The spectral analysis of the Hodrick–Prescott filter
推导了HP滤波中周期成分的自协方差函数和谱，揭示了其依赖于单位根过程新息的自协方差、弱平稳过程的自协方差以及样本中部的权重分量，并扩展至协整序列的交叉协方差与交叉谱。
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.