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MacroBoost Hybrid (MBH) Filter

Source: R/05_algo_mbh.R
mbh_filter.Rd

Decomposes a time series into trend and cycle using a robust boosting algorithm. Unlike the HP filter, MBH uses the Huber loss function to automatically downweight outliers (like the COVID-19 shock), preventing them from distorting the trend.

Usage

mbh_filter(
  x,
  knots = NULL,
  mstop = 500L,
  d = NULL,
  nu = 0.1,
  df = 4L,
  select_mstop = FALSE,
  boundary.knots = NULL
)

Arguments

x

Numeric vector, ts, xts, or zoo object.

knots

Integer. Number of interior knots for the P-Spline. If NULL (default), it is calculated as max(20, floor(n / 2)). High knot density is required for the trend to be flexible enough.

mstop

Integer. Maximum number of boosting iterations (default 500). If select_mstop = TRUE this is the upper bound; the actual stopping point is chosen by AICc.

d

Numeric or NULL. The delta parameter for Huber loss. If NULL (default), it is auto-calibrated as the Median Absolute Deviation (MAD) of the first differences of the series (mad(diff(y))).

Scale-mismatch warning: When x is a log-level series, diff(y) returns inter-period growth rates (typical scale 0.001–0.02), whereas the output gap (the residual the filter must explain) has a much larger scale (typical scale 0.01–0.05). Using mad(diff(y)) as d therefore sets the Huber threshold far too low: the filter treats normal business-cycle oscillations as outliers, truncates their gradients, and blocks learning. The trend becomes over-smooth and the cycle absorbs too much long-run variance.

Recommendation: For a quick preliminary calibration use d = mad(hp_filter(x)$cycle), which sets the threshold on the residual scale. Supply an explicit numeric value to override the automatic fallback entirely.

nu

Numeric. The learning rate (shrinkage) for boosting (default 0.1).

df

Integer. Effective degrees of freedom per boosting step for the P-Spline base learner (default 4). This enforces the weak-learner constraint of Bühlmann & Hothorn (2007): each boosting step contributes only a small, smooth update so that the trend is built up gradually over many iterations rather than fitted in one pass.

End-point instability warning: Higher df values cause the B-spline basis matrix to shift drastically when the sample size changes by even one observation (the "rubber-band effect"). The last few data points pull the estimated trend non-smoothly, producing unreliable end-of-sample estimates. Keep df = 4 (the default) unless you have a specific reason to deviate.

select_mstop

Logical. If TRUE, the optimal number of boosting iterations is selected automatically via AICc (corrected AIC), following Bühlmann & Hothorn (2007). The mstop argument acts as the search upper bound. Default FALSE.

AICc underfitting warning: In the combination of Huber quasi-likelihood + P-splines, AICc penalises model complexity hyper-aggressively. In practice the algorithm stops at iteration ~5–15 instead of the intended ~500. The resulting trend is nearly a straight line; all long-run variance is pushed into the cycle component, defeating the purpose of the filter. Treat select_mstop = TRUE as an experimental option and validate visually before relying on it.

boundary.knots

A numeric vector of length 2 specifying the global domain for the B-spline basis (e.g., c(1, T_max)). If NULL (default), the range of time_idx is used. For real-time stability, fix this to the full-sample domain so the basis does not shift as the sample grows.

Value

A macrofilter object with trend, cycle, data, and meta components. The meta list contains method = "MBH", knots, d, mstop, nu, df, select_mstop, and compute_time.

Details

The model estimated is an additive model: $$y_t = \text{Linear}(t) + \text{Smooth}(t) + \epsilon_t$$

It is fitted using mboost::mboost() with:

  • Base Learners: A linear time trend (mboost::bols()) to capture the global path, plus a B-spline (mboost::bbs()) to capture local curvature.

  • Loss Function: Huber loss (mboost::Huber()) with parameter d. This is the key to robustness.

The default parameters (knots = n/2, mstop = 500) are calibrated to mimic the flexibility of a standard HP filter while retaining the robustness of the Huber loss.

Calibration Guidance

Three failure modes were discovered through empirical stress-testing. The defaults guard against all three:

1. Huber delta scale mismatch (d)

The automatic fallback mad(diff(y)) operates on the scale of growth rates, not the output gap. For log-level input this sets d one to two orders of magnitude too small, causing ordinary business-cycle swings to be treated as outliers. If the estimated cycle looks implausibly large or the trend is nearly linear, override with d = mad(hp_filter(x)$cycle) as a starting point.

2. AICc underfitting (select_mstop)

AICc + Huber quasi-likelihood + P-splines stops boosting at iteration ~5–15. The trend degenerates to a near-straight line and the cycle absorbs all long-run variance. Leave select_mstop = FALSE (the default) and set mstop explicitly instead.

3. End-point instability (df)

Values above 4 shift the B-spline basis matrix non-smoothly as the sample grows, producing a "rubber-band" distortion in the final observations. Keep df = 4 (the default) for real-time applications.

Examples

# Quarterly GDP-like series
set.seed(42)
y <- ts(cumsum(rnorm(200)), start = c(2000, 1), frequency = 4)
result <- mbh_filter(y)
print(result)
#> -- MacroFilter [MBH] --
#>    Observations : 200
#>    Parameters   : knots = 100, d = 0.9445, mstop = 500, mstop_initial = 500, nu = 0.1, df = 4, select_mstop = FALSE
#>    Cycle range  : [-4.17, 5.05]  sd = 1.721
#>    Compute time : 0.173 s