I started to study Time Series Analysis and have stumbled on a roadblock.

When introducing the autocovariance function, the instructor mentions that we assume stationarity in the data that we are working with. That's fine; this should mean

  1. No systematic changes in either mean or variance
  2. No seasonality

But suppose if the autocovariance of $X_t$ and $X_{t-3}$ is pretty high, then that means that the $t$th data point depends strongly on the $(t-3)$th data point. MY QUESTION: shouldn't this set up some kind of seasonality in the data, thus contradicting our assumption of stationarity?

I'll explain: Say the autocovariance of $X_t$ and $X_{t-3}$ is pretty high. Then, taking a time series, we will have the following dependences: say the value for December is high. Then so will be values for September, June and March. Similarly, suppose that values for January are low: so will be those for October, July, and April. Thus when we plot the data out, we should see a clear increase from Jan $\rightarrow$ March, then a drop to April, then increases and subsequently drops in the regions April $\rightarrow$ June, July $\rightarrow$ September and October $\rightarrow$ December. Isn't this a seasonal pattern? Thus, if our data is stationary, should the autocovariance function always be zero?


1 Answer 1


In theory to properly model time series $x_1,x_2,\dots,x_T$ you need to estimate the joint distribution $F(x_1,x_2,\dots,x_T)$. Note, that each observation $x_t$ is a sample of the random variable $X_t$. So, we're dealing with a single sample from a set of $T$ random variables. This is an impossible construct to deal with.

Hence, we hope that in fact we don't need this joint distribution, and instead we want to work with auto covariance function (ACF) $\gamma_h$ where $\gamma_h=cov[x_t,x_{t+h}]$. Note, that it doesn't depend on $t$ anymore. That's the kind of stationarity we're talking about, that auto covariance function exists and depends only on the distance in time between observations, but not the time itself.

Full joint distribution $F$ is impossible to estimate, because we literally have one sample from it. In contrast, ACF can be estimated at least partially, because we can treat $x_1, x_2$ and $x_2,x_3$ as two samples to estimates $\gamma_1$. That's why we like ACF and want it to exist.

  • $\begingroup$ That, while insightful and helpful to me (as a beginner in TSA), doesn't really answer my question. Doesn't a high autocovariance function $\gamma _k$ for some $k$ imply some sort of seasonality in the data? $\endgroup$ Oct 28, 2023 at 5:21
  • $\begingroup$ @insipidintegrator, we look at the shape of the ACF for clues about the lag structure of the process. for instance a spike at monthly lag 4 may indicate a quarterly seasonality etc. ACF by itself doesn't conclusively prove much. $\endgroup$
    – Aksakal
    Oct 28, 2023 at 19:54
  • $\begingroup$ Hi, that is $exactly$ my point. A spike would imply seasonality, but our data is stationary. Isn't that contradictory? $\endgroup$ Oct 29, 2023 at 5:39
  • $\begingroup$ @insipidintegrator, thus my answer. the stationarity is about the lack of $t$ in ACF $\gamma_h$ $\endgroup$
    – Aksakal
    Oct 29, 2023 at 21:31
  • 1
    $\begingroup$ @insipidintegrator Hamilton is a standard tex, not sure if I'd say it's good $\endgroup$
    – Aksakal
    Nov 7, 2023 at 3:44

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