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?
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,
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
t2 fits the stationarity requirement for the regressors. Since the trend components in
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.)