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System ARDL (SUR-ECM)

pyardl.system

Every other estimator in the library is single-equation: one y, possibly repeated across individuals in a panel (pyardl.panel), but never several dependent variables simultaneously linked through their errors. SystemARDL covers that case: two or more ARDL equations whose residuals are correlated contemporaneously, estimated jointly by Feasible Generalized Least Squares — Seemingly Unrelated Regressions (Zellner 1962).

Distinct from Johansen. pyardl.cointegration.johansen estimates a full VAR system where every variable is endogenous and the number of cointegrating relations is itself estimated. SystemARDL instead takes already-specified single-equation ECMs (each cointegrating relation assumed known, and tested separately with the bounds test), linked only through the contemporaneous correlation of their residuals — a more restrictive model, but one that never needs a system-wide cointegration rank chosen.

SystemARDL(equations, det="const", iterate=True, max_iter=50, tol=1e-8)

from pyardl.system import SystemARDL

res = SystemARDL({
    "money": (y1, x1, (1, 1)),
    "credit": (y2, x2, (1, 1)),
}, det="const", iterate=True).fit()
Argument Meaning
equations {name: (y, x, order)}, one entry per equation
det applied to every equation
iterate repeat the FGLS round to convergence (equivalent to ML under normality) rather than a single GLS pass

Every y must share the same length and time index — a system with partially disjoint dates between equations is out of scope.

Estimation algorithm

  1. Fit every equation separately by plain OLS, reusing pyardl.core.ardl.ARDL unchanged, with a common hold_back so all equations share the same estimation sample.
  2. Estimate the residual covariance Sigma_hat from the OLS residuals.
  3. Re-estimate the whole system by GLS, whitening the stacked design by Sigma_hat^{-1/2} (a Cholesky-based M x M solve — small relative to the sample size) and solving the whitened system with a single numpy.linalg.lstsq call.
  4. Repeat steps 2-3 to convergence when iterate=True.

If every equation shares the exact same regressor matrix, SUR reduces to equation-by-equation OLS exactly (the classical Zellner 1962 result) — pyardl reproduces this to numerical precision (tests/unit/system/test_system_ardl.py::TestIdenticalRegressorsIdentity).

SystemARDLResults

Member Returns
equations {name: ARDLResults} — the per-equation OLS fit, for comparison
sigma Sigma_hat, the residual covariance, as a pandas.DataFrame
fgls_params / fgls_se {name: pandas.Series} — FGLS coefficients and standard errors per equation
n_iter FGLS iterations run
names {name: [term, ...]} — parameter names in the order they appear in the stacked system

.efficiency_gain(name)

Ratio of the single-equation OLS standard errors to the FGLS ones, for one equation. Above 1 means FGLS is more precise — the gain Zellner's method promises, larger with stronger residual correlation and more different regressors between equations. Not guaranteed to exceed 1 in every finite sample (the SUR efficiency gain can be small or negative when Sigma_hat is poorly estimated relative to the number of equations); this reports what was measured on the fitted data, not what theory promises asymptotically.

.test_cross_equation_restriction(r_matrix, r)

Wald test of H0: R theta = r on the raw stacked UECM coefficients (adjustment speeds, level coefficients, short-run terms) — a restriction like lambda_1 = lambda_2 is linear in these and directly testable, generalising the single-equation long-run restriction test (pyardl.restrictions) to a system.

!!! warning "Scope: raw parameters, not the long-run ratio" A restriction on the long-run coefficient theta = -gamma/lambda across equations (rather than on gamma or lambda directly) would need the delta method extended to the stacked system's covariance — not implemented. See docs/DEVIATIONS.md.

Returns WaldResult(stat, df, pvalue).

.summary()

Publication-style report: residual correlation matrix and per-equation FGLS coefficients with standard errors.

References

Zellner, A. (1962). An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. Journal of the American Statistical Association, 57(298), 348-368.