# Augmented Dickey-Fuller Questions

I've been searching in bibliography about this test applied to an AR(p) model. $$Q(L)(Y_{t})=c+\epsilon_{t}$$

Where L represent the Lag Operator and $Q=1-\phi_{1}x-.....-\phi_{p}x^{p}$ is the polynomial expression associated to the model.

I know that if $Q(r)=0$ implies $|r|>1$, then the process is stationary (at least in weak sense).

My question is: Why the Null Hypothesis of Augmented Dickey-Fuller test is stated as: "$r=1$ is a root of the polynomial"? Rejecting that hypothesis implies that every single root of Q lies outside the unit circle??

I'm new at this area so every recommendation or suggestion will be useful. Thanks.

• This is a purely statistical question, you might get the best answers here: stackexchange.com – Richard Jan 14 '16 at 7:11
• You may also try here: stats.stackexchange.com. – Gordon Jan 14 '16 at 16:28
• I'm voting to close this question as off-topic because it belongs on stats.stackexchange.com – chollida Jan 14 '16 at 21:11

• The point is that you do not reject in the tail where $r>1$ but only in the tail where $r<1$. Your statement If r>1 (not in absolute value) the series follows and explosive and therefore is not stationary is correct by itself but it is redundant in the context of the augmented Dickey-Fuller test. Therefore rejecting the null is only compatible with stationarity (and not with explosivity). – Richard Hardy Feb 28 '16 at 22:25