hp_filter: Hodrick-Prescott Filter (Sparse Matrix Implementation)

In MacroFilters: Robust Trend-Cycle Decomposition for Macroeconomic Time Series

Hodrick-Prescott Filter (Sparse Matrix Implementation)

Description

Decomposes a time series into trend and cycle components by solving the HP penalized least-squares problem using a sparse Cholesky factorization. This avoids the dense O(n^3) inversion used by other implementations and scales linearly in the number of observations.

Usage

hp_filter(x, lambda = NULL, freq = NULL, boot_iter = 0, block_size = "auto")

Arguments

x	Numeric vector, ts, xts, or zoo object.
lambda	Smoothing parameter. If NULL (default), it is auto-detected using the Ravn-Uhlig rule (6.25 * freq^4).
freq	Numeric frequency override (1 = annual, 4 = quarterly, 12 = monthly). Used only when lambda is NULL and the frequency cannot be inferred from x.
boot_iter	Non-negative integer. Number of block-bootstrap iterations for uncertainty quantification (default 0, bootstrap disabled). When ⁠> 0⁠, the function adds ⁠$trend_lower⁠ and ⁠$trend_upper⁠: a 95% normal-approximation band, ⁠trend +/- 1.96 * sd(bootstrap trends)⁠, centred on the estimated trend. The bootstrap sd is used instead of empirical percentiles because it is smooth and stable at practical boot_iter. Each bootstrap refit uses the same mstop as the base fit, so larger boot_iter raises cost linearly. See also block_size.
block_size	Positive integer or "auto". Block length for the moving-block bootstrap (used only when boot_iter > 0). If "auto" (default), it is set to 2 * stats::frequency(x) (two full cycles), bounded above by floor(length(x) / 3) to keep at least three blocks.

Details

The HP filter minimises

\sum (y_t - \tau_t)^2 + \lambda \sum (\Delta^2 \tau_t)^2

which admits the closed-form solution

(I + \lambda D'D)\,\tau = y

where D is the second-difference operator.

The implementation builds D as a banded sparse matrix (Matrix::bandSparse()) and solves the symmetric positive-definite system with a sparse Cholesky decomposition (Matrix::solve()).

When lambda is not supplied the Ravn-Uhlig (2002) rule is applied: lambda = 6.25 * freq^4, yielding 6.25 (annual), 1600 (quarterly), and 129 600 (monthly).

Value

A list of class c("macrofilter", "list") with trend, cycle, data, and meta. When boot_iter > 0 it also carries trend_lower and trend_upper (95% normal-approximation bootstrap band).

References

Hodrick, R.J. and Prescott, E.C. (1997). Postwar U.S. Business Cycles: An Empirical Investigation. Journal of Money, Credit and Banking, 29(1), 1–16.

Ravn, M.O. and Uhlig, H. (2002). On Adjusting the Hodrick-Prescott Filter for the Frequency of Observations. Review of Economics and Statistics, 84(2), 371–376.

Examples

# Quarterly GDP-like series
y <- ts(cumsum(rnorm(200)), start = c(2000, 1), frequency = 4)
result <- hp_filter(y)
print(result)

MacroFilters documentation built on June 12, 2026, 1:06 a.m.