Why does the ACF of an AR(1) contains sometimes a sinusoid-like pattern? and what does it mean?

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I think the time series is fit to AR(1). As I understand it, in an AR model, the value of x at time t is a linear function of the value of x at time t–1.

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If wt is random, then we see a random Figure in correlogram. If not, we can see a pattern in correlogram, is it correct? If yes why do we see here a sinusoid pattern? In this case has the wt (Residual) a constant value?

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    $\begingroup$ "Sine" and "sinusoid" are the usual English words. It implies periodicity. In your case, the peaks at 9, 18, 27, 36, ... imply a period of 9. What does AR(1) have to do with your question? $\endgroup$ – Nick Cox Dec 11 '13 at 10:01
  • $\begingroup$ I have edited my post $\endgroup$ – TangoStar Dec 11 '13 at 10:18
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    $\begingroup$ Presumably wt is just the name of your variable in whatever software you are using. You call it a residual, but don't explain what the model is. I am not a time series expert, but my understanding is that AR(1) itself won't show periodicity in the acf, but just a decline with lag. No autocorrelation like that is consistent with a constant value. $\endgroup$ – Nick Cox Dec 11 '13 at 10:27
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    $\begingroup$ If $\phi_1$ is negative you can get an alternating and damping pattern, see this page but that's not sinusoidal. I am also not a time-series expert but I think you'd need a more complex model to get the pattern you show. $\endgroup$ – Peter Flom - Reinstate Monica Dec 11 '13 at 11:42
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    $\begingroup$ It's important to distinguish between the empirical ACF, especially from a short series (you have what? 65, 70 observations?) & the theoretical one - the expectation. You certainly can get wavy patterns from an AR(1) process, but the expected ACF exponentially decays. In any case the strength & persistence of the pattern in your plot suggests a periodic (seasonal) effect inconsistent with an AR(1) process. $\endgroup$ – Scortchi - Reinstate Monica Dec 11 '13 at 20:34

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