Is the Dickey Fuller test one-sided or two-sided?

Question

The Dickey Fuller test tests whether a unit root is present in an AR model. Specifically, we have

$$ X_t = \phi X_{t-1} + \varepsilon_t $$ where $\varepsilon_t$ is a Gaussian noise term.

Now the Dickey Fuller test tests the Null Hypothesis H0: $\phi=1$, or equivalently H0:$\gamma=0$ in the differenced equation. This is how the test is described in text books, for example, in Pesaran (2015, page 332) or Enders (2012, page 114). Enders does not specify an alternative hypothesis, Pesaran does state that H1:$|\phi|<1$. However, to me this leaves a few questions about the DF-test unanswered:

1. The H0 and H1 stated above combined do not comprise the real line. This raises the question of what to do with values $\phi \leq -1$? Those would reject the null hypothesis, but clearly we would also have a unit root. I assume therefore, that the test statistic is $|\phi|$ and not $\phi$? This is the only explanation that makes sense to me. On the other hand that would mean that all these text books made a mistake, which I find unlikely. Or do economists simply handwave the $\phi<0$ cases away?

2. Assuming the test statistic $|\phi|$, if the test were two-sided, then we could reject H0 if we observed $|\phi|>1$, which does not make sense, since we have a unit root for those cases. Because of that, I assume that the DF-test is a one-sided test, which can only be rejected by test-statistics $|\phi|<1$. Is this correct?

Answer 1

First, $\phi$ is the true coefficient of the process, not the test statistic.

The test statistic that is something else than $\phi$, namely the default t-statistic for the hypothesis that the coefficient on $X_{t-1}$ in a regression of $\Delta X_t$ on $X_{t-1}$ is zero. See e.g. Why is Dickey-Fuller test applied on the difference operator and not on the variable directly?

As to 1., indeed, values of $\phi\leq-1$ are "oscillatingly explosive" and hence not really plausible in economics (nor any other discipline I am aware of).

x <- filter(rnorm(500), -1.01, "recursive"); plot(x):

As to 2., no, values of $|\phi|>1$ are not "unit root" in that $1-\phi\cdot1\neq0$, i.e., 1 is not a solution to the characteristic polynomial.

But since we reject when the DF test statistic takes sufficiently negative values, the test has no power when $\phi>1$ as we then also tend to have positive coefficients on the estimated coefficient on $X_{t-1}$ and hence positive test statistics and hence no rejections.

When $\phi\leq-1$, by the same argument, the test statistic will be (strongly) negative, and hence the DF test statistic would indeed produce rejections of the null that $\phi=1$. (You could verify this by repeatedly running summary(lm(diff(x)~head(x,-1)-1))$coefficients[1,3] on new iterations of x above.)

But then, 1. kicks in again, where I argue that such processes are not very practically relevant (although examples are mentioned in the comments below).

Hence, to summarize, this is the sense in which the DF test is seen as a one-sided test of the null that $\phi=1$ against the "stationary" alternative that $|\phi|<1$.