Overlaying Time Scales and Persistence Estimation of Financial Volatility Data

A common finding in the empirical literature is that financial volatility exhibits high persistence, or slow mean reversion of the order of months. We present evidence that financial volatility data contains more than a single time scale. When occasional parameter changes are not accounted for in global GARCH(1,1) estimations, they lead to an estimated persistence far above the average data-generating persistence. For parameter changes within realistic ranges for stock-price volatility we obtain global estimates close to integration while the average data-generating mean reversion is of the order of a few days. We find that the sensitivity of the maximum likelihood estimation of the GARCH(1,1) model to parameter changes is so high that it can be used to detect them. Using synthetic data, we compare this heuristic method to an established changepoint detector. We apply the heuristic to the annualized daily volatility series of the Dow Jones Industrial Average and the S&P500 index between 1985 and 2001 and identify segmentations that compare to each other and reduce the estimated average mean-reversion time to about a week. Spectral analysis supports our findings, it reveals a short time scale of the magnitude of 5-10 days present in the Dow Jones and in the S&P500. We propose a two-scale extension of the analogue to GARCH(1,1) in continuous time. ∗It is a pleasure to thank George Papanicolaou for his invitation to Stanford and the intense cooperation. I thank Knut Sølna, UC Irvine, and Jonathan Mattingly, Stanford, for many helpful discussions. Any remaining errors are mine.