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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.

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    $\begingroup$ Hi: It's an all or nothing type of thing. Note though that some stationarity definitions are stronger than others. There's weak, strict and possibly others which I can't recall at the moment. $\endgroup$
    – mlofton
    Commented Aug 4, 2020 at 1:45
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    $\begingroup$ You might be interested in looking at regime change models, in which some parameters are allowed to change at discrete points in time. The classic example is the change between bull and bear markets in financial models: theses models have one set of parameters that describe a bull market, then a regime change occurs after which a different set of parameters for the bear market. These don't fit the definition of stationarity, but you could have something similar, where the distribution is constant on an interval of time, and then nonconstant on the next interval of time, etc. $\endgroup$
    – Eric Perkerson
    Commented Aug 4, 2020 at 2:15
  • $\begingroup$ @ericperkerson Thanks, sounds worth taking a look at! $\endgroup$
    – Michael Howell
    Commented Aug 5, 2020 at 2:06

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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 $\tau=-\frac 1 {\ln \phi_1}$

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  • $\begingroup$ I thought the variances of all ARIMA process was assumed to be proportional to the variance of the error term, not just random walks. Am I mistaken? $\endgroup$
    – Michael Howell
    Commented Aug 5, 2020 at 2:11
  • $\begingroup$ @MichaelHowell, I'm talking about the variance of the prediction. It is proportional to the error variance but also scales with time initially, just like a random walk. The difference is that it eventually slows down and converges to a constant, while in RW it keeps scaling with time to infinity $\endgroup$
    – Aksakal
    Commented Aug 5, 2020 at 2:39
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    $\begingroup$ Hi: if you are interested in looking further at what Aksakal is referring to, they are called "near unit root processes" in the literature. As far as the mention of regime changes, I would argue that a regime change is still indicative of non-stationarity. If the regime change implies a change in the mean or a change in the variance, then stationarity ( both kinds ) still does not hold. $\endgroup$
    – mlofton
    Commented Aug 5, 2020 at 3:28
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    $\begingroup$ I do not know the theory, I have not seen this addressed, but in practice a line may have a trend in it at some points and not others. So when you say a line is stationary, or not, you really have to define at what point this is true or not. Probably if a set of data is looked over a long enough period of time it is never stationary. $\endgroup$
    – user54285
    Commented Aug 6, 2020 at 16:15
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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.

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  • $\begingroup$ So if the underlying data generating process changes then a stationary or non-stationary model would simply become more appropriate? $\endgroup$
    – Michael Howell
    Commented Aug 5, 2020 at 2:08

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