The ARDL bounds test checks for a long-run level relationship using a joint F-test and a t-test on the lagged dependent level, compared against I(0)/I(1) critical bounds — valid whether the regressors are I(0), I(1) or mixed.
The bounds procedure fits a single conditional error-correction model (the unrestricted ECM): Δy is regressed on lagged levels of y and the regressors, plus short-run differences. Cointegration is tested as H₀: the lagged-level coefficients are jointly zero (no level relationship), using an F-statistic whose distribution is non-standard and depends on whether the regressors are I(0) or I(1).
Pesaran, Shin & Smith give two critical-value bounds: a lower bound assuming all regressors are I(0) and an upper bound assuming all are I(1). If the F-statistic exceeds the upper bound there is a level relationship; below the lower bound there is none; between the bounds the test is inconclusive. A companion t-test on the speed-of-adjustment term (cases I, III, V) provides corroborating evidence. Long-run coefficients are recovered as −δ/λ and the error-correction speed is λ.
One numeric dependent series + ≥ 1 numeric regressor (row order = time), a deterministic case (I–V), and lag orders (auto-selected by AIC/BIC or set manually).
The conditional ECM coefficients; the bounds F-statistic and t-statistic with the I(0)/I(1) bounds at 10/5/1% and a cointegration verdict; the long-run coefficients; and the error-correction speed λ.
Read the F verdict first: 'inconclusive' means the outcome depends on the unknown integration orders — inspect unit-root tests. A significant, negative λ (and a t-statistic beyond the t-bounds) confirms error correction.
The critical values are asymptotic; for small T the bounds are slightly liberal (see Narayan 2005 for small-sample values).