Question about unit root testing and non-stationarity of time series with a trend using ADF test for further making regression I am new to time series analysis and I have the following question
The purpose of my research is to build a regression between two time series.
The length of my sample is 21 observations.
I have a situation like this:
The ADF test with trend and constant shows that the time series is stationary (although it has a trend).
The question that interests me:
Do I need to take the first differences of such a time series? Or can a regression be done with the original time series values (i.e. without the first differences)?
If yes, then if there is a trend in the time series, then it is necessary to use the ADF test without a trend?
 A: Issue 1: Should you believe the ADF test?
Due to the very small sample size, the ADF test will have pretty low power, i.e. the ability to reject $H_0$ when it is incorrect. Your test does reject $H_0$, so power does not seem to be an issue. But this may go hand in hand with a possible size distortion. Instead of the nominal 5% significance level, the actual significance level may be very different (e.g. much higher) in a small sample.
Another thing is the specification of the model used in the ADF test. The test results are sensitive to the number of lags and inclusion/omission of the trend. The choices you make thus have to be considered carefully.
In sum, you should not be too confident in the test result.
Issue 2: Should you difference a time series that is stationary around a deterministic trend?
No. This is not advisable as it would remove a linear trend but at the same time introduce an artificial moving average component with a unit root into the resulting series. The phenomenon is known as overdifferencing. Hence, better not difference series with deterministic trends unless they have a unit root, too.
What you can do is model the time series by including the trend among the regressors. Note that the estimator of the slope coefficient will be superconsistent (converge to the estimand faster than usual). This property allows for an alternative approach: detrend the series first and then model the resultant stationary component further.
