Boosting the HP filter for trending time series with long-range dependence
将HP滤波器和提升HP滤波器的渐近理论扩展到长程依赖时间序列，证明提升迭代可一致估计分数布朗运动极限过程，并提供数据驱动的停止准则。
This article extends recent asymptotic theory developed for the Hodrick Prescott (HP) filter and boosted HP (bHP) filter to long-range dependent time series that have fractional Brownian motion (fBM) limit processes after suitable standardization. Under general conditions, it is shown that the asymptotic form of the HP filter is a smooth curve, analogous to the finding in Phillips and Jin for integrated time series and series with deterministic drifts. Boosting the filter using the iterative procedure suggested in Phillips and Shi leads under well-defined rate conditions to a consistent estimate of the fBM limit process or the fBM limit process with an accompanying deterministic drift when that is present. A stopping criterion is used to automate the boosting algorithm, giving a data-determined method for practical implementation. The theory is illustrated in simulations and two real data examples that highlight the differences between simple HP filtering and the use of boosting. The analysis is assisted by employing a uniformly and almost surely convergent trigonometric series representation of fBM.