Augmented Dickey–Fuller test

The augmented Dickey–Fuller (ADF) statistic, used in the test, is a negative number. The more negative it is, the stronger the rejection of the hypothesis that there is a unit root at some level of confidence.

Testing procedure

The procedure for the ADF test is the same as for the Dickey–Fuller test but it is applied to the model

${\displaystyle \mathrm{\Delta }{y}_t=\alpha +\beta t+\gamma y_{t-1}+{\delta }_1\mathrm{\Delta }{y}_{t-1}+\cdots +{\delta }_{p-1}\mathrm{\Delta }{y}_{t-p+1}+{\epsilon }_t,}$

where ${\displaystyle \alpha }$ is a constant, ${\displaystyle \beta }$ the coefficient on a time trend and ${\displaystyle p}$ the lag order of the autoregressive process. Imposing the constraints ${\displaystyle \alpha =0}$ and ${\displaystyle \beta =0}$ corresponds to modelling a random walk and using the constraint ${\displaystyle \beta =0}$ corresponds to modeling a random walk with a drift. Consequently, there are three main versions of the test, analogous those of the Dickey–Fuller test. (See that article for a discussion on dealing with uncertainty about including the intercept and deterministic time trend terms in the test equation.)

By including lags of the order p, the ADF formulation allows for higher-order autoregressive processes. This means that the lag length p must be determined in order to use the test. One approach to doing this is to test down from high orders and examine the t-values on coefficients. An alternative approach is to examine information criteria such as the Akaike information criterion, Bayesian information criterion or the Hannan–Quinn information criterion.

The unit root test is then carried out under the null hypothesis ${\displaystyle \gamma =0}$ against the alternative hypothesis of ${\displaystyle \gamma <0.}$ Once a value for the test statistic

${\displaystyle {\mathrm{D}\mathrm{F}}_{\tau }=\frac{\hat{\gamma }}{\mathrm{SE}(\hat{\gamma })}}$

is computed, it can be compared to the relevant critical value for the Dickey–Fuller test. As this test is asymmetric, we are only concerned with negative values of our test statistic ${\displaystyle {\mathrm{D}\mathrm{F}}_{\tau }}$. If the calculated test statistic is less (more negative) than the critical value, then the null hypothesis of ${\displaystyle \gamma =0}$ is rejected and no unit root is present.