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I am new to Time Series Analysis. Say, we have a time series $(y_{t})_{t}$ that we want to filter with a moving average filter. I have been told that we should choose the window size $L$ of the filter to be uneven. Apparently, if it were even, this would introduce another cyclical component into the data. My question is: why? Why would an even window size introduce a cyclical component, while an uneven window size would not?

There seems to be no mathematical justification for this...

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  • $\begingroup$ This is not a direct answer, but even-ness/odd-ness can easily create cycles in pretty trivial mathematical models. For example $y_t = \alpha + -1^{i+1} \rho y_{t-1} + \varepsilon_{t}$, where $i$ is an integer creates cycles when $i$ is even (and $|\rho|$ is far enough from zero that you can see equilibration occuring after a perturbation), and no cycles when $i$ is odd. $\endgroup$
    – Alexis
    Commented Jun 30, 2020 at 19:44

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