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For a weakly stationary process with mean $m$ and nonzero variance, put $\gamma (h)=\mathrm{Cov}(X_{t+h},X_t)$ and $\rho (h)=\gamma (h)/\gamma (0)$. The function $\rho$ is its autocorrelation function. For observations $x_1,\dots ,x_N$, one common sample autocorrelation function isA correlogram plots these estimates against lag; the population analogue plots $\rho (h)$.

The lag-$h$ partial autocorrelation function measures the correlation between $X_t$ and $X_{t-h}$ after removing their linear projections on the intervening observations. Its sample counterpart can be calculated by the Yule-Walker equations: solveand take $\hat{\alpha }(h)={\hat{\varphi }}_{h,h}$, assuming the matrix is nonsingular. This defines a standard sample partial autocorrelation function.

For a minimal causal autoregressive model of order $p$, the autocorrelation typically decays geometrically, possibly with damped oscillations, whereas the partial autocorrelation function is zero beyond lag $p$: after those $p$ preceding values have been projected out, the innovation is orthogonal to older values. For a moving-average model of order $q$, nonoverlapping sets of driving white noise give $\gamma (h)=0$ for $|h|>q$; its partial autocorrelation function generally tails off when the representation is invertible. Thus an ACF cutoff suggests an MA order; a PACF cutoff suggests an AR order. An ARMA model generally has neither cutoff. These are population properties; sample noise makes cutoffs approximate, so fitted residuals and uncertainty should also be inspected. Under independent white noise, fixed-lag sample correlations have approximate standard error $N^{-1/2}$, giving a rough diagnostic band rather than a universal simultaneous test.

For a series with constant mean $\mu$, write $X_t=x_t-\mu$. An autoregressive moving-average process has the formwhere $w_t$ is zero-mean white noise with variance ${\sigma }_w^2>0$. The backshift operator is $B{X}_t=X_{t-1}$. Define the autoregressive polynomial and the moving-average polynomial byThe autoregressive operator and moving-average operator give the concise equationOrders are normally specified minimally, with nonzero last coefficients and no common polynomial factor. A displayed unreduced equation can have larger orders than the process actually needs.

Causality means that the stationary process can be recovered from present and past noise:Invertibility means that the noise can be recovered from present and past observations:The causality and invertibility root criteria for an ARMA model, for the reduced representation, areIndeed the transfer series are $\sum {\psi }_j{z}^j=\Theta (z)/\Phi (z)$ and $\sum {\pi }_j{z}^j=\Phi (z)/\Theta (z)$. When all denominator roots lie outside the closed unit disc, each rational function is analytic on a larger disc, so its Taylor coefficients decay geometrically and are absolutely summable. Conversely, a genuine denominator zero in the closed unit disc gives a pole incompatible with such a convergent filter. The reduced-representation proviso prevents a cancelled root from being mistaken for an obstruction.

For order identification by autocorrelation cutoffs, first estimate the sample autocorrelation function and sample partial autocorrelation function from a suitably stationary, centred series. The autocorrelation function is $\rho (h)=\mathrm{Cov}(X_t,X_{t-h})/\mathrm{Var}(X_t)$. The lag-$h$ partial autocorrelation function is the last coefficient in the best linear regression of $X_t$ on $X_{t-1},\dots ,X_{t-h}$, equivalently the correlation left after removing the intervening lags.

For a pure moving-average model of order $q$, observations more than $q$ lags apart share no noise, so the population ACF is zero beyond $q$, while the PACF usually tails off. For a pure autoregressive model of order $p$, the PACF is zero beyond $p$, while the ACF usually decays, possibly with damped oscillation. Thus a clear sample ACF cutoff suggests an MA order and a clear PACF cutoff suggests an AR order. For a mixed ARMA process both usually tail off, so these plots guide a small set of candidate orders rather than determine $p,q$ uniquely. Sampling variation blurs cutoffs; compare fitted candidates and check that their residual ACF resembles white noise. Common-factor cancellation can also disguise the orders in a displayed equation.

Use the sample autocorrelation function and sample partial autocorrelation function to look for approximate cutoffs. For a minimal causal autoregressive model of order $p$, the partial autocorrelation function is zero beyond $p$, while the autocorrelation function usually tails off, possibly with damped oscillation. For an invertible moving-average model of order $q$, the autocorrelation function is zero beyond $q$, while the partial autocorrelation function tails off. In a mixed autoregressive moving-average model, both generally tail off.

Thus the last visibly nonzero partial correlation suggests an autoregressive order, and the last visibly nonzero autocorrelation suggests a moving-average order. Finite samples do not produce exact zeros. Approximate white-noise reference bands can help flag clearly nonzero lags, but they are not universal confidence intervals for arbitrary correlated processes. Fit a few suggested orders and check residual autocorrelation, rather than treating these plots as a proof of the model order. This is order identification by autocorrelation cutoffs.