# Hodrick–Prescott filter

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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.[1] However, it was first proposed much earlier by E. T. Whittaker in 1923.[2]

## The equation

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 ${\displaystyle y_{t}\ =\tau _{t}\ +c_{t}\ +\epsilon _{t}\,}$.[3] Given an adequately chosen, positive value of ${\displaystyle \lambda }$, there is a trend component that will solve

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

The first term of the equation is the sum of the squared deviations ${\displaystyle d_{t}=y_{t}-\tau _{t}}$ which penalizes the cyclical component. The second term is a multiple ${\displaystyle \lambda }$ 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 ${\displaystyle \lambda }$, the higher is the penalty. Hodrick and Prescott suggest 1600 as a value for ${\displaystyle \lambda }$ for quarterly data. Ravn and Uhlig (2002) state that ${\displaystyle \lambda }$ should vary by the fourth power of the frequency observation ratio; thus, ${\displaystyle \lambda }$ should equal 6.25 for annual data and 129,600 for monthly data.[4]

## Drawbacks to the Hodrick–Prescott filter

The Hodrick–Prescott filter will only be optimal when:[5]

• 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 normally distributed.
• Analysis is purely historical and static (closed domain). The filter causes misleading predictions when used dynamically since the algorithm changes (during iteration for minimization) the past state (unlike a moving average) of the time series to adjust for the current state regardless of the size of ${\displaystyle \lambda }$ used.

The standard two-sided Hodrick–Prescott filter is non-causal as it is not purely backward looking. Hence, it should not be used when estimating DSGE models based on recursive state-space representations (e.g., likelihood-based methods that make use of the Kalman filter). The reason is that the Hodrick–Prescott filter uses observations at ${\displaystyle t+i,i>0}$ to construct the current time point ${\displaystyle t}$, while the recursive setting assumes that only current and past states influence the current observation. One way around this is to use the one-sided Hodrick–Prescott filter.[6]

Exact algebraic formulas are available for the two-sided Hodrick–Prescott filter in terms of its signal-to-noise ratio ${\displaystyle \lambda }$.[7]

## References

1. ^ Hodrick, Robert; Prescott, Edward C. (1997). "Postwar U.S. Business Cycles: An Empirical Investigation". Journal of Money, Credit, and Banking. 29 (1): 1–16. JSTOR 2953682.
2. ^ Whittaker, E. T. (1923). "On a New Method of Graduation". Proceedings of the Edinburgh Mathematical Association. 41: 63–75. doi:10.1017/S001309150000359X. - as quoted in Philips 2010
3. ^ Kim, Hyeongwoo. "Hodrick–Prescott Filter" March 12, 2004
4. ^ Ravn, Morten; Uhlig, Harald (2002). "On adjusting the Hodrick–Prescott filter for the frequency of observations". The Review of Economics and Statistics. 84 (2): 37.
5. ^ French, Mark W. (2001). "Estimating Changes in Trend Growth of Total Factor Productivity: Kalman and H-P Filters versus a Markov-Switching Framework". FEDS Working Paper No. 2001-44. SSRN .
6. ^ Stock; Watson (1999). "Forecasting Inflation". Journal of Monetary Economics. 44: 293–335. doi:10.1016/s0304-3932(99)00027-6.
7. ^ McElroy (2008). "Exact Formulas for the Hodrick-Prescott Filter". Econometrics Journal. 11: 209–217. doi:10.1111/j.1368-423X.20008.00230.x.