Methods — five tests, one verdict.
Most papers use ADF in isolation. Real desks stack tests with opposing nulls and complementary power. This page runs ADF + Engle-Granger, KPSS (opposite null), Variance Ratio (Lo-MacKinlay), Hurst (R/S), CUSUM (Brown-Durbin-Evans), and Johansen (full system) on the same spread, then summarises the verdict.
Verdict
MEGA-WHENStrong evidence: ADF rejects (small τ), Engle-Granger cointegrated, KPSS does notreject (η below cv), VR(5) below 1 with negative z, Hurst < 0.5, CUSUM stable, Johansen trace rejects r=0. A pair that passes 6 of 7 is robust; anything below 4 is noise.
Variance Ratio profile (Lo-MacKinlay)
VR(q) at multiple horizonsVR(q) < 1 across multiple horizons is strong evidence of mean reversion; negative z-statistic at the same horizons makes it statistically reliable.
CUSUM of mean-shifts (Brown-Durbin-Evans 1975)
structural break detectedBrownian-motion-like envelope under H0 (constant β). The CUSUM crossing a band ⇒ the relationship has shifted and the spread is no longer the same statistical object.
Hurst exponent — R/S regression
slope = H = 0.838, R² = 0.997log(R/S) on log(n). A 45° slope means H = 0.5 (random walk). H < 0.5 is mean reversion — the property we want for a tradable spread.
Half-life sensitivity to window length
OU fit on trailing windowsA robust pair shows similar half-life across windows. Wide variance ⇒ the mean-reversion speed is itself drifting (regime-shifting), and the OU model is being asked to do too much.
Johansen — the multivariate sibling of Engle-Granger
2-variable trace + max-eigenvalueJohansen estimates all cointegration vectors at once (no "regress y on x" asymmetry) and is the right tool when you have ≥ 3 assets. With 2 assets it should agree with Engle-Granger; with broken pairs it's typically the more forgiving test.