Hodrick–Prescott filter

The Hodrick–Prescott filter (also known as Hodrick–Prescott decomposition) is a mathematical tool used in macroeconomics, especially in real business cycle theory, to remove the cyclical component of a time series from raw data. It is used to obtain a smoothed-curve representation of a time series, one that is more sensitive to long-term than to short-term fluctuations. The adjustment of the sensitivity of the trend to short-term fluctuations is achieved by modifying a multiplier ${\displaystyle \lambda }$. The filter was popularized in the field of economics in the 1990s by economists Robert J. Hodrick and Nobel Memorial Prize winner Edward C. Prescott. However, it was first proposed much earlier by E. T. Whittaker in 1923.

The reasoning for the methodology uses ideas related to the decomposition of time series. Let ${\displaystyle y_{t}\,}$ for ${\displaystyle t=1,2,...,T\,}$ denote the logarithms of a time series variable. The series ${\displaystyle y_{t}\,}$ is made up of a trend component, denoted by ${\displaystyle \tau \,}$ and a cyclical component, denoted by ${\displaystyle c\,}$ such that ${\displaystyle y_{t}\ =\tau _{t}\ +c_{t}\ +\epsilon _{t}\,}$. Given an adequately chosen, positive value of ${\displaystyle \lambda }$, there is a trend component that will solve

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