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 λ {\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, though it was first proposed much earlier by E. T. Whittaker in 1923., see Whittaker-Henderson smoothing. The Hodrick–Prescott filter is a special case of a smoothing spline.
The equation The reasoning for the methodology uses ideas related to the decomposition of time series. Let y t {\displaystyle y_{t}\,} for t = 1 , 2 , . . . , T {\displaystyle t=1,2,...,T\,} denote the logarithms of a time series variable. The series y t {\displaystyle y_{t}\,} is made up of a trend component τ t {\displaystyle \tau _{t}} and a cyclical component c t {\displaystyle c_{t}} such that y t = τ t + c t {\displaystyle y_{t}\ =\tau _{t}\ +c_{t}\,} . Given an adequately chosen, positive value of λ {\displaystyle \lambda } , there is a trend component that will solve
min τ ( ∑ t = 1 T ( y t − τ t ) 2 + λ ∑ t = 2 T − 1 [ ( τ t + 1 − τ t ) − ( τ t − τ t − 1 ) ] 2 ) . {\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 d t = y t − τ t {\displaystyle d_{t}=y_{t}-\tau _{t}} , which penalizes the cyclical component. The second term is a multiple λ {\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 λ {\displaystyle \lambda } , the higher is the penalty. Hodrick and Prescott suggest 1600 as a value for λ {\displaystyle \lambda } for quarterly data. Ravn and Uhlig (2002) state that λ {\displaystyle \lambda } should vary by the fourth power of the frequency observation ratio; thus, λ {\displaystyle \lambda } should equal 6.25 (1600/4^4) for annual data and 129,600 (1600*3^4) for monthly data; in practice, λ = 100 {\displaystyle \lambda =100} for yearly data and λ = 14 , 400 {\displaystyle \lambda =14,400} for monthly data are commonly used, however. The Hodrick–Prescott filter is explicitly given by
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