# 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?

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.

• I disagree with this answer. The series is not even weakly stationary (a.k.a. wide-sense stationary) because $E[Y_t] = \sin(t)$ is not a constant. It is what is sometimes referred to as covariance-stationary because the covariance $\operatorname{cov}(Y_{t_1},Y_{t_2})$ depends only on the difference $t_1-t_2$ between the time instants. The series is, of course, not strictly stationary in any sense of the word. – Dilip Sarwate Oct 16 '15 at 12:57
• Determinism, that is, lack of randomness, is not what is relevant here; it is the definition of stationarity (or weak stationarity since time series folks seem to use stationary to mean weakly stationary or wide-sense stationary) that is relevant, and by the usual definitions, your answer is incorrect. See, for example this more recent question where the issue is discussed in detail and the accepted answer there (by @Silverfish) is a contradiction of your answer here. – Dilip Sarwate Oct 16 '15 at 13:58
• Considering the academic definition I agree with DilipSarwate. The WSS definition is defined over the unconditional mean of the process, not the conditional mean. Moreover if you claim that we can strip away the deterministic trend in some cases, therefore we may conclude that a process is stationary, by same logic I can claim that random walk is stationary because I can differentiate it and achieve a stationary process. But we know that this is a wrong twist. – Cowboy Trader Oct 16 '15 at 14:52
• @Aksakal You are not reading what I am writing correctly. I don't claim that random walk is stationary. I said you cannot claim that a process is stationary because a modified version of it is stationary. Random walk is non-stationary because its unconditional variance is growing, however if we follow your logic of conditioning it has constant conditional variance. In general you are wrong in the definition of WSS. – Cowboy Trader Oct 16 '15 at 15:13
• @Aksakal Nowhere in your answer do you mention trend stationarity or covariance stationarity or any other form of stationarity (weak sense, to order $n$, ot whatever. If $t$ is assumed to be an integer, then $\sin(0), \sin(1), \sin(2), \sin(3),\cdots$ is not even a periodic sequence (in the sense that $x_{n+N} = x_n$ for all integer $n$) or an ultimately periodic sequence either. Thus, your example, acceptable though it is to the OP, is not by any means an answer to the question asked. I am downvoting your answer: it in not just "not useful", it is downright misleading. Poor OP. – Dilip Sarwate Oct 16 '15 at 18:48

A seasonal pattern that remains stable over time does not make the series non-stationary. A non-stable seasonal pattern, for example a seasonal random walk, will make the data non-stationary.

A stable seasonal pattern is not stationary in the sense that the mean of the series will vary across seasons and, hence, depends on time; but it is stationary in the sense that we can expect the same mean for the same month in different years.

A stable seasonal pattern may therefore fit in the concept of a cyclostationary process, i.e., a process with a periodic mean and a periodic autocorrelation function.

The above does not apply to a non-stable seasonal pattern.

• +1 for bringing up the concept of cyclostationary processes. – Dilip Sarwate Oct 17 '15 at 13:58

IMHO, persistent seasonality, by definition, is a type of non-stationarity: the mean of a seasonal process varies with the season, E[z(t*s+j)] = f(j), where s is the number of seasons, j is a particular season (j=1,...,s), and t is specific period (typically a year). Thus, E[y(t)] = E[sin(t)+u(t)] = sin(t) is not a stable mean, although it is deterministic: you could group observations with different means.

Luis

• +1 I agree with your statement that seasonality is a type of non-stationarity. – Dilip Sarwate Oct 16 '15 at 13:11