Help understanding Autocorrelation and Partial Autorcorrelation (PACF)

Question

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?

Answer 1

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.