The Engle-Granger Two-Step Procedure

Two stock prices can each wander around forever like a random walk while still being tied together — think of two shares of the same company listed on different exchanges, or two similar utility companies. The Engle-Granger procedure is the classic, two-step recipe for testing whether that kind of long-run tether, called cointegration, actually exists between two price series.

Step one: regress one price series on the other, $P_1 = \alpha + \beta P_2 + \epsilon$, using ordinary least squares, and keep the residuals $\epsilon$ — the leftover part of $P_1$ that $\beta P_2$ doesn't explain. Step two: run a unit-root test (typically the augmented Dickey-Fuller test) on those residuals. If the residuals are stationary — meaning they hover around a constant level rather than drifting off — then the two series share a common stochastic trend and are cointegrated, because whatever combination of them produces a residual that keeps snapping back to a fixed level. If the residuals themselves look like a random walk, there's no evidence of a stable long-run relationship, and any apparent correlation between the two series' prices could be spurious.

The well-known catch is that the critical values for the unit-root test in step two aren't the standard Dickey-Fuller values — because the residuals came from an estimated regression rather than being observed directly, the test needs its own, wider critical value table (tabulated by Engle and Granger and later refined by MacKinnon), and using the ordinary Dickey-Fuller table understates how easily "significant" stationarity can appear by chance.