In the augmented Dickey-Fuller test, what does failing to reject the null mean?

## What the test is actually testing The augmented Dickey-Fuller test asks whether a series contains a unit root. The crucial design choice is which side of the hypothesis pair non-stationarity sits on: - **Null hypothesis:** the series has a unit root, and is therefore non-stationary. - **Alternative hypothesis:** the series is stationary — around a constant, or around a deterministic trend, depending on the specification. That orientation drives every interpretation. Rejecting is affirmative evidence for stationarity. Failing to reject is not affirmative evidence for a unit root; it is the ordinary asymmetry of significance testing, where a non-rejection means the data did not supply enough evidence to overturn the assumed default. ## The regression behind it Start from `X_t = phi * X_{t-1} + e_t` and subtract `X_{t-1}` from both sides: `dX_t = (phi - 1) * X_{t-1} + e_t = gamma * X_{t-1} + e_t` A unit root, `phi = 1`, is exactly `gamma = 0`. So the test regresses the period-to-period change on the previous level and asks whether that coefficient is zero. The full specification adds deterministic terms and lagged changes: `dX_t = alpha + beta*t + gamma*X_{t-1} + d_1*dX_{t-1} + ... + d_p*dX_{t-p} + e_t` The intuition for `gamma` is mean reversion. If `gamma` is meaningfully negative, a high level today pulls the next change downward — the series is pulled back toward a centre. If `gamma` is zero, the level carries no information about the next change, which is the random-walk case. ## What the augmentation adds The original Dickey-Fuller test assumes the errors are white noise. Real series are rarely that clean: even after accounting for the level, the changes are usually autocorrelated. The **augmented** version adds `p` lagged difference terms whose only job is to absorb that serial correlation so the residuals behave and the statistic follows its intended distribution. Those lag coefficients are nuisance parameters — you never interpret them. Choosing `p` matters: too few leaves autocorrelation in the errors and distorts the test size; too many wastes degrees of freedom and lowers power. ## Why the critical values are special Under a unit root the regressor `X_{t-1}` is itself non-stationary, and the usual asymptotics do not apply. The estimator of `gamma` has a non-standard limiting distribution — the Dickey-Fuller distribution — that is shifted left relative to a normal. Its critical values are more negative than ordinary t critical values, and they differ depending on whether you included a constant, a constant and a trend, or neither. Comparing an ADF statistic against a standard t-table is a classic error and rejects far too often. The test is one-sided in the negative direction, because only a negative `gamma` corresponds to mean reversion. ## Choosing the deterministic terms The specification is a modelling decision, not a default. Including no constant assumes the series has mean zero. Including a constant allows a non-zero level. Including a constant and a trend makes the alternative *trend stationarity* — stationary fluctuation around a sloping line. Omitting a trend when the data plainly have one biases the test toward not rejecting, because the unmodelled slope looks like persistence. Adding an unnecessary trend costs power. Always state which version was run alongside the result. ## Where power fails The honest weakness of this test is power. A stationary series with a coefficient of 0.95 is hard to distinguish from one with a coefficient of exactly 1.00 unless the sample is long: the two produce very similar-looking paths over a few hundred points. On short samples the test therefore fails to reject nearly everything, and a non-rejection carries very little information. This is not a flaw to work around by re-running the test on subsamples until it rejects; it is a limit that argues for pairing the test with one whose null is the opposite, and for weighing the plot and the domain knowledge alongside both. ## How to report it A usable report says which deterministic terms were included, the lag order used for the augmentation, the statistic, the critical value it was compared against, and the conclusion phrased in the right direction: "we reject the unit root at the 5% level, so the evidence supports stationarity", or "we cannot reject a unit root; the data are consistent with non-stationarity, and given the sample length that is weak evidence". Never write "the test proves the series is non-stationary".