Normalized least-squares estimation in time-varying ARCH models

TL;DR

This paper introduces a simple, efficient kernel normalized-least-squares estimator for time-varying ARCH models, improving small-sample performance and enabling accurate financial data modeling and forecasting.

Contribution

It proposes a novel kernel-NLS estimator with a closed form, outperforming existing methods in small samples and providing a practical approach for modeling time-varying volatility.

Findings

Kernel-NLS estimator has the same convergence rate as kernel-QML.

Estimator performs well with small samples and under mild assumptions.

Application to financial data yields accurate fits and forecasts.

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.