Hodrick-Prescott filter

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 $lambda$. The filter was first applied by economists Robert J. Hodrick and recent Nobel Prize winner Edward C. Prescott. [Hodrick, Robert, and Edward C. Prescott (1997), "Postwar U.S. Business Cycles: An Empirical Investigation," "Journal of Money, Credit, and Banking".] Though Hodrick and Prescott popularized the filter in the field of economics, it was first proposed by CEV Leser. [Leser, C. E. V. 1961. A Simple Method of Trend Construction. Journal of the Royal Statistical Society. Series B (Methodological), 23, 91–107.] [ [http://epub.ub.uni-muenchen.de/304/1/schlicht-HP-3-DP.pdf Ekkehard Schlicht (2004), Estimating the Smoothing Parameter in the so-called Hodrick-Prescott Filter, Discussion Paper 2004-2, Dept. Economics, U Munich] ]

The equation

The reasoning for the formula is as follows: Let $y\_t,$ for $t\; =\; 1,\; 2,\; ...,\; T,$ denote the logarithms of a time series variable. The series $y\_t,$ is made up of a trend component, denoted by $au,$ and a cyclical component, denoted by $c,$ such that $y\_t\; =\; au\_t\; +\; c\_t,$ [Kim, Hyeongwoo. " [http://business.auburn.edu/~hzk0001/hpfilter.pdf Hodrick-Prescott Filter] " March 12, 2004] . Given an adequately chosen, positive value of $lambda$, there is a trend component that will minimize

:min $sum\_\{t\; =\; 1\}^T\; \{(y\_t\; -\; au\; \_t\; )^2\; \}\; +\; lambda\; sum\_\{t\; =\; 2\}^\{T\; -\; 1\}\; \{\; [(\; au\; \_\{t+1\}\; -\; au\; \_t)\; -\; (\; au\; \_t\; -\; au\; \_\{t\; -\; 1\}\; )]\; ^2\; \}.,$

The first term of the equation is the sum of the squared deviations $d\_t=y\_t-\; au\_t$ which penalizes the cyclical component. The second term is a multiple $lambda$ 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 $lambda$, the higher is the penalty. Hodrick and Prescott advise that, for quarterly data, a value of $lambda=1600$ is reasonable.

Drawbacks to H-P filter

The Hodrick-Prescott filter will only be optimal when: [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]
*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).

References

ee also

* Kalman filter
* Band-pass filter

External links

* [http://www.web-reg.de/hp_addin.html "a freeware Hodrick Prescott Excel Add-In"]
* [http://dge.repec.org/codes/prescott/hpfilter.for "Prescott's Fortran code"]