# Dickey-Fuller test for stationarity

The Dickey-Fuller test tests an AR(1) series for stationarity.

An AR(1) series can be written as:

$$x_t = \phi x_{t-1} + \epsilon_t$$

with $$\phi$$ constant and $$\epsilon_t$$ white noise.

The series is stationary only if $$\phi<1$$.

This series can be written as:

$$\Delta x_t = \beta x_{t-1} + \epsilon_t$$

with $$\beta = \phi - 1$$

The null and alternative hypothesis of the DF test are respectively:

H0: $$\beta=0$$ (i.e. $$\phi=1$$) -> non-stationary

H1: $$\beta<0$$ (i.e. $$\phi<1$$) -> stationary

I do not understand why the case $$\phi>1$$ (which should corresponds to 'non-stationary') is left out in the hypothesis.

It is because when $$\phi > 1$$ it's clearly non-stationary, as it has trend. In this case, the failure of stationarity can be detected by eye, so the ADF test is not necessary.
More importantly, the ADF test is really testing whether or not there's a unit root. So the null hypothesis is "yes, there's a unit root" and $$\phi > 1$$ does not fit into that camp. Note that a time series can pass the ADF test, meaning there's no unit root, and still fail to be stationary. It's a unit root test, not an "if and only if" test for stationarity.