# Confusion about stationarity because of contradiction in unit root test and graphical analysis

I am confused about the output of a unit root test I performed. After taking data with a seasonal effect in seasonal differences, the time-plot and the correlogram (shown below) both suggest non-stationarity.

But when performing the unit root test, the null hypothesis of non-stationarity is rejected.

So now I don't know what to assume; stationarity or not? When I take the data in first differences, the same problem stays.

Can you guys maybe help?

• Could you explain just how those plots suggest non-stationarity? – whuber Oct 31 '17 at 17:04
• Stationarity has the condition that var[Yt] is the same for all t which is clearly not the case when analyzing the time-plot. Another condition is that Cov(Yt, Yt-k) is same for all t, which is rejected by the correlogram? Or am I wrong in this graphical analysis? – J.B Oct 31 '17 at 17:11
• A process can be non-stationary, and also not have unit root. For example, it could be a long memory process (with a coefficient of lagged $y$ an absolute value close to, but definitely less than 1), or a short memory process (with a coefficient of lagged $y$ an absolute value definitely greater than 0, but also less than 1). "Unit root" simply means a memory process that is undifferentiable from infinitely long memory (with a coefficient of lagged $y$ not significantly different than 1)—with respect to the time-span of the data set, at any rate. – Alexis Oct 31 '17 at 18:53
• The correlogram couldn't even be interpreted without assuming the covariance is stationary! I think you might be confusing the lag $k$, plotted on the horizontal axis on the correlogram, with $t$ itself, which appears nowhere there. And yes, the time plot is not stationary, but it's understood that before you look any further and construct a correlogram or conduct unit root tests you will remove any such grossly apparent seasonality or trend. Otherwise your tests merely confirm the obvious and fail to provide the subtler information they are designed to reveal. – whuber Oct 31 '17 at 19:07