How does the Augmented Dickey-Fuller test detect unit roots?

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AcadiFi · Accepted Answer

The Augmented Dickey-Fuller (ADF) test examines whether a time series has a unit root, implying non-stationarity. The null hypothesis is presence of a unit root; rejection means the series is stationary. The augmentation adds lagged difference terms to the regression to absorb autocorrelation in the residuals, making the test valid under broader conditions than the original Dickey-Fuller. The test statistic is the t-ratio on the lagged level coefficient; critical values follow a non-standard distribution (MacKinnon values) rather than Student-t.

Regression form with constant and trend: Δyt = α + βt + γ yt-1 + Σφi Δyt-i + et. Test γ = 0 versus γ < 0. If calculated t-statistic is more negative than critical value (e.g., -3.45 at 5% with trend), reject null—series is stationary. Example: testing quarterly US GDP, the ADF statistic is -1.8 with 5% critical value of -3.41, so fail to reject—GDP has a unit root and is I(1). First differences of GDP produce a stationary series with ADF typically below -4.

Choice of lag length k for augmentation matters: too few lags leaves autocorrelation biasing the test; too many reduces power. Information criteria (AIC, BIC) or the Schwert rule (k = 12(T/100)^0.25) guide selection. Always include constant; include trend if the series has visible drift. ADF has low power against near-unit-root alternatives, so combine with KPSS for robustness.

How does the Augmented Dickey-Fuller test detect unit roots?

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