Hodrick-Prescott filter

The Hodrick-Prescott filter is a macroeconomic mathematical tool for analyzing business cycles . It is used to adjust a time series so that it is less dependent on short-term fluctuations. The Hodrick-Prescott filter separates the trend of a time series from the cyclical component. With it you can stationaryize the time series .

The formula

Let it be with the logarithm of the time series. Given a positive , there is a trend component , that minimizes the following term ${\displaystyle y_{t}\,}$ ${\displaystyle t=1,2,...,T\,}$ ${\displaystyle \lambda }$ ${\displaystyle \tau \,}$

${\displaystyle \sum _{t=1}^{T}{(y_{t}-\tau _{t})^{2}}+\lambda \sum _{t=2}^{T-1}{[(\tau _{t+1}-\tau _{t})-(\tau _{t}-\tau _{t-1})]^{2}}.}$

The first term is the sum of the squared deviations of . The second term is a multiple of the sum of the squared second difference of the trend component. This term penalizes the variation in the trend component. The higher , the greater the punishment . Robert J. Hodrick and Edward C. Prescott use for quarterly numbers . The Hodrick-Prescott filter is a two-sided filter. It can compensate for the trend more flexibly than a simple linear filter. ${\displaystyle d_{t}=y_{t}-\tau _{t}}$ ${\displaystyle \lambda }$ ${\displaystyle \lambda }$ ${\displaystyle \lambda =1600}$

The filter was named after Hodrick and Prescott, but the mathematician John von Neumann probably invented a similar device earlier.

Although the HP filter is the most popular means of eliminating a cyclical component, it has received widespread criticism. Above all, the filter shows a distortion at the edges, since estimated values ​​have to be used for the edge values ​​in order to be able to form differences. In addition, the filter cannot show longer trend deviations by design. By choosing from , it is fixed at a certain maximum length of the business cycle . A recession of more than three years, for example, would count as an already downward trend. Ultimately, this remains a freely chosen parameter that has no theoretical foundation. ${\displaystyle \lambda }$ ${\displaystyle \lambda }$

modification

In order to solve the problem of distortion at the edges, Switzerland, which is dependent on this filter method in the context of the debt brake , uses a modified version, the so-called modified Hodrick-Prescott filter . The weights of the last periods are reduced so that the distortion is neutralized.

Web links

• Freeware Hodrick Prescott Excel Add-In
• Stefan Stamfort: Calculation of trend-adjusted indicators for Germany with the help of filter procedures. Deutsche Bundesbank, Discussion Paper Series 1: Economic Studies No. 19/2005