Autocorrelation ($\rho (h)$)

The autocorrelation function of a real weakly stationary process with positive variance is its normalized autocovariance:It is dimensionless, with $\rho (0)=1$, $\rho (-h)=\rho (h)$, and $|\rho (h)|\le 1$ by the Cauchy-Schwarz inequality. It is unchanged by replacing $X_t$ with $a{X}_t+b$ for fixed $a\ne 0$. This makes autocorrelation useful when comparing records measured on different scales. The ACF is a property of the model; the sample autocorrelation function estimates it and will generally not reproduce theoretical zeros exactly.

An AR(1) with coefficient $\varphi =0.7$ has autocorrelations $0.7,0.49,0.343$ at lags one through three. With $\varphi =-0.7$ these become $-0.7,0.49,-0.343$: the same diminishing magnitude, alternating in sign. An MA(1) with $\theta =0.7$ instead has lag-one autocorrelation $0.7/1.49=70/149\simeq 0.4698$ and zero at every higher lag. Equal numerical coefficients in an autoregressive model and a moving-average model do not mean equal dependence. See Bartlett, lecture 3: theoretical ACFs and prediction.

The bars show exact model autocorrelations, not estimates from the illustrated sample paths in time series. The AR(1) retains a diminishing trace of a disturbance at arbitrarily large lags; the MA(1) stops sharing driving noise after lag one. A zero autocorrelation means uncorrelated random variables, and does not generally imply independent random variables. For a Gaussian process it does imply independence of the two coordinates.

There is a direct prediction interpretation. Among affine predictions based on just $X_t$, the best linear predictor of $X_{t+h}$ is $\mu +\rho (h)(X_t-\mu )$, with mean squared prediction error $\gamma (0)[1-\rho (h{)}^2]$. Thus negative autocorrelation is also useful predictive information. A prediction using the whole observed past can do better; best linear prediction from a finite past describes that problem. Interpreting the affine prediction as a conditional expectation requires additional assumptions, such as a joint normal distribution.

A correlation time measures how rapidly a stationary stochastic process loses temporal dependence. Specify whether it means an exponential decay time, an integral of the normalized autocorrelation, or another width. For $R(t)=R(0)e^{-|t|/\tau }$, the exponential time and the one-sided integral ${\int }_0^{\infty }R(t)dt/R(0)$ both equal $\tau$.

The partial autocorrelation at lag $k$ is the correlation of $X_t$ and $X_{t-k}$ after their linear projections onto the intervening observations are removed. A causal autoregressive model of order $p$ has zero partial correlations after lag $p$. This complements the finite cutoff of the autocorrelation function for a moving-average model.

Replace the autocovariance in the finite-lag linear prediction equations by the sample autocovariance function. The last regression coefficient at order $k$ estimates the lag-$k$ partial autocorrelation function.

The sample autocorrelation function replaces the mean and lagged covariance in $\rho (h)$ by their empirical counterparts.

A correlogram plots the estimated autocorrelation function against lag. The plot helps distinguish the finite autocorrelation cutoff of a moving-average model from the decaying dependence of a causal autoregressive model. Estimated cutoffs are diagnostics, not exact finite-sample identities.