Let's say you have a variable with a time series going back ten years. In the first 5 years, it clearly trends from 5 to 10. And, in the next 5 years, it trends downward from 10 back down to 5. When you test it for Unit Root using either Dickey-Fuller or the ADF test, should you use the ADF test with a trend or not?

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    What do you mean by "clearly trends"? Unit root processes are infamous for displaying seemingly obvious trends, while these "trends" are purely stochastic. If this is really the case here, an ADF test with no trend would make sense. If you believe that there might be a linear trend which has changed, then you should do some kind of structural break/broken trend test. Perhaps this will lead you to something useful: scholar.harvard.edu/files/stock/files/… – hejseb Oct 25 '15 at 18:51
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    See also section 2 here (might be a little more accessible) mobile.businessandeconomics.mq.edu.au/our_departments/Economics/… – hejseb Oct 25 '15 at 19:15
up vote 1 down vote accepted

I think the best test specification would be neither ADF with no trend nor ADF with a linear trend, because clearly none of the alternatives adequatly reflects the actual trend in the data.

You may consider using covariate-augmented Dickey-Fuller (CADF) test proposed in Hansen "Rethinking the Univariate Approach to Unit Root Testing: Using Covariates to Increase Power" (1995). Hansen's own R code for the test is available here. There is also an R package "CADFtest" by Claudio Lupi with a vignette and a reference manual which may be more readily usable than Hansen's code.

For the CADF test you would supply two regressors, t1=c(1:br,rep(0,T-br)) and t2=c(rep(0,br),1:(T-br)) to account for the two linear components of the trend, where br is the last point of the upward-trending period and T is the lenght of the data sample.

However, I am unsure how the use of t1 and t2 fits the stationarity requirement for the regressors. Since the trend components in t1 and t2 are nonstationary, they might mess up the null distribution of the parameter of interest in the CADF test regression. That could be a good argument for not using the CADF test in this situation.

If so, you could perhaps just split your sample into two parts and use the regular ADF test with a trend for each of them. It should be better than using the ADF test for the whole sample regardless of inclusion or exclusion of a linear trend. Doing the latter might well induce the ADF test to suggest presence of a unit root even if the process around this "broken trend" is actually stationary.

The last option for someone who is good at unit root asymptotics would be to derive the appropriate null distribution of the CADF test in this "broken trend" setting.

(Here is a somewhat related post.)

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    What do you suggest is the meaning of the divided-sample ADF test? I highly doubt that this is an advisable approach. Moreover, this would imply that the break date is exogenous and given. Usually, when there are structural breaks that is not the case. – hejseb Oct 25 '15 at 18:48
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    My concern is that while the root of interest is the same, you are estimating it very inefficiently if you split it. You also run the risk of accepting one test but rejecting another. The trend is just a nuisance, but the root is the same in the entire model. – hejseb Oct 25 '15 at 19:14
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    I can only agree. However, I was trying to choose the best option from what options there are, not that the best option would be very good. My argumentation was: (1) chances are that the ADF test with no trend will suggest I(1) regardless of whether the detrended data is I(1) or I(0) because of the neglected deterministic trend; (2) the same for ADF test with trend; (3) CADF will not work because the linear trend component before the break as well as the component after the break are nonstationary variables. Only splitting the sample will allow using correctly specified ADF test regressions. – Richard Hardy Oct 25 '15 at 19:23
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    @hejseb (continued) Some power would be lost, true, but isn't that better than using clearly misspecified tests? Regarding the break point: intuitively, I would not be too worried about the detection of the break point. If its location is somewhat unclear, a few observations around the potential break point could be skipped. That would harm the power again, I must admit. Also, often (or at least sometimes) we know the cause of the break and can clearly identify the date. But what would your advice be? – Richard Hardy Oct 25 '15 at 19:25
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    My advice would be to refer to the literature where there are suggested remedies). Including a (single) trend is clearly inappropriate here, but since power is usually very low in the ADF test I think one should not throw it away so easily here. Withou having researched the literature in this particular instance, a trend change is like one of the most basic structural breaks. Instead of using a second-best (but still, I'm guessing, a quite poor) option using standard tests, finding the appropriate adjustment in the literature I believe is better. – hejseb Oct 25 '15 at 19:50

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