The hodrick-prescott filter: A special case of penalized spline smoothing

Robert L. Paige, A. Alexandre Trindade

Research output: Contribution to journalArticlepeer-review

25 Scopus citations


We prove that the Hodrick-Prescott Filter (HPF), a commonly used method for smoothing econometric time series, is a special case of a linear penalized spline model with knots placed at all observed time points (except the first and last) and uncorrelated residuals. This equivalence then furnishes a rich variety of existing data-driven parameter estimation methods, particularly restricted maximum likelihood (REML) and generalized cross-validation (GCV). This has profound implications for users of HPF who have hitherto typically relied on subjective choice, rather than estimation, for the smoothing parameter. By viewing estimates as roots of an appropriate quadratic estimating equation, we also present a new approach for constructing confidence intervals for the smoothing parameter. The method is akin to a parametric bootstrap where Monte Carlo simulation is replaced by saddlepoint approximation, and provides a fast and accurate alternative to exact methods, when they exist, e.g. REML. More importantly, it is also the only computationally feasible method when no other methods, exact or otherwise, exist, e.g. GCV. The methodology is demonstrated on the Gross National Product (GNP) series originally analyzed by Hodrick and Prescott (1997). With proper attention paid to residual correlation structure, we show that REML-based estimation delivers an appropriate smooth for both the GNP series and its returns.

Original languageEnglish
Pages (from-to)856-874
Number of pages19
JournalElectronic Journal of Statistics
StatePublished - 2010


  • Econometric smoothing
  • Gross national product
  • Parametric bootstrap confidence interval
  • Saddlepoint approximation
  • Semiparametric model


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