Augmented Dickey–Fuller test


In statistics and econometrics, an augmented Dickey–Fuller test tests the null hypothesis that a unit root is present in a time series sample. The alternative hypothesis is different depending on which version of the test is used, but is usually stationarity or trend-stationarity. It is an augmented version of the Dickey–Fuller test for a larger and more complicated set of time series models.
The augmented Dickey–Fuller 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 testing procedure for the ADF test is the same as for the Dickey–Fuller test but it is applied to the model
where is a constant, the coefficient on a time trend and the lag order of the autoregressive process. Imposing the constraints and corresponds to modelling a random walk and using the constraint corresponds to modeling a random walk with a drift. Consequently, there are three main versions of the test, analogous to the ones discussed on Dickey–Fuller test
By including lags of the order p the ADF formulation allows for higher-order autoregressive processes. This means that the lag length p has to be determined when applying the test. One possible approach 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 against the alternative hypothesis of Once a value for the test statistic
is computed it can be compared to the relevant critical value for the Dickey–Fuller test. As this test is asymmetrical, we are only concerned with negative values of our test statistic. If the calculated test statistic is less than the critical value, then the null hypothesis of is rejected and no unit root is present.

Intuition

The intuition behind the test is that if the series is characterised by a unit root process then the lagged level of the series will provide no relevant information in predicting the change in besides the one obtained in the lagged changes. In this case the and null hypothesis is not rejected. In contrast, when the process has no unit root, it is stationary and hence exhibits reversion to the mean - so the lagged level will provide relevant information in predicting the change of the series and the null of a unit root will be rejected.

Examples

A model that includes a constant and a time trend is estimated using sample of 50 observations and yields the statistic of −4.57. This is more negative than the tabulated critical value of −3.50, so at the 95 percent level the null hypothesis of a unit root will be rejected.

Alternatives

There are alternative unit root tests such as the Phillips–Perron test or the ADF-GLS test procedure developed by Elliott, Rothenberg and Stock.

Implementations in statistics packages