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.
Arguments
- x
Numeric vector,
ts,xts, orzooobject.- 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
lambdaisNULLand the frequency cannot be inferred fromx.- boot_iter
Non-negative integer. Number of block-bootstrap iterations for uncertainty quantification (default
0, bootstrap disabled). When> 0, the function adds$trend_lowerand$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 practicalboot_iter. Each bootstrap refit uses the samemstopas the base fit, so largerboot_iterraises cost linearly. See alsoblock_size.- block_size
Positive integer or
"auto". Block length for the moving-block bootstrap (used only whenboot_iter > 0). If"auto"(default), it is set to2 * stats::frequency(x)(two full cycles), bounded above byfloor(length(x) / 3)to keep at least three blocks.
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).
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).
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.