Bootstrap-based bandwidth choice for log-periodogram regression

Abstract

The choice of the bandwidth in the local log-periodogram regression is of crucial importance for estimation of the memory parameter of a long memory time series. Different choices may give rise to completely different estimates, which may lead to contradictory conclusions, for example about the stationarity of the series. We propose here a data-driven bandwidth selection strategy that is based on minimizing a bootstrap approximation of the mean-squared error (MSE). Its behaviour is compared with other existing techniques for optimal bandwidth selection in a MSE sense, revealing its better performance in a wider class of models. The empirical applicability of the proposed strategy is shown with two examples: the widely analysed in a long memory context Nile river annual minimum levels and the input gas rate series of Box and Jenkins. Over the last years, the log-periodogram regression, first proposed by Geweke and Porter-Hudak (1983) and analysed in detail by Robinson (1995a), has become one of the most popular tools for the estimation of the memory parameter d in long memory time series. It has been widely applied for statistical inference in empirical research owing to its simple implementation, pivotal asymptotic normality and robustness as a result of the local condition. The log- periodogram regression estimation in the version of Robinson (1995a; LPE hereafter) is based on a simple least squares regression of the logarithm of the periodogram over the logarithm of the m Fourier frequencies closest to the origin, providing that m goes to infinity but more slowly than the sample size such that the band of frequencies used in the estimation shrinks to zero. The parameter m, known as the bandwidth in a local or semi-parametric memory parameter estimation context, plays an important role on the performance of the LPE. A large m reduces the variance at the cost of a higher bias, which in some situations can render meaningless estimates, as for example in the presence of a significant short memory component. On the contrary a low m guarantees a small bias but with larger variability. From an empirical perspective, the estimates of the memory parameter usually vary significantly with the choice of m. Figures 9b and