Overlaying Time Scales and Persistence Estimation in GARCH(1,1) Models

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. After showing that the expectation of the sum of the estimates of the autoregressive coefficients of a GARCH(1,1) model is one when there are unknown parameter changes, we explore the phenomenon in simulations. 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. Spectral analysis of the Dow Jones Industrial Average and the S&P500 index between 1985 and 2001 reveals a short time scale of the magnitude of 5-10 days present in the data. Thus, two different time scales exist in the data, one of the order of months corresponding to different volatility regimes, and one of the order of days corresponding to the average mean reversion within regimes. ∗It is a pleasure to thank George Papanicolaou for his invitation to Stanford and many discussions. I am grateful to Caio Almeida, Arnold Kim, Mordecai Kurz, Jonathan Mattingly, and Knut Sølna for many helpful comments. Any remaining errors are mine.