often observed in financial time series, which warrants the further study of parameter estimation methods for the model. Since the parameters are changing over time, a successful estimator needs to perform well for small samples. We propose a kernel normalized-least-squares (kernel-NLS) estimator which has a closed form, and thus outperforms the previously proposed kernel quasi-maximum likelihood (kernel-QML) estimator for small samples. The kernel-NLS estimator is simple, works under mild moment assumptions and avoids some of the parameter space restrictions imposed by the kernel-QML estimator. Theoretical evidence shows that the kernel-NLS estimator has the same rate of convergence as the kernel-QML estimator. Due to the kernel-NLS estimator’s ease of computation, computationally intensive procedures can be used. A prediction-based cross-validation method is proposed for selecting the bandwidth of the kernel-NLS estimator. Also, we use a residual-based bootstrap scheme to bootstrap the tvARCH process. The bootstrap sample is used to obtain pointwise confidence intervals for the kernel-NLS estimator. It is shown that distributions of the estimator using the bootstrap and the “true” tvARCH estimator asymptotically coincide. We illustrate our estimation method on a variety of currency exchange and stock index data for which we obtain both good fits to the data and accurate forecasts.

Suggested Citation

  • Fryzlewicz, Piotr & Sapatinas, Theofanis & Subba Rao, Suhasini, 2008. "Normalized least-squares estimation in time-varying ARCH models," LSE Research Online Documents on Economics 25187, London School of Economics and Political Science, LSE Library.
  • Handle: RePEc:ehl:lserod:25187
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    References listed on IDEAS

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    Cited by:

    1. Rohan, Neelabh, 2013. "A time varying GARCH(p,q) model and related statistical inference," Statistics & Probability Letters, Elsevier, vol. 83(9), pages 1983-1990.
    2. Bonsoo Koo & Oliver Linton, 2010. "Semiparametric Estimation of Locally Stationary Diffusion Models," STICERD - Econometrics Paper Series 551, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    3. Kejin Wu & Sayar Karmakar, 2023. "A model-free approach to do long-term volatility forecasting and its variants," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 9(1), pages 1-38, December.
    4. Frazier, David T. & Koo, Bonsoo, 2021. "Indirect inference for locally stationary models," Journal of Econometrics, Elsevier, vol. 223(1), pages 1-27.
    5. David T. Frazier & Bonsoo Koo, 2020. "Indirect Inference for Locally Stationary Models," Monash Econometrics and Business Statistics Working Papers 30/20, Monash University, Department of Econometrics and Business Statistics.
    6. Khismatullina, Marina & Vogt, Michael, 2023. "Nonparametric comparison of epidemic time trends: The case of COVID-19," Journal of Econometrics, Elsevier, vol. 232(1), pages 87-108.
    7. David Gabauer & Rangan Gupta & Sayar Karmakar & Joshua Nielsen, 2022. "Stock Market Bubbles and the Forecastability of Gold Returns (and Volatility)," Working Papers 202228, University of Pretoria, Department of Economics.
    8. Chen, Bin & Hong, Yongmiao, 2016. "Detecting For Smooth Structural Changes In Garch Models," Econometric Theory, Cambridge University Press, vol. 32(03), pages 740-791, June.
    9. Stefan Birr & Stanislav Volgushev & Tobias Kley & Holger Dette & Marc Hallin, 2017. "Quantile spectral analysis for locally stationary time series," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(5), pages 1619-1643, November.
    10. Dennis Kristensen & Young Jun Lee, 2019. "Local Polynomial Estimation of Time-Varying Parameters in Nonlinear Models," Papers 1904.05209, arXiv.org, revised Aug 2023.
    11. Feiyu Jiang & Dong Li & Ke Zhu, 2019. "Adaptive inference for a semiparametric generalized autoregressive conditional heteroskedasticity model," Papers 1907.04147, arXiv.org, revised Oct 2020.
    12. Kejin Wu & Sayar Karmakar, 2021. "Model-Free Time-Aggregated Predictions for Econometric Datasets," Forecasting, MDPI, vol. 3(4), pages 1-14, December.
    13. Sayar Karmakar & Marek Chudy & Wei Biao Wu, 2020. "Long-term prediction intervals with many covariates," Papers 2012.08223, arXiv.org, revised Sep 2021.
    14. Fryzlewicz, Piotr & Oh, H. S., 2011. "Thick pen transformation for time series," LSE Research Online Documents on Economics 37663, London School of Economics and Political Science, LSE Library.
    15. Kejin Wu & Sayar Karmakar, 2023. "GARHCX-NoVaS: A Model-free Approach to Incorporate Exogenous Variables," Papers 2308.13346, arXiv.org.
    16. Karmakar, Sayar & Richter, Stefan & Wu, Wei Biao, 2022. "Simultaneous inference for time-varying models," Journal of Econometrics, Elsevier, vol. 227(2), pages 408-428.
    17. Cizek, P., 2010. "Modelling Conditional Heteroscedasticity in Nonstationary Series," Discussion Paper 2010-84, Tilburg University, Center for Economic Research.
    18. Roueff, François & von Sachs, Rainer, 2011. "Locally stationary long memory estimation," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 813-844, April.
    19. Mishra Anuj & Ramanathan Thekke Variyam, 2017. "Nonstationary autoregressive conditional duration models," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 21(4), pages 1-22, September.
    20. Xuejie Feng & Chiping Zhang, 2020. "A Perturbation Method to Optimize the Parameters of Autoregressive Conditional Heteroscedasticity Model," Computational Economics, Springer;Society for Computational Economics, vol. 55(3), pages 1021-1044, March.
    21. Jiang, Feiyu & Li, Dong & Zhu, Ke, 2021. "Adaptive inference for a semiparametric generalized autoregressive conditional heteroskedasticity model," Journal of Econometrics, Elsevier, vol. 224(2), pages 306-329.
    22. 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.

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    More about this item

    Keywords

    cross-validation; (G)ARCH models; kernel smoothing; least-squares estimation; locally stationary models;
    All these keywords.

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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