# Unit roots Dickey Fuller test question

I've been searching in bibliography about this test applied to an ARMA(p,q) model, and find out that every single book states the null hypothesis as "1 is a root of the operator". I was wondering if that was merely a simplification, or that's the test?

I'm asking this question because it seems to me that the null hypothesis must be "there exists a root with absolute value less or equal to one".

I'm new in this subject, and any information or book recommendation is appreciated.

• So are you wondering if the root can be >|1|? – gung - Reinstate Monica Jan 13 '16 at 18:40
• No, I'm wondering if rejecting the null hypothesis "1 is a root of the operator" implies that the operator has all the roots outside the unit circle, – Ivan Rey Jan 13 '16 at 19:15
• Thanks for your reply. My question is about the maths involved in the test. I don't understand why rejecting the Null Hypothesis $H_{0}:\gamma=0$ implies that the poynomial expression associated to the model has all roots outside the unit circle. – user100635 Jan 14 '16 at 0:06

Consider a simple special case of the augmented Diskey-Fuller test equation:

$$\Delta x_t = \gamma x_{t-1} + \varepsilon_t$$

(I have skipped all other potential right-hand-side terms since they are just nuisance with respect to your question).

You reject the null hypothesis of the test when $\gamma$ divided by its standard error is sufficiently negative. If $\gamma$ indeed is negative, the process is stationary. If $\gamma$ is zero, the process has a unit root. If $\gamma$ is positive, the process is explosive. Hence, when the null hypothesis is not rejected, the process may be either a unit-root process or an explosive one. (For the sake of formality I have to note that we do not accept a null hypothesis, we may just fail to reject it.) Addressing your actual question,

I'm wondering if rejecting the null hypothesis "1 is a root of the operator" implies that the operator has all the roots outside the unit circle

the answer is yes, the null hypothesis is rejected when the roots are outside the unit circle (rather than the more general case that they are not on the unit circle).

• Thanks for the reply Richard, but maybe I didn't express myself correctly. My Null Hypothesis is $H_{0}: " |\gamma+1| \geq 1 "$. While I was looking for the test in the internet and books, I saw that a $\gamma=0$ is being used to run the test. The question is. Why is that correct? – Ivan Rey Jan 13 '16 at 21:13
• I guess your question is more about formal references than the nature of the test. For me the test equation and the test statistic reveals all I need to know about it. But OK, it is still a valid question. – Richard Hardy Jan 13 '16 at 21:27