3.13 Long-Memory Stochastic Volatility Model

More recently, the SV model is further extended to allow for long memory in volatility, using the idea of fractional difference. As stated in Chapter 2, a time series is a long-memory process if its autocorrelation function decays at a hyperbolic, instead of an exponential, rate as the lag increases. The extension to long-memory models in volatility study is motivated by the fact that the autocorrelation function of the squared or absolute-valued series of an asset return often decays slowly, even though the return series has no serial correlation; see Ding, Granger, and Engle (1993). Figure 3.10 shows the sample ACF of the daily absolute returns for IBM stock and the S&P 500 index from July 3, 1962, to December 31, 2003. These sample ACFs are positive with moderate magnitude but decay slowly.

Figure 3.10 Sample ACF of daily absolute log returns for (a) S&P 500 index and (b) IBM stock for period from July 3, 1962, to December 31, 2003. Two horizontal lines denote asymptotic 5% limits.

A simple long-memory stochastic volatility (LMSV) model can be written as

(3.41)

where σ > 0, the ϵt are iid N(0, 1), the ηt are iid and independent of ϵt, and 0 < d < 0.5. The feature of long memory stems from the fractional difference (1 − B)d, which implies that the ACF of ut decays slowly at a hyperbolic, instead of an exponential, rate as the lag increases. For model (3.41), we have

Thus, the series is a Gaussian long-memory signal plus a non-Gaussian white noise; see Breidt, Crato, and de Lima (1998). Estimation of the LMSV model is complicated, but the fractional difference parameter d can be estimated by using either a quasi-maximum-likelihood method or a regression method. Using the log series of squared daily returns for companies in the S&P 500 index, Bollerslev and Jubinski (1999) and Ray and Tsay (2000) found that the median estimate of d is about 0.38. For applications, Ray and Tsay (2000) studied common long-memory components in daily stock volatilities of groups of companies classified by various characteristics. They found that companies in the same industrial or business sector tend to have more common long-memory components (e.g., big U.S. national banks and financial institutions).