How should my auto-correlation plot look like for a mean reverting process? From what I have recently learned, auto-correlation should be low and should decay fast enough. But when I run the following command in R

`acf(sin(1:100)) #mean reverting process `

I see that auto-correlation is very different from zero and intuitively makes a lot of sense, when values are high it becomes negative and when values are low it becomes positive. So I'm confused which one is correct: zero auto-correlation or similar profile as above.

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#### Best Answer

First, $y(t) = sin(t)$ is not a mean reverting process. It *appears* as mean reverting process at discrete sampling $t_i=i$, but it is not. Think of this $E[y(t)]=sin(t)$, i.e. your mean is not reverting anywhere it is exactly $sin(t)$ and it's time varying.

You get mean reversion when there are random shocks, and *something* pulls your process back to the *mean*.

For instance, consider this process: $$y(t)=y(t-Delta t)+thetacdotleft(sin(t)-y(t-Delta t)right)+varepsilon(t)$$ When $y(t-Delta t)$ was below its *time varying mean* $sin(t)$ something pushes it upwards at speed $theta$. In other words, $y(t)$ is reverting to its mean.

Secondly, mean reverting processes are often modeled as ARMA(p,q) processes. AR(p) process will shows exponential decay in modulus of ACF. One could interpret your ACF as such, where the decay is very slow (in absolute value of lag coefficients). The partial ACF or PACF is cut off at lag p for AR(p) process.

MA(q) process is exactly the opposite: PACF exponentially decays, while ACF cuts of at q lags.

ARMA(p,q) has ACF and PACF oscillating. You ACF looks not unlike such.

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