Sample ACF and Sample PACF#

Stage 4 · 🎯 Prediction & the Sample ACF / PACF · Lesson 10 of 18 · intermediate

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From process to sample#

In practice you never see the true ACF — you estimate it from one finite record. The sample autocovariance and sample ACF are

\[\hat{\gamma}(h) = \frac{1}{n}\sum_{t=1}^{n-|h|} (x_t - \bar{x})(x_{t+|h|} - \bar{x}), \qquad \hat{\rho}(h) = \frac{\hat{\gamma}(h)}{\hat{\gamma}(0)}.\]

Both grow noisier at large lags, where few pairs contribute — a common rule keeps \(n \ge 50\) and lags \(h \le n/4\).

Significance bands#

How large is “large”? If the series were white noise, then for big \(n\) each \(\hat{\rho}(h)\) is approximately \(\mathcal{N}(0, 1/n)\), so correlations beyond

\[\pm \frac{1.96}{\sqrt{n}}\]

are unlikely by chance — the dashed bands on every correlogram. For a genuinely correlated series the bands widen, following Bartlett’s formula \(\operatorname{Var}(\hat{\rho}_k) \approx \frac{1}{n}\big(1 + 2\sum_{j<k}\rho_j^2\big)\).

The sample PACF#

The sample PACF applies the Durbin–Levinson recursion to the sample autocovariances, producing \(\hat{\phi}_{hh}\) at each lag. It shares the same white-noise bands \(\pm 1.96/\sqrt{n}\), so significant spikes stand out in the same way.

Reading them together#

Identification is a two-plot habit. An ACF that cuts off after lag \(q\) alongside a PACF that tails off suggests MA(q); a PACF that cuts off after lag \(p\) with a tailing ACF suggests AR(p); both tailing off suggests ARMA. In statsmodels these plots are plot_acf and plot_pacf.

Hint