A supreme test for periodic explosive GARCH

TL;DR

This paper introduces a uniform statistical test to detect and locate explosive periods in GARCH processes, which are crucial for understanding volatility behavior in financial time series.

Contribution

It develops a novel double supreme test for detecting and dating parameter changes in GARCH models, with theoretical guarantees and practical algorithms.

Findings

Test performs well in simulations with good size and power.

Successfully applied to Apple and Bitcoin return data.

Provides a new tool for volatility analysis in finance.

Abstract

We develop a uniform test for detecting and dating explosive behavior of a strictly stationary GARCH$(r,s)$ (generalized autoregressive conditional heteroskedasticity) process. Namely, we test the null hypothesis of a globally stable GARCH process with constant parameters against an alternative where there is an 'abnormal' period with changed parameter values. During this period, the change may lead to an explosive behavior of the volatility process. It is assumed that both the magnitude and the timing of the breaks are unknown. We develop a double supreme test for the existence of a break, and then provide an algorithm to identify the period of change. Our theoretical results hold under mild moment assumptions on the innovations of the GARCH process. Technically, the existing properties for the QMLE in the GARCH model need to be reinvestigated to hold uniformly over all possible periods of change. The key results involve a uniform weak Bahadur representation for the estimated parameters, which leads to weak convergence of the test statistic to the supreme of a Gaussian Process. In simulations we show that the test has good size and power for reasonably large time series lengths. We apply the test to Apple asset returns and Bitcoin returns.