Can stationarity occur over different time periods? According to what I've read on stationarity it seems like it is either an all or nothing property. Basically, if we have a time series with a unit root then it progresses as a random walk with no stable mean or variance.
But is it possible for a time series to be stationary but only over long periods? Could it have periods of nonstationarity followed by stationarity? If so, how could this be detected? Thank you for your help.
 A: I would think if you have a structural break and this changed the pattern of your data then you could have a non-stationary time series become stationary or the other ways around. Data can go from having a trend to not having a trend over time.
A: Generally, the answer is NO, because nothing in the definition of stationarity suggests a finine period, so it must apply to all time line. However, if you examine a simple AR(1) process, it behaves very much like a random walk in short time scale. So much that it can be indistinguishable from RW if a sample is small and relatively short.
Here's AR(1) model
$$x_t=\phi_1x_{t-1}+e_t\\e_t\sim IID(0,\sigma_e^2)$$
$$\Delta x_t=(\phi_1-1)x_{t-1}+e_t$$
If you look at the variance of the process in short time frame, i.e. when $1-\phi_1\approx 0$, it's proportional to time $var[x_t]\approx t\sigma^2_e$, just like a unit root process. Obviously, in a long run the variance converges to a constant unlike a random walk process.
Note, the time scale of AR(1) process is $\tau=-\frac 1 {\ln \phi_1}$
