Augmented Dickey-Fuller Test (ADF)

The augmented Dickey-Fuller test (ADF) is a statistical test used to decide whether a time series contains a unit root. A series with a unit root wanders without returning to a fixed level, as a random walk does, and is non-stationary. The test is the most widely used check for stationarity in econometrics and quantitative finance.

It extends the original Dickey-Fuller test of 1979 by adding lagged changes of the series to the test regression. The extra terms absorb serial correlation that would otherwise distort the result. Traders apply the test to prices, and more often to the spread between related prices, to judge whether a series tends to revert to its mean.

The test estimates the following regression:

• $\Delta y_t$ is the change in the series from time $t-1$ to $t$.
• $\alpha$ is a constant and $\beta t$ is an optional time trend.
• $\gamma$ is the coefficient on the lagged level $y_{t-1}$, the quantity being tested.
• $\delta_i$ are coefficients on $p$ lagged changes, the terms that make the test augmented.
• $\varepsilon_t$ is the error term.

The null hypothesis is $\gamma = 0$: the level of the series has no pull on its next change, so there is a unit root. The alternative hypothesis is that $\gamma$ is negative. A negative value means that when the series is high its next change tends to be down, which is mean-reverting behavior.

The test statistic is the estimated coefficient divided by its standard error:

It is calculated like a t-statistic but does not follow the usual t-distribution when the null is true. It must be compared with special Dickey-Fuller tables. For a large sample and a regression with a constant and no trend, the critical value is about −3.43 at the 1 percent level, −2.86 at 5 percent and −2.57 at 10 percent. The null is rejected when the statistic is more negative than the critical value.

Suppose a regression on 500 daily observations of a price spread gives $\hat{\gamma} = -0.12$ with a standard error of 0.03. The statistic is −0.12 ÷ 0.03 = −4.0. This is below −3.43, so the unit root is rejected at the 1 percent level and the spread is treated as stationary. A statistic of −1.5 would not be below −2.57, and the unit root could not be rejected.

Two settings affect the outcome. The first is the deterministic part: no constant, a constant only, or a constant and a trend. Each has its own table of critical values, and the choice should match the series. A spread expected to hover around a fixed level calls for a constant. The second is the number of lags $p$. Too few leave serial correlation in the errors, and too many reduce the power of the test. The lag length is usually picked by an information criterion or by dropping lags that are not significant.

Statistical software reports a p-value computed from the Dickey-Fuller distribution, which can be compared directly with the chosen significance level.

Asset prices themselves almost never pass the test, and their returns almost always do. The practical interest lies in combinations of assets. In pairs trading, one price is regressed on another and the ADF test is run on the residual. If the residual is stationary, the two prices show cointegration and the gap between them is a candidate for a mean reversion strategy. Because the residual comes from an estimated regression, this two-step procedure uses stricter critical values than the standard tables. For more than two assets, the Johansen test is the usual tool.

The test is also a routine step before fitting an ARIMA model, where it guides how many times a series must be differenced.

The ADF test has low power against alternatives that are close to a unit root. A series that reverts slowly may fail to reject the null in a short sample, even though it is stationary. The opposite error also occurs: a search over hundreds of candidate pairs will turn up some that pass by chance, and those relationships often break down afterward. Results are sensitive to the sample period, to the lag choice, and to structural breaks, which can make a stationary series look like a random walk.

Failing to reject the null is not proof of a unit root. For that reason analysts often pair the ADF test with a test whose null is stationarity, such as the KPSS test, and treat agreement between the two as stronger evidence than either alone.