On the Log Periodogram Regression Estimator of the Memory Parameter in Long Memory Stochastic Volatility Models

Abstract

We consider semiparametric estimation of the memory parameter in a long memory stochastic volatility model+ We study the estimator based on a log periodogram regression as originally proposed by Geweke and Porter-Hudak~1983, Journal of Time Series Analysis 4, 221-238!+ Expressions for the asymptotic bias and variance of the estimator are obtained, and the asymptotic distribution is shown to be the same as that obtained in recent literature for a Gaussian long memory series+ The theoretical result does not require omission of a block of frequencies near the origin+ We show that this ability to use the lowest frequencies is particularly desirable in the context of the long memory stochastic volatility model+

Properties of the Sample Autocorrelations of Nonlinear Transformations in Long-Memory Stochastic Volatility Models

“…However, whether these statistics form a suitable basis for model identification for long-memory time series has not been investigated yet. On the other hand, it is also worth pointing out that there are some popular semiparametric estimators of the parameter d in LMSV models, based on the sample autocorrelations of log-squared, squared, and absolute returns, which are typically negatively biased [see, e.g., Andersen and Bollerslev (1997), Bollerslev and Wright (2000), Deo and Hurvich (2001), and Crato and Ray (2002)]. The biases found in these estimators might be related to the negative biases of the sample autocorrelations reported in this article.…”

Long Memory in Nonlinear Processes

“…This rules out most LMSV/LMSD models, since {log v 2 t } is typically non-Gaussian. For the LMSV/LMSD model, results analogous to those of [DH01] were obtained by [Art04] for the GSE estimator, based once again on {log X 2 t }. The use of GSE instead of GPH allows the assumption that {h t } is Gaussian to be weakened to linearity in a Martingale difference sequence.…”

“…This upper bound, m = o[n 4d/(4d+1) ], where n is the sample size, becomes increasingly stringent as d approaches zero. The results in [DH01] assume that d > 0 and hence rule out valid tests for the presence of long memory in {h t }. Such a test based on the GPH estimator was provided and justified theoretically by [HS02].…”

“…The first term in (19) imposes a lower bound on the allowable value of m, requiring that m tend to ∞ faster than n 4d/(4d+1) . It is interesting that [DH01], under similar smoothness assumptions, found that for m 1/2 (d GP H − d) to be asymptotically normal with mean zero, whered GP H is the GPH estimator, the bandwidth m must tend to ∞ at a rate slower than n 4d/(4d+1) . Thus for any given d, the optimal rate of convergence for the local Whittle estimator is faster than that for the GPH estimator.…”

“…This provides some justification for the use of GPH for estimating long memory in volatility. Nevertheless, it can also be seen from Theorem 1 of [DH01] that the presence of the noise term {u t } induces a negative bias in the GPH estimator, which in turn limits the number m of Fourier frequencies which can be used in the estimator while still guaranteeing √ m-consistency and asymptotic normality. This upper bound, m = o[n 4d/(4d+1) ], where n is the sample size, becomes increasingly stringent as d approaches zero.…”