Unit root tests, stationarity, and the null hypothesis

I was reading about unit root test, when I started to get slightly confused about the setting for the Null hypothesis vs Alternative hypothesis, and so I thought of asking the experts opinion.

In the augmented Dickey-Fuller test, the null hypothesis is that there IS a unit root. My confusion comes from the fact that I think the null hypothesis should be that there is NO unit root. Allow me to explain:

The reason why I think so (and I know that I am probably wrong but I am hoping someone might point out my error), is more philosophical rather than mathematical. By this I mean: Accepting the null hypothesis, implies that what it says MIGHT be true (statistically), and rejecting the null hypothesis based on the observed data, means that there is a a very little (tiny) chance (p-value) that the null hypothesis is true, given the data, but very very unlikely (and hence we reject the null hypothesis).

But if we accept the null hypothesis, and we transform the data (by differencing it) to get rid of the unit root, then we have acted on what MIGHT be true, and as result we would be modeling a different time series.

IF (and that is a big if) the null hypothesis was that there is no unit root, then after running (my hypothetical) unit root test, I would only transform the data, if there is a very little chance that the magnitude of the root is less than 1.

Thanks for correcting my wrong thoughts in advance.

• You mix statistical testing logic with post-testing decision making. When you retain Ho you do it because of little evidence that it is unlikely. When you choose to difference, you assume that the retained is true. – ttnphns Sep 23 '11 at 6:49
• While others (@Karl, @ttnphns) have directly answered your question, let me answer a question that you did not ask. As an alternative or supplement to the Dickey-Fuller test, you may consider the KPSS test: en.wikipedia.org/wiki/KPSS_test, which has stationarity as its null hypothesis. – Charlie Sep 23 '11 at 15:00

The null hypothesis is "the differences, $y_{t+1} - y_t$, are stationary". You're suggesting switching it to the opposite, but one won't be able to carry out such a test, as very-close-to-stationary will look just like stationary.

But what you are really saying is that one should only take differences and act as if they're stationary if there is good evidence that they are stationary.

You might satisfy this concern by being less stringent about the conclusion of non-stationarity of the differences.

• (@charlie) thank you for ur advice. (@ttnphns, @Karl): the slight discomfort that i do have with this is that: Not having enough evidence to reject H0 is not the "same" as having evidence to support H0. I guess this relate a bit to type I and type II error. – Lalas Sep 23 '11 at 15:37

I think the null hypothesis should be that there is NO unit root.

I'm assuming that you're advocating to create a test for this null hypothesis. If that's not the case, then we have a problem, because you don't slap on the null hypothesis on an existing test. You try to come up with a test that tests a given null hypothesis usually. The other way is you have a model, and see what you can test. Your particular model will allow for tesing some hypotheses and not the others.

Case in point, Dickey-Fuller test. Take a look at its model: $$\Delta y_t = \alpha + \beta t + \gamma y_{t-1} + \delta_1 \Delta y_{t-1} + \cdots + \delta_{p-1} \Delta y_{t-p+1} + \varepsilon_t$$

We can't test your preferred null, i.e. NO unit root. Testing for this null would mean to check whether $\beta\ne 0$. Unfortunately, we can't test for this, but we can test whether $\beta=0$ holds. That is the reason why the null is "there is unit root", and not the philosophical preferences of Mr. Dickey and Mr. Fuller. In other words they test the hypothesis that they can test, not the one the would probably like to test.

Now, going back to your original point. If you can come up with the test that tests your null hypothesis of "NO unit" root, then go ahead and do it! By all means. Nobody would mind to have such a test, I believe. More tests - better for all of us.

The null hypothesis has to be something that has a specific statistical model that can be used to compute the probability of the observed data. A unit root is a specific model (root = 1) thus it is a testable null hypothesis. No unit root is a much more difficult (impossible?) thing to specify as a model to evaluate the probability of the data thus it is not used as the null hypothesis.

Note that we can only reject (data is unlikely to have come from the model). Failure to reject can arise from the null being true or from having insufficient data.

But one should always remember that a statistical test is just a tool that compares (measures) variability against the size of a departure from null. In the end, it is the size that matters, not the incidental fact that we are able to measure it.

If you reject the $H_0$ so there significantly enough evidence to say that the time series has not a unit root..

If you cannot reject the $H_0$, you did not proof anything.. Therefore $H_0$= unit root

Moreover, if your times series contain unit root, you have to change your inference for many statistical procedures. Hope this helps you