The Annals of Statistics

Normalized least-squares estimation in time-varying ARCH models

Piotr Fryzlewicz, Theofanis Sapatinas, and Suhasini Subba Rao

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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.

Article information

Ann. Statist. Volume 36, Number 2 (2008), 742-786.

First available: 13 March 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62P20: Applications to economics [See also 91Bxx]

Cross-validation (G)ARCH models kernel smoothing least-squares estimation locally stationary models


Fryzlewicz, Piotr; Sapatinas, Theofanis; Subba Rao, Suhasini. Normalized least-squares estimation in time-varying ARCH models. The Annals of Statistics 36 (2008), no. 2, 742--786. doi:10.1214/07-AOS510.

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