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 $\lambda$. 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 $y_t\,$ 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 $\tau\,$ and a cyclical component, denoted by $c\,$ such that $y_t\ = \tau_t\ + c_t\ + \epsilon_t\,$.^[3] Given an adequately chosen, positive value of $\lambda$, there is a trend component that will solve

$\min_{\tau}\left(\sum_{t = 1}^T {(y_t - \tau _t )^2 } + \lambda \sum_{t = 2}^{T - 1} {[(\tau _{t+1} - \tau _t) - (\tau _t - \tau _{t - 1} )]^2 }\right).\,$

The first term of the equation is the sum of the squared deviations $d_t=y_t-\tau_t$ 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 suggest 1600 as a value for $\lambda$ for quarterly data. Ravn and Uhlig (2002) state that $\lambda$ should vary by the fourth power of the frequency observation ratio; thus, $\lambda$ 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 $\lambda$ 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 $t+i, i>0$ to construct the current time point $t$, 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 $\lambda$.^[7]

See also

• Band-pass filter
• Kalman filter

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 293105.
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.

Further reading

• Enders, Walter (2010). "Trends and Univariate Decompositions". Applied Econometric Time Series (Third ed.). New York: Wiley. pp. 247–7. ISBN 978-0470-50539-7.
• Favero, Carlo A. (2001). Applied Macroeconometrics. New York: Oxford University Press. pp. 54–5. ISBN 0-19-829685-1.
• Mills, Terence C. (2003). "Filtering Economic Time Series". Modelling Trends and Cycles in Economic Time Series. New York: Palgrave McMillan. pp. 75–102. ISBN 1-4039-0209-7.

External links

• a freeware Hodrick Prescott Excel Add-In
• Prescott's Fortran code
• Hodrick–Prescott filter in matlab
• One-sided Hodrick–Prescott filters in matlab
• HP filter in R with package 'mFilter'
• HP filter online app