R-Squared-Bootstrapping for Gegenbauer-Type Long Memory

Abstract

The autocorrelation of long memory processes decays much slower than that of short memory processes, e.g. autoregressive processes. Fractional integration and Gegenbauer are basic models that show long memory behavior. Many tests for long memory of the fractional integration type (at f = 0) have been developed based on test statistics (e.g. the R/S-type statistics) or parameter estimators (e.g. the GPH estimator). But, so far, no work has shown reasonable power on testing for cyclic long memory. The authors investigate a parametric bootstrap procedure with an R-squared statistic as an assessment of cyclic long memory behavior. According to simulation results, the R-squared-bootstrapping method performs excellently in detecting Gegenbauer-type processes (e.g. with long memory behavior associated with frequencies $$f\in (0,0.5]$$ ) while at the same time controlling observed significance levels. The R-squared-bootstrapping test is also applied to the Lynx data and the result suggests the presence of long range dependence.