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. However, it was first proposed much earlier by E. T. Whittaker in 1923.

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 $y_{t}\ =\tau _{t}\ +c_{t}\ +\epsilon _{t}\,$. 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.

Drawbacks to the Hodrick–Prescott filter

The Hodrick–Prescott filter will only be optimal when:

• 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.

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

See also

• Band-pass filter
• Kalman filter

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