# How to determine ADF test criterion?

I tried ADF test and there are 3 criterions.

1. intercept
2. intercept and trend
3. none

I usually decide the criterion after looking at the graph.
(For example, if the data is around 0 then choose intercept, and if the data has trend then choose trend, if the data has both then choose intercept and trend..)
I wonder if my method is right or not.
How do you choose ADF criterion?

• "I searched some statistical websites and they say usually that three results should be the same" Can you please edit your question and include a link to such a site? Dec 20, 2021 at 8:48
• @Giskard I've visited that site long time ago so it took a while to find the link. Reading it again, I think maybe I misunderstood the sentence but still it seems that there is still no exact standard for the ADF test.(researchgate.net/post/…) Dec 20, 2021 at 13:04
• @guest, if I were after high-quality advice on ADF and related matters, I would search Cross Validated before going to ResearchGate. Dec 20, 2021 at 15:18

1. what you describe is not criterion, but model specification. Criteria are used to determine lag order (e.g. you have Akaike criterion etc).

2. You should choose the specification that best matches your data. If you choose wrong specification your model will be misspecified and that typically leads to bias in the coefficient values. On the other hand if you include something extra, you won't run into misspecification issue but you will lower the power of your test because every extra coefficient sacrifices a degree of freedom in your model. Hence you want model specification that is as appropriate as possible (see discussion in Verbeek, A Guide to Modern Econometrics Ch8).

You can determine whether to include trend or not by testing whether there is significant linear time trend in the series (just run auxiliary regression where you regress $$t$$ on $$y$$.

When it comes to intercept you can decide by testing whether the average of $$y$$ is statistically different from zero, and also by checking if there is no drift in the series (e.g. does it look like mean is changing over time). If mean is not zero and it looks like it is shifting over time then you need to include intercept.

3. I searched some statistical websites and they say usually that three results should be the same(How could it be possible?).

This is not possible, at the minimum the DF $$t$$-stat has to change at least marginally (since you have different degrees of freedom). So it is impossible that results are exactly the same.

However, maybe the site you read was talking about the conclusion based on the result (i.e. is there unit root or is the series stationary?). There it would be true that sometimes all 3 specifications lead to rejection of null hypothesis, or vice versa. However, that does not mean you can just run any model you want. In any research you want to have the best possible results not just half-assed results that might turn out to be incorrect once someone runs proper specification of your test.