The SC-MIDAS framework¶

SC-MIDAS (Staggered-Combination MIDAS) is a two-layer nowcasting recipe that turns a heterogeneous panel of monthly and quarterly indicators into a single best-estimate forecast for the quarterly target \(y_t\) (typically GDP growth) at several horizons.

The three building blocks¶

1. Indicator models¶

Each monthly indicator \(x^{(i)}\) is paired with the target through a direct-forecast MIDAS regression:

\[ y_{t+h} \;=\; \alpha^{(i)}_h \;+\; \beta^{(i)}_h \sum_{j=0}^{K_i-1} w_h(j;\theta^{(i)}_h)\, x^{(i)}_{t,j} \;+\; \mathbf{d}_{t+h}^{\!\top}\gamma^{(i)}_h \;+\; \varepsilon_{t+h}. \]

Optional outlier dummies \(\mathbf{d}_{t+h}\) enter at the target frequency. Estimation uses OLS for linear-in-parameter weighting schemes (Almon, U-MIDAS) and Levenberg-Marquardt non-linear least squares with closed-form analytic Jacobians for the rest (exponential Almon, Beta).

Each quarterly hard indicator \(z^{(k)}\) uses a plain OLS counterpart:

\[ y_{t+h} \;=\; a^{(k)}_h \;+\; \mathbf{z}^{(k)\,\top}_t c^{(k)}_h \;+\; \mathbf{d}_{t+h}^{\!\top}\gamma^{(k)}_h \;+\; \varepsilon_{t+h}. \]

In both cases, the pipeline stores one model per horizon \(h\) in MidasCombo.midas_models_[var][h] and MidasCombo.ols_models_[var][h].

2. Layer 1 — soft combination¶

The indicator fits feed into one or more first-layer ComboSpec nodes whose combination weights are derived from rolling-window residuals. Three inverse-error variants are available (mean absolute error, mean squared error, root mean squared error) plus equal-weight average.

Each source uses its own prior residuals. Lower errors receive more weight; window selects how many residuals to use and discount_rate controls the influence of older errors. With a finite window, sources without enough residuals receive an equal share. See Weighting schemes for the calculation and observation-count rules.

minimum_sample_size excludes sources with too few finite fitted observations before weighting. The default is 10.

3. Layer 2 — soft × hard merging¶

The second layer pools the Layer-1 combo with a quarterly hard regressor via constrained regression (fit_regression_weights) with method='constrained_ls':

\[ \min_{\mathbf{w}\,\ge 0,\;\mathbf{1}^{\!\top}\mathbf{w}=1} \sum_t \Bigl( y_t - \mathbf{w}^{\!\top}\hat{\mathbf y}_t \Bigr)^2. \]

ComboSpec(method='regression', window=None) uses an expanding window over the full sample, which is the natural Layer-2 set-up. When there are exactly two sources (soft combo + hard) this reduces to the EViews-style convex combination \(w \in [0, 1]\).

End-to-end recipe¶

from nowcast_midas import MidasCombo, MidasSpec, OLSSpec, ComboSpec

midas_monthly_1 = MidasSpec("monthly_1", method="almon", n_lags=5)
midas_monthly_2 = MidasSpec("monthly_2", method="almon", n_lags=5)
midas_monthly_3 = MidasSpec("monthly_3", method="unrestricted", n_lags=3)
ols_quarterly_1 = OLSSpec("quarterly_1", n_lags=1)

soft = ComboSpec(
    "soft",
    sources=[midas_monthly_1, midas_monthly_2, midas_monthly_3],
    method="mse",
    window=8,
    discount_rate=0.95,
)
final = ComboSpec(
    "final", sources=[soft, ols_quarterly_1], method="regression", window=None
)

model = MidasCombo(combo_specs=final, horizons=3)
model.fit(target=target_df, regressors=regressors_df)
forecasts = model.forecast()

Relation to the EViews reference¶

Single-vintage numerical equivalence is a reference-validation target. The remaining gap is the recursive OOS estimation loop (re-fitting at every historical vintage), which is a workflow-level feature outside the scope of the estimation API.