0
$\begingroup$

I am rather confused about the alternative hypothesis of the Dickey-Fuller test (at least from a practical perspective).

I understand the mathematical details behind the DF test, but I am struggling to understand the consequences of rejecting the null hypothesis.

Say that we have a time series with an upward facing (linear) trend. We don't know a priori if this is a stationary process around a time trend or a random walk with a drift.

If we use the following specification of the DF test:

$$\Delta Y(t)=\beta_1+\gamma\cdot Y(t-1)+e(t)$$

and the null hypothesis is rejected then the alternative is that the process $Y(t)$ is stationary with non zero mean $\frac{\beta_1}{1+\gamma}$.

The problem is that this test is not comparing "deterministic trend" versus "stochastic trend". Like, if you reject the null hypothesis I mentioned above, the alternative is not "suitable" since we know that the series has an upward facing trend.

Am I missing something?

Improve this question
$\endgroup$
4
  • $\begingroup$ If you want to test the null of a random walk with drift against stationarity around a linear time trend, you should include a trend in your test regression, see stats.stackexchange.com/questions/210885/… for further details. $\endgroup$
    – Christoph Hanck
    Commented Jul 17 at 8:04
  • $\begingroup$ I haven't looked at what Christoph linked to but, IIRC, Hamilton does a nice job of explaining this. All that happens if you reject the null is that you are saying that there is evidence to reject the null. You're not saying anything about what the alternative is specifically. One thing you could google for is "trend stationary versus difference stationary". The famous paper on that is Nelson and Plosser, 1982. I think it's JASA. I'll find it and put it in another comment. $\endgroup$
    – mlofton
    Commented Jul 17 at 8:35
  • $\begingroup$ I had the year right and the journal wrong. This spurred a whole bunch of research into trend-stationary versus difference stationary so there is absolutely more recent work. But it's a good entry into the topic and it was groundbreaking at the time hedibert.org/wp-content/uploads/2015/03/nelson-plosser-1982.pdf $\endgroup$
    – mlofton
    Commented Jul 17 at 8:38
  • $\begingroup$ Christoph: Your link looks interesting at a glance. I'll read it tomorrow because it's too late for my brain. $\endgroup$
    – mlofton
    Commented Jul 17 at 8:42

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.