Hodrick–Prescott filter

The Hodrick–Prescott filter is a mathematical tool used in macroeconomics, especially in real business cycle theory to separate 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 λ. The filter was first applied by economists Robert J. Hodrick and Nobel Prize(2004) winner Edward C. Prescott. Though Hodrick and Prescott popularized the filter in the field of economics, it was first proposed by E. T. Whittaker (Whittaker, E. T. (1923). On a new method of graduation, Proceedings of the Edinburgh Mathematical Association, 78, 81-89.)

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 $y_t\ = \tau_t\ + c_t\,$. Given an adequately chosen, positive value of λ, there is a trend component that will minimize

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

The first term of the equation is the sum of the squared deviations d_t = y_t − τ_t 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.

Drawbacks to the H–P 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 has misleading predictive outcome 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 λ used.

See also

• Kalman filter
• Band-pass filter

External links

• a freeware Hodrick Prescott Excel Add-In
• Prescott's Fortran code
• Hodrick–Prescott filter in matlab