# 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?