AR ACF/PACF Signature

What happens to the ACF/PACF of an AR(\(p\)) process after \(p\) lags?

Answer

For an AR(\(p\)) process, the ACF tails off and PACF cuts off after \(p\) lags, i.e.

\[ \phi_{kk} = 0 \qquad \text{for all } k > p, \]

where \(\phi_{kk}\) denotes the PACF at lag \(k\).

Intuition: For any lag \(k>p\), for example \(k=p+1\), the lagged value \(X_{t-(p+1)}\) may still be correlated with \(X_t\), but only indirectly through the intermediate lags.

Once we control for the intermediate lags \(X_{t-1},\dots,X_{t-p}\), the older lag \(X_{t-(p+1)}\) contains no additional (direct) information about \(X_t\). Therefore, the PACF is zero for all \(k>p\).

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