ADF test, PP test, KPSS test: Which test to prefer? If a time series is tested for Unit Root (by ADF, PP, KPSS,...) problem is detected with some tests and not found by others. Which one is preferred? For example if ADF says us that there is a Unit root and PP says that is not problem. Which one of them would be preferred?
 A: Generally, if there are multiple statistical tests that apply for the same thing and work differently, then a single significant result leads to rejection. It follows from the logic of hypothesis testing theory that failing to reject means that there has not been enough evidence. This is how normality is often tested.
A: I answered two verisimilar question:


*

*Contradictory results of ADF and KPSS unit root tests

*What is the difference between a stationary test and a unit root test? 
(I summarized the most relevant part of this answer below)



How unit-root test and stationarity-test complement each other
If you have a time series data set how it usually appears in econometric time series I propose you should apply both a Unit root test: (Augmented) Dickey Fuller or Phillips-Perron depending on the structure of the underlying data and a KPSS test.
Case 1 Unit root test: you can’t reject H0; KPSS test: reject H0. Both imply that series has unit root.
Case 2 Unit root test: Reject H0. KPSS test: don`t reject H0. Both imply that series is stationary.
Case 3 If we can’t reject both test: data give not enough observations.
Case 4 Reject unit root, reject stationarity: both hypothesis are component hypothesis – heteroskedasticity in series may make a big difference; if there is structural break it will affect inference.
Power problem: if there is small random walk component (small variance σ 2 u ), we can’t reject unit root and can’t reject stationarity.
Economics: if the series is highly persistence we can’t reject H0 (unit root) – highly persistent may be even without unit root but it also means we shouldn’t treat/take data in levels.
General rule about statistical testing You cannot proof a null hypothesis you can only affirm it. However if you reject a null hypothesis you can be very sure that the null hypothesis is really not true. Thus alternative hypothesis is always a stronger hypothesis than the null hypothesis.

