Unit root test Unemployment I want to analyze the unemployment rate in Austria from 1999 to 2017, quaterly data.
Here's the code for the time series:    
data<-ts(c(4.7,3.5,3.2,3.5,4.7,3.1,3.1,3.2,4.0,3.4,3.3,3.7,4.9, 
      3.6,3.6,3.8,4.8,4.2,3.9,4.3,5.8,5.4,5.3,5.4,5.7,5.7,5.6,5.6,6.0, 
      5.2,4.9,4.9,5.2,5.0,4.9,4.4,4.5,3.7,4.0,4.3,5.0,5.3,5.7,5.3, 
      5.2,4.9,4.8,4.4,5.1,4.4,4.0,4.8,4.7,4.8,5.1,4.9,5.7,5.0,5.3, 
      5.4,6.0,5.4,5.6,5.6,5.8,5.8,5.6,5.7,6.3,6.1,6.1,5.6,6.0,5.4), start=c(1999,1), end=c(2017,2), frequency=4)

What I did first was a graph to catch some features of the data generating process:
What I'm wondering is: this time series comes from an I(1) is a I(0). Basically Do I need to differentiate?
I would expect not to differentiate, cause I suppose that the unemployment rate should be around the natural rate of unemployment. Maybe that's not the case and there's a structural change in the economy which shift the NA.r.u. up.
The identify if it's need a differention I performed an ADF- Augmented with constant. The result is: it inclued no delay and p-value is slightly above 5% (0.05415). So I can't reject the null hypothesis of unit root. So this indicates me to differentiate.
But first I want to see the original correlogram: 

According to this graph and my knowledge and I'm not worried for unit roots. Maybe it's only higly persistent.
So I would implement an ARMA(1,0)(2,0) which provides a good residual correlogram but not so good forecasting:

Maybe Should I drop the first observation (till 2004) cause it seems to be more seasonl effected than the other part of time series.
Should I differentiate? And, overall, my reasonings are right?
 A: I am not familiar with the Austrian economy so I cannot assess the claim about the unemployment rate being close to the natural level; I understand that, should that be the case, that claim would be equivalent to asserting that there is a cointegrating relationship between the actual and the natural rates of unemployment. With that in mind, such a relationship would not disqualify the possibility that both series are explosive, in a 'similar' way.
Additionally, the recent financial crisis and its aftermath would probably justify some considerations about breaks in structural parameters of the european economy in general and Austria in particular; of course that is a tentative assumption; more testing is needed.
Taking a look at the formal tests you provide, I personally would not be comfortable treating the unemployment rate as stationary series; there seems to be indications for AR(1) components from the correlograms and the ADF does not reject the Null.
Depending on what you intend to do, I think that you'd be better off exploring some other specifications of the ADF tests and/or consider the output of stationarity tests like KPSS. 
