Sequential Monitoring for Changes in GARCH(1,1) Models Without Assuming Stationarity

Abstract

In this article, we develop two families of sequential monitoring procedure to (timely) detect changes in the parameters of a GARCH(1,1) model. Our statistics can be applied irrespective of whether the historical sample is stationary or not, and indeed without previous knowledge of the regime of the observations before and after the break. In particular, we construct our detectors as the CUSUM process of the quasi-Fisher scores of the log likelihood function. To ensure timely detection, we then construct our boundary function (exceeding which would indicate a break) by including a weighting sequence which is designed to shorten the detection delay in the presence of a changepoint. We consider two types of weights: a lighter set of weights, which ensures timely detection in the presence of changes occurring “early, but not too early” after the end of the historical sample; and a heavier set of weights, called “Rényi weights” which is designed to ensure timely detection in the presence of changepoints occurring very early in the monitoring horizon. In both cases, we derive the limiting distribution of the detection delays, indicating the expected delay for each set of weights. Our methodologies can be applied for a general analysis of changepoints in GARCH(1,1) sequences; however, they can also be applied to detect changes from stationarity to explosivity or vice versa, thus allowing to check for “volatility bubbles”, upon applying tests for stationarity before and after the identified break. Our theoretical results are validated via a comprehensive set of simulations, and an empirical application to daily returns of individual stocks.