Filtering Time Series with Penalized Splines

The decomposition and filtering of time series is an important issue in economics and econometrics and related fields. Even though there are numerous competing methods on the market, in application one often meets one of the few favorites. The first method to mention in this selection is the so called Hodrick & Prescott (1997)-filter (HP-filter hereafter). The idea is to decompose a time series yt, say, into a smooth path gt and remaining deviations (residuals or shocks) εt, respectively. To achieve smoothness a penalty is imposed on gt such that second order differences are penalized. The idea traces back to Leser (1961) and Whittaker (1923) and is simple in its numerical implementation. The application requires the specification of a penalized parameter λ, say, which steers the smoothness of the fitted path ĝt. Hodrick