Hodrick-Prescott filter with automatically selected jumps

Reference: Maranzano, P. and Pelagatti, M. (2025), "A Hodrick-Prescott filter with automatically selected breaks", Economic Modelling 150, 107132, doi:10.1016/j.econmod.2025.107132.

This R package implements our novel method to supplement the classical HP filter with jumps and, possibly, regressors. The method is based on the following state-space representation

$$y_t = x_t^\top \beta + \mu_t + \varepsilon_t$$

$$\mu_{t+1} = \mu_t + \nu_t$$

$$\nu_{t+1} = \nu_t + \zeta_t,$$

where $y_t$ is the observable time series, $\mu_t$ is the level component, $\nu_t$ is the slope component, $\varepsilon_t$ and $\zeta_t$ are white noise sequences with variances $\sigma^2_\varepsilon$ and $\sigma^2_\zeta$, respectively. The smoother, that is, the linear projection of $\mu_t$ on the span of the observations ${y_1,\ldots,y_n}$, coincides with the HP filter, where the smoothing constant $\lambda$ is given by $\sigma^2_\varepsilon / \sigma^2_\zeta$. Finally, $x_t$ is a vector of regressors, and $\beta$ is a vector of regression coefficients. These regressors are mainly used to model seasonal patterns in the data and should have a zero mean to not alter the interpretation of the HP filter as a trend extractor.

Smoothing splines with discontinuities

This part of the package implements two novel methods for fitting cubic smoothing splines that allow for a finite number of discontinuities at unknown locations. The key idea is to replace the physical spacing $\delta_i = x_{i+1} - x_i$ between consecutive observations with a latent spacing $\tau_i = \delta_i + \gamma_i$, where $\gamma_i \ge 0$ is a non-negative domain-warping variable. A large $\gamma_i$ stretches the latent domain between $x_i$ and $x_{i+1}$, compressing a smooth latent transition into a tiny physical interval and thereby producing an apparent jump. The warping variables are constrained by a budget: $\gamma_i \ge 0$ and $\sum_i \gamma_i \le M$.

The objective function to be minimised over $f$ and $\boldsymbol{\gamma}$ is

$$J(f, \boldsymbol{\gamma}) = \sum_{i=1}^n (y_i - f(x_i))^2 + \lambda \sum_{i=1}^{n-1} \int_0^{\tau_i} \left(\frac{d^2 f_i(s)}{ds^2}\right)^2 ds$$

subject to $\gamma_i \ge 0$ and $\sum_i \gamma_i \le M$, where the roughness penalty is evaluated in the latent domain. For $\boldsymbol{\gamma} = \mathbf{0}$ the problem reduces to the standard cubic smoothing spline.

Two estimation methods are provided.

MLE via Kalman filter and smoother. Following Wecker and Ansley (1983), the cubic smoothing spline coincides with the signal extracted from a state-space model whose transition matrix depends on the spacing $\delta_i$. Replacing $\delta_i$ with $\tau_i = \delta_i + \gamma_i$ yields a state-space model for the discontinuous case. The likelihood is maximised jointly over $(\sigma, \lambda, \boldsymbol{\gamma})$ using constrained optimisation (CCSA/NLopt) with analytical scores derived from the Kalman filter and smoother. The smoothed state provides the estimated function. This approach is implemented in ssj_mle(), which estimates $\lambda$ by maximum likelihood for a given budget $M$, and in auto_ssj_mle(), which selects $M$ automatically by EBIC. When $\lambda$ is supplied by the user, ssj_kfs() (and its wrapper auto_ssj_kfs()) solve the same problem with $\lambda$ held fixed; ssj_fix() and auto_ssj_fix() implement the earlier variant in which the additional variances, rather than the spacings, are the free parameters.

Alternating Time-Warping (ATW) algorithm. As an alternative, a two-step block coordinate descent algorithm with $O(n \log n)$ complexity per iteration is provided. The f-step solves the banded linear system $(\mathbf{S} + \lambda \mathbf{R})\mathbf{z} = \mathbf{y}^*$ via a block-tridiagonal Cholesky decomposition in $O(n)$ operations. The $\gamma$-step performs a gradient descent on the roughness energy followed by a projection onto the simplex ${\boldsymbol{\gamma} \ge 0,, \sum_i \gamma_i \le M}$. The smoothing parameter $\lambda$ can be supplied or piloted from a preliminary ssj_mle() fit; the budget $M$ is selected by EBIC over a grid. This approach is implemented in ssj_atw() (fixed $\lambda$ and $M$) and auto_ssj_atw() (automatic selection of $M$).

Cubic smoothing splines with discontinuities (CSSD)

The package also implements the cubic smoothing spline with discontinuities of Storath and Weinmann (2024, JCGS 33(2), 651-664), which penalises the number of discontinuities directly rather than through a warping budget:

$$\min_{f, J} p \sum_i \left(\frac{y_i - f(x_i)}{\delta_i} ight)^2 + (1-p) \int (f''(t))^2,dt + \gamma,|J|.$$

The global minimiser is found by dynamic programming with incremental QR updates (Givens rotations). The relevant functions are ssj_cssd_dp() for a given pair $(p, \gamma)$, auto_ssj_cssd() for K-fold cross-validated selection of both parameters, and predict_cssd_dp() to evaluate the fitted piecewise cubic spline at arbitrary locations. The algorithm is $O(n^2)$ in time and memory, so it is best suited to samples of at most a few thousand points; ssj_cssd() provides a much faster piecewise-constant (total-variation) approximation for larger problems.

Installation

install.packages("jumps")
# development version:
# remotes::install_github("matteopelagatti/jumps")