Estimating long memory in volatility

We consider semiparametric estimation of the memory parameter in a model that includes as special cases both long-memory stochastic volatility and fractionally integrated exponential GARCH (FIEGARCH) models. Under our general model the logarithms of the squared returns can be decomposed into the sum of a long-memory signal and a white noise. We consider periodogram-based estimators using a local Whittle criterion function. We allow the optional inclusion of an additional term to account for possible correlation between the signal and noise processes, as would occur in the FIEGARCH model. We also allow for potential nonstationarity in volatility by allowing the signal process to have a memory parameter d* ≥ 1/2. We show that the local Whittle estimator is consistent for d* ∈ (0, 1). We also show that the local Whittle estimator is asymptotically normal for d* ∈ (0, 3/4) and essentially recovers the optimal semiparametric rate of convergence for this problem. In particular, if the spectral density of the short-memory component of the signal is sufficiently smooth, a convergence rate of n^2/5 for d* ∈ (0, 3/4) can be attained, where n is the sample size and δ > 0 is arbitrarily small. This represents a strong improvement over the performance of existing semiparametric estimators of persistence in volatility. We also prove that the standard Gaussian semiparametric estimator is asymptotically normal if d* = 0. This yields a test for long memory in volatility.