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A test of the [null] hypothesis that a time series has a unit root (ie, that it is non-stationary).

== Testing procedure == From Wikipedia's ADF 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

$$\Delta y_t = \alpha + \beta t + \gamma y_{t-1} + \delta_1 \Delta y_{t-1} + \cdots + \delta_{p-1} \Delta y_{t-p+1} + \varepsilon_t,$$

where $$\alpha$$ is a constant, $$\beta$$ the coefficient on a time trend and $$p$$ the lag order of the autoregressive process. Imposing the constraints $$\alpha = 0$$ and $$\beta = 0$$ corresponds to modelling a random walk and using the constraint $$\beta = 0$$ corresponds to modeling a random walk with a drift. Consequently, there are three main versions of the test, analogous to 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 $$\gamma = 0$$ against the alternative hypothesis of $$\gamma < 0.$$ Once a value for the test statistic

$$DF_\tau = \frac{\hat{\gamma}}{SE(\hat{\gamma})}$$

is computed it can be compared to the relevant critical value for the Dickey–Fuller Test. If $$DF_\tau>DF_{\text{crit. val.}}$$ where $$DF_{\text{crit. val.}}$$ is the relevant critical value, then the null hypothesis of $$\gamma = 0$$ is rejected and no unit root is present. (This test is non symmetrical so we do not consider the statistic's absolute value.)

The ADF-test is often used in conjunction with the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test for stationarity and detrending (ADF/KPSS) of time series as in the KPSS-test the [null] and alternate hypotheses are switched or opposite those of the ADF test, i.e., one the two tests will be significant.