Electronic Journal of Statistics

The Hodrick-Prescott Filter: A special case of penalized spline smoothing

Robert L. Paige and A. Alexandre Trindade

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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.

Article information

Electron. J. Statist. Volume 4 (2010), 856-874.

First available in Project Euclid: 15 September 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F25: Tolerance and confidence regions
Secondary: 62G09: Resampling methods

Semiparametric model parametric bootstrap confidence interval saddlepoint approximation econometric smoothing gross national product


Paige, Robert L.; Trindade, A. Alexandre. The Hodrick-Prescott Filter: A special case of penalized spline smoothing. Electron. J. Statist. 4 (2010), 856--874. doi:10.1214/10-EJS570. http://projecteuclid.org/euclid.ejs/1284557751.

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