Estimating Long Memory in Volatility
26 Pages Posted: 3 Nov 2008
Date Written: February 2002
We consider semiparametric estimation of the memory parameter in a modelwhich includes as special cases both the long-memory stochasticvolatility (LMSV) and fractionally integrated exponential GARCH(FIEGARCH) models. Under our general model the logarithms of the squaredreturns can be decomposed into the sum of a long-memory signal and awhite noise. We consider periodogram-based estimators which explicitlyaccount for the noise term in a local Whittle criterion function. Weallow the optional inclusion of an additional term to allow for acorrelation between the signal and noise processes, as would occur inthe FIEGARCH model. We also allow for potential nonstationarity involatility, by allowing the signal process to have a memory parameter d1=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 estimatoris asymptotically normal for d 2 (0; 3=4), and essentially recovers theoptimal semiparametric rate of convergence for this problem. Inparticular if the spectral density of the short memory component of thesignal is suficiently smooth, a convergence rate of n2=5-δ for d 2(0; 3=4) can be attained, where n is the sample size and δ > 0is arbitrarily small. This represents a strong improvement over theperformance of existing semiparametric estimators of persistence involatility. We also prove that the standard Gaussian semiparametricestimator is asymptotically normal if d = 0. This yields a test forlong memory in volatility.
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