Augmented Dickey–Fuller test
In statistics and econometrics, an augmented Dickey–Fuller test (ADF) 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 (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 testing procedure for the ADF test is the same as for the Dickey–Fuller test but it is applied to the model where formula_2 is a constant, formula_3 the coefficient on a time trend and formula_4 the lag order of the autoregressive process. Imposing the constraints formula_5 and formula_6 corresponds to modelling a random walk and using the constraint formula_6 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 (see that  page 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" 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 formula_8 against the alternative hypothesis of formula_9 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 formula_11. If the calculated test statistic is less (more negative) than the critical value, then the null hypothesis of formula_8 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 (formula_13) will provide no relevant information in predicting the change in formula_14 besides the one obtained in the lagged changes (formula_15). In this case the formula_8 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 formula_11  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 (PP) or the ADF-GLS test procedure (ERS) developed by Elliott, Rothenberg and Stock (1996).