Does a seasonal time series imply a stationary or a non stationary time series

If I have a time series that has got seasonality, does that automatically make the series non stationary? My intuition (probably off) is that it does not.

Seasonality means that the series goes up and down around a constant value….something like a sine wave. So by this logic a time series with seasonality is a (weakly) stationary series (constant mean).

Is this wrong? Why?

Answer

Seasonality does not make your series non-stationary. The stationarity applies to the errors of your data generating process, e.g. $y_t=sin(t)+\varepsilon_t$, where $\varepsilon_t\sim\mathcal{N}(0,\sigma^2)$ and $Cov[\varepsilon_s,\varepsilon_t]=\sigma^21_{s=t}$ is a stationary process, despite having a periodic wave in it, because the errors are stationary.

Seasonality does not make your process stationary either. Consider the same process but $\varepsilon_t\sim\mathcal{N}(0,t\sigma^2)$, in this case the error variance is non-stationary and seasonality has nothing to do with it.

Attribution
Source : Link , Question Author : Victor , Answer Author : Aksakal

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