Estimating long memory in volatility

We consider semiparametric estimation of the memory parameter in a model which includes as special cases both the long-memory stochastic volatility (LMSV) 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 which explicitly account for the noise term in a local Whittle criterion function. We allow the optional inclusion of an additional term to allow for a 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 2 (0; 1). We also show that a modi ed version of the local Whittle estimator is asymptotically normal for d 2 (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 suÆciently smooth, a convergence rate of n Æ for d 2 (0; 3=4) can be attained, where n is the sample size and Æ > 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.