Hodrick-Prescott filter
From Wikipedia, the free encyclopedia
The Hodrick-Prescott filter is a mathematical tool used in macroeconomics, especially in real business cycle theory. It is used to obtain a smoothed non-linear 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 λ.
Contents |
[edit] The formula
The reasoning for the formula is as follows: Let 
for 
denote the logarithms of a time series variable. The series is made up of a trend component, denoted by 
and a cyclical component, denoted by 
such that 
[1]. Given an adequately chosen, positive value of λ, there is a trend component that will minimize
- min
![\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 }.\,](../../../../math/3/0/2/302de546828ebee8262fe6ea95501bcc.png)
The first term of the equation is the sum of the squared deviations dt = yt − τt which penalizes the cyclical component. The second term is a multiple λ of the sum of the squares of the trend component's second differences. This second term penalizes variations in the growth rate of the trend component. The larger the value of λ, the higher is the penalty. Hodrick and Prescott advise that, for quarterly data, a value of λ = 1600 is reasonable.
The filter was first applied by economists Robert J. Hodrick and recent Nobel Prize winner Edward C. Prescott.[2] Though Hodrick and Prescott were the first to make use of the filter in the field of economics, it is believed that others, such as the mathematician John von Neumann, had already employed similar versions in the past.
[edit] Drawbacks to H-P filter
The Hodrick-Prescott filter will only be optimal when:[3]
- Data exists in a I(2) trend.
- If one-time permanent shocks or split growth rates occur, the filter will generate shifts in the trend that do not actually exist.
- Noise in data is approximately Normal~(0,σ²)(White Noise).
[edit] Notes
- ^ Kim, Hyeongwoo. "Hodrick-Prescott Filter" March 12, 2004
- ^ Hodrick, Robert, and Edward C. Prescott (1997), "Postwar U.S. Business Cycles: An Empirical Investigation," Journal of Money, Credit, and Banking.
- ^ French, Mark. "[http://www.federalreserve.gov/pubs/feds/2001/200144/200144pap.pdf Estimating changes in trend growth of total factor productivity: Kalman and H-P filters versus a Markov-switching framework]" September 6th, 2001
