Volume 7, Issue 2 -

This paper develops a new test of true versus spurious long memory, based on log-periodogram estimation of the long memory parameter using skip-sampled data. A correction factor is derived to overcome the bias in this estimator due to aliasing. The procedure is designed to be used in the context of a conventional test of significance of the long memory parameter, and a composite test procedure is described that has the properties of known asymptotic size and consistency. The test is implemented using the bootstrap, with the distribution under the null hypothesis being approximated using a dependent-sample bootstrap technique to approximate short-run dependence following fractional differencing. The properties of the test are investigated in a set of Monte Carlo experiments. The procedure is illustrated by applications to exchange rate volatility and dividend growth series.

This paper discusses tests for the cointegration rank of integrated vector autoregressions when the series are recursively adjusted for deterministic components. To this end, the asymptotic properties of recursive, or adaptive, procedures for the removal of general additive deterministic components are analyzed in two different, complementary, situations. When the stochastic component of the examined time series is weakly stationary (as would be the equilibrium errors), the effect of recursive adjustment vanishes with increasing sample size. When the suitably normalized stochastic component converges weakly to some limiting continuous-time process with integrable paths (as would be the case with the common stochastic trends), recursive adjustment has a permanent effect even asymptotically: the normalized recursively adjusted process converges weakly to a recursively adjusted version of the limiting process. The null limiting distributions of the cointegration rank tests can be expressed in terms of recursively adjusted Brownian motions. Moreover, the finite-sample properties of the cointegration rank tests with recursive adjustment are examined in cases of empirical relevance: the considered deterministic components are a constant, and a constant and a linear trend, respectively. Compared to the likelihood ratio tests or the tests with generalized least squares adjustment, improvements in terms of empirical rejection frequencies under the null are found in finite samples; improvements are found under the alternative as well, with the likelihood ratio test performing increasingly better as the magnitude of the initial condition increases. Regarding rank selection, a very simple combination of the three testing procedures with different adjustments performs best.

The paper uses the daily stock market index returns of Fama–French to attempt a comparative forecasting analysis of different volatility models. The comparison naturally pre-requests the specification of the competing volatility frameworks and therefore the paper among other issues deals with dilemmas about whether volatility–return relations hold. As expected the analysis focuses on FIEGARCH-M models that extend the basic long memory volatility framework of Bollerslev and Mikkelsen (1996. “Modeling and Pricing Long Memory in Stock Market Volatility.” Journal of Econometrics 73:151–84) with the introduction of a volatility in mean effect. Taking also into consideration the work of Christensen, Nielsen, and Zhu (2007. “The Effect of Long Memory in Volatility of Stock Market Fluctuations.” Review of Economics and Statistics 89:684–700)for the existence of spillover effects when conditional in mean equations hold a stationary and a long memory component the analysis estimates the filter long memory volatility models FIEGARCH-MG and FIEGARCH-MH presented in Christensen, Nielsen, and Zhu (2010. “Long Memory in Stock Market Volatility and the Volatility-in-Mean Effect: The FIEGARCH-M Model.” Journal of Econometrics 155:170–87) in order to test whether such filter adjustments can improve volatility forecasting. Although there is no particular reason to assume that the stationary inputs in the return equations will necessarily follow the normal distribution that Christensen, Nielsen, and Zhu (2010. “Long Memory in Stock Market Volatility and the Volatility-in-Mean Effect: The FIEGARCH-M Model.” Journal of Econometrics 155:170–87) assume, the paper follows this path but nevertheless enriches this aspect of the analysis by introducing alternative distributional assumptions. The results indicate the existence of a statistically significant mean effect when both filter models are estimated under the assumption of t-student distribution, although as far as volatility forecasting is concerned both filtered models cannot outperform in terms of forecasting criteria the parsimonious FIEGARCH version that dominates filter and non-filter volatility models under various forecasting horizons.

In this paper, we investigate how close a fractional process can be to a random walk with drift in terms of the sample path. Given the innovation sequence, we calculate the distance to the closest random walk with drift in the sum of squares sense. We also derive the expected distance between the processes under the assumption of white noise normal innovations. A local approximation formula for this distance is given in terms of the sample size, showing that it increases with the sample size more rapidly than the square of the number of observations. Two empirical examples illustrate the results.