Weighted bootstrap in GARCH models

TL;DR

This paper demonstrates that a simple bootstrap method for GARCH models provides reliable confidence intervals with smaller sample sizes, bridging the gap between theory and practical application in financial volatility analysis.

Contribution

The paper introduces and validates a bootstrap methodology for GARCH parameter estimation, showing its effectiveness in finite samples and its equivalence to the original estimator's limit distribution.

Findings

Bootstrap method has the same limit distribution as the original estimator.

Reliable confidence intervals can be constructed with sample sizes as low as 1,000.

Accuracy improves with larger samples, approaching 100,000 observations.

Abstract

GARCH models are useful tools in the investigation of phenomena, where volatility changes are prominent features, like most financial data. The parameter estimation via quasi maximum likelihood (QMLE) and its properties are by now well understood. However, there is a gap between practical applications and the theory, as in reality there are usually not enough observations for the limit results to be valid approximations. We try to fill this gap by this paper, where the properties of a recent bootstrap methodology in the context of GARCH modeling are revealed. The results are promising as it turns out that this remarkably simple method has essentially the same limit distribution, as the original estimatorwith the advantage of easy confidence interval construction, as it is demonstrated in the paper. The finite-sample properties of the suggested estimators are investigated through a simulation study, which ensures that the results are practically applicable for sample sizes as low as a thousand. On the other hand, the results are not 100% accurate until sample size reaches 100 thousands - but it is shown that this property is not a feature of our bootstrap procedure only, as it is shared by the original QMLE, too.