`ridge` – Ridge regression (L2-regularised OLS).#

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Operational op under axis family, sub-layer L4_A_model_selection, layer l4. Standalone callable: mf.functions.ridge_fit.

Function signature#

mf.functions.ridge_fit(
    X: np.ndarray | pd.DataFrame,
    y: np.ndarray | pd.Series,
    *,
    alpha: float = 1.0,
    prior: str enum {"none", "random_walk", "shrink_to_target", "fused_difference"} = 'none',
    coefficient_constraint: str enum {"none", "nonneg"} = 'none',
    vol_model: str enum {"ewma", "garch11"} | None = None,
    random_state: int | None = None,
) -> RidgeFitResult

Parameters#

name	type	default	constraint	description
`X`	`np.ndarray	pd.DataFrame`	—	—
`y`	`np.ndarray	pd.Series`	—	—
`alpha`	`float`	`1.0`	>=0	L2 regularisation strength. Larger values shrink coefficients more aggressively toward zero.
`prior`	`str enum {"none", "random_walk", "shrink_to_target", "fused_difference"}`	`'none'`	—	Coefficient prior. `none` = standard ridge. `random_walk` = Goulet Coulombe (2025 IJF) TVP-as-ridge two-step estimator. `shrink_to_target` = Albacore_comps Variant A (simplex non-neg + target penalty). `fused_difference` = Albacore_ranks Variant B (fused-difference penalty).
`coefficient_constraint`	`str enum {"none", "nonneg"}`	`'none'`	—	Sign / cone constraint. `nonneg` enforces β >= 0 via augmented NNLS (Assemblage Regression, Coulombe et al. 2024). Ignored when `prior` is `shrink_to_target` or `fused_difference` (those priors handle non-negativity internally).
`vol_model`	`str enum {“ewma”, “garch11”}	None`	`None`	only used when prior=‘random_walk’
`random_state`	`int	None`	`None`	—

Returns#

RidgeFitResult — frozen dataclass with fit results.

Attribute	Type	Description
`.coef_`	`np.ndarray`	Fitted coefficient vector, shape (n_features,).
`.intercept_`	`float`	Fitted intercept scalar.
`.alpha`	`float`	Regularisation strength used.
`.predict(X)`	`np.ndarray`	Predictions for new data X, shape (n_samples,).
`.summary()`	`str`	Human-readable text table of fit results.

Behavior#

Closed-form ridge: β = (X'X + αI)⁻¹ X'y. Shrinks coefficients toward zero proportional to the regularisation strength α (params.alpha).

Default α = 1.0. The cv_path search algorithm uses RidgeCV to pick α from a grid via leave-one-out CV; the grid_search / random_search algorithms can sweep over leaf_config.tuning_grid[‘alpha’].

When to use

High-dimensional macro panels with collinear predictors; standard benchmark.

v0.9 sub-axes (default values preserve standard ridge):

params.prior – prior on the coefficients. none (default) keeps standard ridge.
- random_walk (operational v0.9.1) – Goulet Coulombe (2025 IJF) ‘Time-Varying Parameters as Ridge Regressions’ two-step closed-form estimator with a random-walk kernel on coefficient deviations. Yields per-time β path via the cumulative-sum reparametrisation β_k = C_RW · θ_k. Helper _TwoStageRandomWalkRidge.
- shrink_to_target (operational v0.9.1) – Maximally Forward-Looking Core Inflation Albacore_comps Variant A (Goulet Coulombe / Klieber / Barrette / Goebel 2024). arg min ‖y − Xw‖² + α‖w − w_target‖² s.t. w ≥ 0, w'1 = 1. Solved via scipy SLSQP. Limit cases: α=0 → unconstrained / NNLS; α→∞ → returns w_target. Helper _ShrinkToTargetRidge. Sub-axis params: prior_target (default uniform 1/K), prior_simplex (default True).
- fused_difference (operational v0.9.1) – Maximally FL Albacore_ranks Variant B. arg min ‖y − Xw‖² + α‖Dw‖² s.t. w ≥ 0, mean(y) = mean(Xw), where D is the first-difference operator. Pairs with the L3 asymmetric_trim op (B-6 v0.8.9) for rank-space transformation. Limit cases: α=0 → standard OLS / NNLS; α→∞ → uniform weights (level set by mean equality). Helper _FusedDifferenceRidge. Sub-axis params: prior_diff_order (default 1), prior_mean_equality (default True).
params.coefficient_constraint – sign / cone constraints. none (default) is unconstrained; nonneg (operational v0.8.9) implements the assemblage non-negative ridge.
params.vol_model (random_walk only) – volatility model for the step-2 Ω_ε reconstruction. ewma (default; RiskMetrics λ=0.94; no extra deps) or garch11 (requires arch>=5.0; auto-falls-back to EWMA when missing).

In recipe context#

Set params.family = "ridge" in the relevant layer to activate this op within a recipe:

# Layer L4 recipe fragment
params:
  family: ridge

References#

macroforecast design Part 2, L4: ‘forecasting model is the layer where every authoring iteration ends – pick family, tune, repeat.’
Hoerl & Kennard (1970) ‘Ridge regression: biased estimation for nonorthogonal problems’, Technometrics 12(1).
Goulet Coulombe (2025) ‘Time-Varying Parameters as Ridge Regressions’, International Journal of Forecasting 41:982-1002. doi:10.1016/j.ijforecast.2024.08.006.
Goulet Coulombe / Klieber / Barrette / Goebel (2024) ‘Maximally Forward-Looking Core Inflation’ – Albacore_comps (shrink_to_target Variant A) and Albacore_ranks (fused_difference Variant B).

ridge – Ridge regression (L2-regularised OLS).#