# Why does an (Augmented) Dickey Fuller test apply in an ARMA situation when it (seems to) assume an $AR(p)$ situation?

The set-up of this question may come out a little confused. But that's because I'm just that -- confused! If you can resolve my confusion, then that would be a full and satisfying answer!

To be as precise as possible amidst my confusion, let's break my question down to parts:

1. What is the null hypothesis of the (non-augmented) Dickey Fuller test? Is it that you begin with a time series that satisfies some $AR(1)$ equation such that its characteristic polynomial has no root at $1$? Do you further assume that the time series is stationary? (There are non-stationary time series that satisfy an $AR(1)$ equation with no root at $1$, right?)
2. Same as Question 1, but for the augmented Dickey Fuller. Presumably we assume that it satisfies some $AR(p)$ equation. Do we assume stationarity? (Is it not true that there are time series that satisfy some $AR(p)$ equation but are not stationary? Sure it is.)
3. It seems to me that the (augmented) Dickey Fuller test is often used for situations that are ultimately modeled using an $ARMA$ model. In that case -- why would ADF (augmented Dikcey Fuller) apply? Doesn't it assume that we're in an $AR(p)$ situation?
4. Is there a theorem that reduces an $ARMA$ situation to an $AR$ situation for checking whether the $AR$-characteristic polynomial has a root at $1$? What is it exactly?
5. It is often colloquially said that ADF is a "test for stationarity". Can you word that more precisely?

Any answer to any of these questions will be very welcome! Don't feel obligated to answer all of them in one fell swoop.

Edit: If I'm reading http://faculty.chicagobooth.edu/ruey.tsay/teaching/uts/lec11-08.pdf correctly, doesn't Dickey Fuller assume something much milder than $AR(1)$? Does it actually just assume that $z_t=\pi z_{t-1}+y_t$, where $y_t$ is some stationary process that satisfies very mild conditions? (This seems to subsume $ARMA(1,q)$, no?)

• Related post here. – Richard Hardy Sep 26 '17 at 15:03

1. The null hypothesis for DF test is a type of non-stationarity. $H_0: \phi_1 = 1$ where $X_t - \mu = \phi_1(X_{t-1}-\mu) + Z_t$.

2. The null hypothesis for ADF test is a type of non-stationarity as well, but with an AR(p) model. If your model is $X_t - \mu = \phi_1(X_{t-1}-\mu)+\cdots + \phi_p(X_{t-p} -\mu) + Z_t$, then $H_0: \sum_{i=1}^p \phi_i = 1$.

3. If an ARMA model is invertible, we can approximate it with an AR process. That's because we can divide both sides of $\phi(B)(X_t - \mu) = \theta(B)Z_t$ by the polynomial $\theta(z)$ to get $\frac{\phi(B)}{\theta(B)} (X_t - \mu) = Z_t$.

4. See above.

5. See above.

• Ah, so the crux is your answer to #3! So this only applies to invertible ARMA's? Does every time series that satisfies some ARMA equation also satisfy an invertible ARMA equation? (In that case, is the invertible ARMA equation unique? Is it unique if you fix p and q?) – Andrew NC Sep 26 '17 at 14:23
• @AndrewNC good question; it sounds like a meaty-enough question to warrant it's own thread/question. – Taylor Sep 26 '17 at 14:37
• Alas, I can only ask a new question every 40 minutes. And more often than not, nobody answers my questions. – Andrew NC Sep 26 '17 at 14:43