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Boosted HP Filter

Source: R/04_algo_bhp.R
bhp_filter.Rd

Iteratively applies the Hodrick-Prescott filter on residuals to better capture stochastic trends. At each iteration the HP smoother is applied to the current residual and the resulting trend increment is added to the cumulative trend estimate. Iteration stops according to one of three rules: BIC minimisation (default), ADF stationarity test on residuals, or a fixed number of iterations.

Usage

bhp_filter(
  x,
  lambda = NULL,
  iter_max = 100L,
  stopping = c("bic", "adf", "fixed"),
  sig_level = 0.05,
  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).

iter_max

Integer. Maximum number of boosting iterations (default 100).

stopping

Character. Stopping rule: "bic" (default), "adf", or "fixed".

sig_level

Numeric. Significance level for the ADF test when stopping = "adf" (default 0.05).

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.

Value

A list of class c("macrofilter", "list") with trend, cycle, data, and meta (method = "bHP", lambda, iterations, stopping_rule, compute_time). When boot_iter > 0 it also carries trend_lower and trend_upper (95% normal-approximation bootstrap band); each bootstrap refit runs a fixed iterations passes, conditioning on the complexity selected by the base fit.

Details

The boosted HP filter starts from the standard HP solution and then re-applies the same HP smoother to the residual (cycle) component. The trend increment from each pass is accumulated, and the procedure stops when one of the following criteria is met:

"bic"

Schwarz information criterion computed as \(n \log(\hat\sigma^2) + \log(n)\,\mathrm{tr}(S^m)\), where \(S^m\) is the iterated smoother. Iteration stops when the BIC increases relative to the previous best.

"adf"

Augmented Dickey-Fuller test on the residual. Iteration stops when the residual is stationary at level sig_level.

"fixed"

Runs exactly iter_max iterations.

References

Phillips, P.C.B. and Shi, Z. (2021). Boosting: Why You Can Use the HP Filter. International Economic Review, 62(2), 521–570.

Examples

# Quarterly GDP-like series
y <- ts(cumsum(rnorm(200)), start = c(2000, 1), frequency = 4)
result <- bhp_filter(y)
print(result)
#> -- MacroFilter [bHP] --
#>    Observations : 200
#>    Parameters   : lambda = 1600, iterations = 100, stopping_rule = bic
#>    Cycle range  : [-1.657, 1.673]  sd = 0.6962
#>    Compute time : 0.005 s