Trend elimination and business cycle estimation are analyzed by finite sample and asymptotic methods. An overview history is provided, operator theory is developed, limit theory as the sample size is derived, and filtered series properties are studied relative to smoothing parameter () behavior. Simulations reveal that limit theory with delivers excellent approximations to the HP filter for common sample sizes but fails to remove stochastic trends, contrary to standard thinking in macroeconomics and thereby explaining “spurious cycle” effects of the HP filter. The findings are related to the long run effects of the global financial crisis.

中文翻译：

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商业周期，趋势消除和HP过滤器

趋势消除和业务周期估计通过有限样本和渐近方法进行分析。提供了概述历史，发展了算子理论，推导了样本量的极限理论，并研究了与平滑参数（）行为有关的滤波后的序列性质。模拟显示，极限理论可以为常见样本量的HP滤波器提供出色的近似值，但无法消除随机趋势，这与宏观经济学中的标准思维相反，从而可以解释HP滤波器的“虚假周期”效应。这些发现与全球金融危机的长期影响有关。