Testing for Long Memory in Volatility

Book excerpt: We consider the asymptotic behavior of log-periodogram regression estimators ofthe memory parameter in long-memory stochastic volatility models, under the nullhypothesis of short memory in volatility. We show that in this situation, if theperiodogram is computed from the log squared returns, then the estimator is asymptoticallynormal, with the same asymptotic mean and variance that would holdif the series were Gaussian. In particular, for the widely used GPH estimator dGPHunder the null hypothesis, the asymptotic mean of mAtilde;𓂬irc;½dGPH is zero and the asymptoticvariance is piAtilde;𓂬irc;²/24 where m is the number of Fourier frequencies used inthe regression. This justifies an ordinary Wald test for long memory in volatilitybased on the log periodogram of the log squared returns.