This article explores a simple property of the Hodrick–Prescott (HP) filter: when the HP filter is applied to a series, the cyclical component is equal to the HP-filtered trend of the fourth difference of the series, except for the first and last two observations, for which different formulas are needed. We use this result to derive small sample results and asymptotic results for a fixed smoothing parameter. We first apply this property to analyze the consequences of a deterministic break. We find that the effect of a deterministic break on the cyclical component is asymptotically negligible for the points that are away from the break point, while for the points in the neighborhood of the break point, the effect is not negligible even asymptotically. Second, we apply this property to show that the cyclical component of the HP filter when applied to series that are integrated up to order 2 is weakly dependent, while the situation for series that are integrated up to order 3 or 4 is more subtle. Third, we characterize the behavior of the HP filter when applied to deterministic polynomial trends and show that in the middle of the sample, the cyclical component reduces the order of the polynomial by 4, while the end point behavior is different. Finally, we give a characterization of the HP filter when applied to an exponential deterministic trend, and this characterization shows that the filter is effectively incapable of dealing with a trend that increases this fast. Our results are compared with those of Phillips and Jin (2015, Business cycles, trend elimination, and the HP filter).

中文翻译：

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HODRICK-PRSCOTT滤波器的特性及其应用

本文探讨了 Hodrick-Prescott (HP) 滤波器的一个简单性质：当 HP 滤波器应用于一个序列时，周期性分量等于该序列的第四个差分的 HP-filtered 趋势，除了第一个和最后两个观察，需要不同的公式。我们使用这个结果来导出固定平滑参数的小样本结果和渐近结果。我们首先应用此属性来分析确定性中断的后果。我们发现，对于远离断点的点，确定性断点对周期性分量的影响是渐近可忽略的，而对于断点附近的点，即使渐近，该影响也不可忽略。第二，我们应用此属性来表明 HP 滤波器的循环分量在应用于积分高达 2 阶的系列时是弱相关的，而积分高达 3 或 4 阶的系列的情况更为微妙。第三，我们描述了 HP 滤波器在应用于确定性多项式趋势时的行为，并表明在样本中间，循环分量将多项式的阶数减少了 4，而端点行为不同。最后，我们给出了 HP 滤波器在应用于指数确定性趋势时的特性，该特性表明滤波器实际上无法处理快速增长的趋势。我们的结果与 Phillips 和 Jin（2015，商业周期、趋势消除和 HP 过滤器）的结果进行了比较。