Unit root tests ambiguous - is time series stationary? I am testing a time series (quarterly) for stationarity. However, using the KPSS test, the ADF test and PP test, I get different results (ADF and PP reject non-stationarity, KPSS rejects stationarity, all of them at a 95% significance level). I am not sure how to handle this now. Can I just go for stationarity, because two of the tests resulted in stationarity?
My time series looks as follows:

 A: ADF and PP tests address a specific form of nonstationarity, i.e. unit-root nonstationarity. Apparently, you reject that form. However, unit-root nonstationarity it is not the only possible form of nostationarity. Hence, you do not conclude that the data is stationary (you only conclude that there is not enough evidence for unit-root nonstationarity). Combine this with an indication of nonstationarity by the KPSS test, and you are even more convinced the data is nonstationary.
Your data might have some level shift (the first half of the sample does not have quite the same level as the second half) and perhaps some heteroskedasticity (the oscilations in the first half of the sample seem a little larger than those in the second half).
So no, I would not feel comfortable "going for stationarity".
A: I am not surprised by these results.  I got them very often.  The KPSS test for some reason is very sensitive, if not overly so, as it rejects the vast majority of variables as stationary.  In other words, it diagnoses almost everything as non-stationary.  Because of that, I have stopped using the KPSS test for stationary diagnostics.  And, I rely on the other tests that seem fairer and more accurate on this issue.  The two other tests you use (PP and ADF) generate far more reliable results on this count.  Also, visually you can tell that your variable appears pretty stationary.  
I am revising my answer from 2017.  The KPSS test after all does not reject stationarity as often as I thought it did.  If you run the test in R, it almost always gives you a p-value of 0.1 making it a bit difficult to accept the null hypothesis that the variable is stationary.  But, underneath the calculation you see a red written warning that states "... p-value is greater than the printed value shown."  It means that whether the p-value is 0.11 or 0.99, the test result will show it as 0.1.  In other words, the KPSS does not reject that a variable is stationary nearly as often as I thought it did.  And, now I pretty much use it again all the time in such circumstance.  It is nice to run a stationarity test that runs in the opposite direction of the others as its null is that the variable is stationary instead of non-stationary as in all the other tests. 
This does not detract that different tests will give you often contradicting results.  I typically run three tests, and if the variable passes 2 out of 3 tests, I deem it as adequate evidence that the variable is stationary. 
Last but not least, when I look at your time series graph your variable appears stationary enough so that it should not render any model that you build using it "spurious."        
A: There are six different unit root test available in Eviews:


*

*The Augmented Dickey-Fuller (ADF) Test   

*Dickey-Fuller Test with GLS Detrending (DFGLS) 

*The Phillips-Perron (PP) Test 

*The Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) Test    

*Elliot, Rothenberg, and Stock Point Optimal (ERS) Test 

*Ng and Perron (NP) Tests


Each test has some assumptions, for example, ADF test should be used is series has auto-correlation but it is homoskedastic. 
and PP test should be used if series has both issues suto-correlation and presence of heteroskedasticity.
It is better to select the suitable test for stationarity, instead of applying all.
You can read more about these test at 
http://www.eviews.com/help/helpintro.html#page/content/advtimeser-Unit_Root_Testing.html
All the best!
