Autocorrelation (ACF) and partial autocorrelation (PACF)

The correlation of a series with its own past at each lag (ACF), and the same after removing the effect of shorter lags (PACF), used to detect dependence, seasonality and AR order.

r_k = \frac{c_k}{c_0}, \qquad c_k = \frac{1}{n}\sum_{t=1}^{n-k}(y_t - \bar y)(y_{t+k} - \bar y)

Sample autocorrelation at lag k (biased autocovariance, divisor n)

$k$: lag (number of periods apart)
$n$: length of the series
$\bar y$: mean of the whole series

\pm \frac{z_{0.975}}{\sqrt{n}} \approx \pm\frac{1.96}{\sqrt{n}}

Approximate 95% limits for white noise

$n$: length of the series

When to use

Checking whether a series (or a model's residuals) is white noise; spotting seasonality (spikes at lags $m, 2m, \ldots$ ); choosing an AR order.
The PACF at lag $k$ is the last coefficient $\phi_{kk}$ of an AR( $k$ ) fitted to the series (computed by the Durbin–Levinson recursion on the ACF).

Assumptions

Equally spaced observations with no missing values; the series should be roughly stationary (trend removed or differenced) for the ACF to be interpretable.
The ±1.96/√n bands assume white noise and apply lag by lag; with many lags, about 1 in 20 will cross by chance.

How to interpret

Slowly decaying ACF: trend or non-stationarity — difference the series.
AR(p): PACF cuts off after lag $p$ , ACF decays gradually.
For residuals, a joint check such as the Ljung–Box test $Q = n(n+2)\sum_{k=1}^{h} r_k^2/(n-k)$ complements the plot.

Common mistakes

Interpreting a single bar that just crosses the band as a real pattern.
Reading the ACF of a trending series as "strong dependence" rather than non-stationarity.
Expecting reliable ACFs from very short series (the bands are wide).

Example

Series 3, 5, 4, 6, 5, 7, 6, 8 ( $n = 8$ ): $r_1 = 0.125$ , $r_2 = 0.472$ , $r_3 = -0.236$ ; PACF at lag 2 = 0.464. The white-noise band is ±1.96/√8 = ±0.693, so no lag is clearly significant — eight points are too few to say much.