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I understand the layman concept of PACF - it is the correlation with the linear dependence of the earlier lags removed.

However, I am confused as to how this relates to Autocorrelation. Consider ACF(2) which is equal to corr(Xt, Xt-2).

Where and how is the correlation between the earlier lags included in the computation of this value? How does Xt-1 come into the picture? I have read explanations involving vectors and residuals but it’s still difficult to comprehend. Could someone explain this more simply?

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  • $\begingroup$ AFAIK, ACF(2) will (in a rough sense) "include" the effect of $X_{t-1}$, if $X_{t-2}$ influences $X_{t-1}$ and $X_{t-1}$ influences ${X_t}$. However, PACF(2) measures the relationship between $X_t$ and $X_{t-2}$ after the effect of $X_{t-1}$ on $X_t$ has been "filtered out". $\endgroup$ Jun 16 '20 at 9:35
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Consider an auto-regressive model such that

$$y_t = \alpha y_{t-1} + \epsilon_t,\qquad \epsilon_t \sim \mathcal{N}(0,\sigma^2)$$

You can rewrite it as

$$y_t = \alpha(\underbrace{\alpha y_{t-2}+\epsilon_{t-1}}_{y_{t-1}}) + \epsilon_t$$

So you can see that $y_t$ and $y_{t-2}$ are "linked" through $y_{t-1}$.

If you compute it:

$Corr(y_t,y_{t-1})=Corr(\alpha y_{t-1} + \epsilon_t,y_{t-1}) =\alpha$

$Corr(y_t,y_{t-2})=Corr(\alpha^2 y_{t-2} + \alpha \epsilon_{t-1}+ \epsilon_t,y_{t-2}) =\alpha^2$

Assume $|\alpha|<1$. You can see that the correlation between the two lags is mitigated by the lag in between.

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  • $\begingroup$ Does the trend continue for AR models? Eg. corr(yt, yt-3) = alpha^3.... $\endgroup$
    – WK Wong
    Jun 17 '20 at 19:36
  • $\begingroup$ Yes. Follow the same logic and you will see it. $\endgroup$
    – Ale
    Jun 18 '20 at 8:30

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