The Hodrick–Prescott (HP) filter is a commonly used tool in macroeconomics to obtain the HP filter trend of a macroeconomic variable. In macroeconomics, the difference between the original series and this trend is called the ‘cyclical component’. In this article, we derive the autocovariance function and the spectrum of the cyclical component of a series that consists of a constant, a linear time trend, a unit root process, and a weakly stationary process. We show that the autocovariance function of the cyclical component of such a series depends on (i) the autocovariance of the innovations of the unit root process; (ii) the autocovariance of the weakly stationary process and; (iii) a component of the weights of the HP filter that is important in the middle of a large sample. The result for the spectrum of the cyclical component matches with earlier results in the literature that were obtained by using an approximate approach. Lastly, we derive the cross-covariance function and the cross-spectrum of the cyclical components of two cointegrated series.

中文翻译：

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Hodrick-Prescott 滤波器的光谱分析

Hodrick-Prescott (HP) 过滤器是宏观经济学中常用的工具，用于获取宏观经济变量的 HP 过滤器趋势。在宏观经济学中，原始序列与这一趋势之间的差异称为“周期性成分”。在本文中，我们推导了由常数、线性时间趋势、单位根过程和弱平稳过程组成的序列的循环分量的自协方差函数和谱。我们证明了这样一个系列的周期性分量的自协方差函数取决于（i）单位根过程创新的自协方差；(ii) 弱平稳过程的自协方差；(iii) HP 过滤器权重的一个组成部分，在大样本的中间很重要。周期性分量谱的结果与文献中使用近似方法获得的早期结果相匹配。最后，我们推导出了两个协整序列的周期分量的互协方差函数和互谱。