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

Abstract

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.

Complexity-Regularized Regression for Serially-Correlated Residuals with Applications to Stock Market Data

“…Next, we examine the trends inferred for example low-and high-frequency trends as we vary the correlation in the residuals, shown in Figure 5. The top panels correspond to a low frequency trend, and Finally, we quantify the performance of each of the data-driven methods using the mean-squared error (42) between the inferred trend and the true trend. We computed the mean-squared error for each of the 1000 realizations across the frequency and residual conditions.…”

“…Several authors have addressed how the choice of the smoothing parameter impacts the correlation structure of the residuals [43][44][45]. Data-driven approaches for choosing the smoothing parameter should be pursued, but as others have discussed [42], and we have demonstrated with our simulation study, care must be taken in the assumptions implicit to the chosen method. Other popular approaches include autoregressive models with a conditional mean that changes linearly in time [46].…”

“…This can either be done in the time-domain, in the state-domain or in a mixture of the two. For time-domain smoothing, one of the most commonly-used tools for detrending data is the Hodrick-Prescott filter [41], which is essentially a special case of the smoothing spline [42]. In their original formulation, Hodrick and Prescott presented a heuristic choice of the smoothing parameter for quarterly data (such as the U.S. gross domestic product).…”

Ridge Regression Representations of the Generalized Hodrick-Prescott Filter

“…The HP filter has been recognized as a type of Whittaker-Henderson method of graduation, which is frequently used in the actuarial literature. 1 Recently, Paige and Trindade (2010) presented a ridge regression (Hoerl and Kennard (1970)) representation of the filter. Let A = [Π, S ] ∈ R T ×T , where the tth row of Π ∈ R T ×2 is [1, t] and the tth row of S ∈ R T ×(T −2) is [(t−2) + , .…”

“…Here, for a number a, (a) + equals a if a is positive and zero otherwise. Paige and Trindade (2010) introduced the following ridge regression:…”