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I am reading an online time series book and get confused with two pictures. According to the author, picture (b) "Daily change in the Google stock price for 200 consecutive days" and g) "Annual total of lynx trapped in the McKenzie River district of north-west Canada" are stationary plots. However, in picture b, there is a big spike, (not constant variance?) and in picture g, it looks like a business cycle (every 40 years) existed. I am not sure if you could call it seasonality. According to Wikipedia, a seasonality is the presence of variations that occur at specific regular intervals less than a year, such as weekly, monthly, or quarterly. As a starter, I would like to get help on how to interpret this kinds of pictures when there are spikes and business cycles in the pictures.

enter image description here

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Series b. This spike is just an outlier. Single outlier does not affect stationarity. Stationarity is about distribution, not about variance constness. An outlier does not affect a distribution of $X_t$, so it does not affect stationarity. This large value is just a random fluctuation, nothing more.

Series g. This series does not have seasonality. It has cycles (you can note, that these spikes have different height). You are right - seasonality implies non-stationarity, but cycles do not, so this time series is stationary.

You can check if the series is stationary using some statistical tests, like Dickey-Fuller test, or KPSS test.

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  • $\begingroup$ Thanks @Georgy, can you explain a little bit about "Stationarity is about distribution, not about variance constness". Constant variance seems to be 1/4 criteria to access the stationarity. $\endgroup$
    – smoothy
    Jan 23, 2021 at 11:00
  • $\begingroup$ @smoothy Well, roughly speaking, process is stationary if $X_t$ has a distribution, that does not depend on $t$. If a value $X_{t_0}$ for specific $t_0$ differs from other in its neighbourhood (like on b) it doesn't implies difference in distribution (and variance too), so the single outlier does not affect stationarity. You can note, that this situation differs form the situation on picture e, where several values in the middle have different variance campared to nearest ones. A single outlier changes a sample variance, but not a population variance. $\endgroup$ Jan 23, 2021 at 11:10

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