HodrickPrescottFilter

The Hodrick-Prescott filter is a widely used method in time series analysis for decomposing a series into a smooth trend and a cyclical component. It is commonly applied in macroeconomics to extract the long-run trend in variables such as gross domestic product (GDP) and to study business cycles.

Mathematically, the filter aims to find a trend τ_t that minimizes the sum of squared deviations from the observed series y_t plus a penalty that enforces smoothness of the trend. Formally, it solves min_τ ∑_t (y_t − τ_t)^2 + λ ∑_t [ (τ_{t+1} − τ_t) − (τ_t − τ_{t−1}) ]^2. The first term measures how well the trend fits the data; the second term penalizes fluctuations in the growth rate of the trend, with the smoothing parameter λ controlling the balance between fidelity to the data and smoothness.

The parameter λ is central to the method. Larger values produce a smoother trend and a smaller cyclical component, while smaller values allow the trend to track the data more closely. Common practice uses different defaults by data frequency: λ ≈ 100 for annual data, λ ≈ 1600 for quarterly data, and λ ≈ 129,600 for monthly data.

Implementation and interpretation: the filter is a linear, nonparametric technique and can be computed by solving a sparse linear system. The resulting components are used for analysis, forecasting, and removing short-term fluctuations before further modeling. End-point bias and potential misallocation of structural changes into the cycle are noted limitations. The Hodrick-Prescott filter is often compared with other filters and with structural time-series models when interpreting cyclical movements.