HPfilter

The Hodrick-Prescott filter, abbreviated HP filter or HPfilter, is a smoothing technique used in time series analysis to separate a data series into a smooth trend component and a cyclical component. It is widely used in macroeconomics to extract the business cycle from gross domestic product and related series.

In its standard form, given a time series y_t, the goal is to find a trend sequence τ_t that minimizes the objective function: sum over t of (y_t - τ_t)^2 + λ sum over t of (Δ^2 τ_t)^2, where Δ^2 τ_t = τ_{t+1} - 2τ_t + τ_{t-1}. The residual cycle is c_t = y_t - τ_t. The smoothing parameter λ controls the trade-off between fidelity to the data and smoothness of the trend; larger λ yields a smoother τ_t and a smaller c_t.

λ is chosen according to data frequency and analyst preference; common guidance cites λ ≈ 1600 for quarterly data and smaller values for higher-frequency data, with larger values for monthly series. The filter assumes a continuous, gradually evolving trend and can be sensitive to endpoint treatment, which may affect the estimated cycle near the series ends.

Used to study business cycles by removing trend, the HP filter is simple and widely implemented in statistical packages. It has limitations: it can impart spurious cycles when the series has structural breaks, non-stationarity, or changing variance; results depend on λ; it does not perform well with outliers or irregularly spaced data; and it assumes a smooth, slowly evolving trend.

Variants include bandpass filters such as Baxter-King and Christiano-Fitzgerald, and alternative detrending methods like linear or polynomial trends. In practice, analysts may compare multiple filters to assess robustness.