We establish some asymptotic properties of a log-periodogram regression estimator for the memory parameter of a long-memory time series. We consider the estimator originally proposed by Geweke and Porter-Hudak (The estimation and application of long memory time series models. J. Time Ser. Anal. 4 (1983), 221±37). In particular, we do not omit any of the low frequency periodogram ordinates from the regression. We derive expressions for the estimator's asymptotic bias, variance and mean squared error as functions of the number of periodogram ordinates, m, used in the regression. Consistency of the estimator is obtained as long as m 3 I and n 3 I with (m log m)an 3 0, where n is the sample size. Under these and the additional conditions assumed in this paper, the optimal m, minimizing the mean squared error, is of order O(n 4a5 ). We also establish the asymptotic normality of the estimator. In a simulation study, we assess the accuracy of our asymptotic theory on mean squared error for ®nite sample sizes. One ®nding is that the choice m n 1a2 , originally suggested by Geweke and Porter-Hudak (1983), can lead to performance which is markedly inferior to that of the optimal choice, even in reasonably small samples.

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+

We study the asymptotic distribution of the sample standardized spectral distribution function when the observed series is a conditionally heteroscedastic martingale di!erence. We show that the asymptotic distribution is no longer a Brownian bridge but another Gaussian process. Furthermore, this limiting process depends on the covariance structure of the second moments of the series. We show that this causes test statistics based on the sample spectral distribution, such as the CrameH r von-Mises statistic, to have heavily right skewed distributions, which will lead to over-rejection of the martingale hypothesis in favour of mean reversion. A non-parametric correction to the test statistics is proposed to account for the conditional heteroscedasticity. We demonstrate that the corrected version of the CrameH r von-Mises statistic has the usual limiting distribution which would be obtained in the absence of conditional heteroscedasticity. We also present Monte Carlo results on the "nite sample distributions of uncorrected and corrected versions of the CrameH r von-Mises statistic. Our simulation results show that this statistic can provide signi"cant gains in power over the Box}Ljung}Pierce statistic against long-memory alternatives. An empirical application to stock returns is also provided.2000 Elsevier Science S.A. All rights reserved.

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Abstract. We consider the problem of selecting the number of frequencies, m, in a log-periodogram regression estimator of the memory parameter d of a Gaussian longmemory time series. It is known that under certain conditions the optimal m, minimizing the mean squared error of the corresponding estimator of d, is given by m (opt) C n 4a5 , where n is the sample size and C is a constant. In practice, C would be unknown since it depends on the properties of the spectral density near zero frequency. In this paper, we propose an estimator of C based again on a logperiodogram regression and derive its consistency. We also derive an asymptotically valid con®dence interval for d when the number of frequencies used in the regression is deterministic and proportional to n 4a5 . In this case, squared bias cannot be neglected since it is of the same order as the variance. In a Monte Carlo study, we examine the performance of the plug-in estimator of d, in which m is obtained by using the estimator of C in the formula for m (opt) above. We also study the performance of a bias-corrected version of the plug-in estimator of d. Comparisons with the choice m n 1a2 frequencies, as originally suggested by Geweke and Porter-Hudak (The estimation and application of long memory time series models.