How to Identify AR, MA, and ARMA Models Using ACF and PACF

AR(p) Process

Identifying a random time‑series model means finding a suitable stochastic process for a stationary series, deciding whether it follows a pure AR, pure MA, or ARMA model using the autocorrelation function (ACF) and the partial autocorrelation function (PACF).

Autocorrelation Function

AR(1) – The lag‑k autocovariance decays exponentially; the ACF shows a tail that approaches zero, called infinite memory. When the coefficient is negative, the decay is oscillatory.

AR(2) – The variance and lag‑1, lag‑2 autocovariances are derived similarly; the ACF again tails to zero, with the shape (monotonic or oscillatory) determined by the nature of the characteristic roots.

General AR(p) – The lag‑k autocovariance can be expressed recursively, leading to an ACF that always tails (never cuts off) as long as the AR polynomial is stable.

Partial Autocorrelation Function

The ACF measures total correlation, which may mask indirect relationships. The PACF removes the influence of intermediate lags, revealing the direct correlation between a variable and its lag after conditioning on all intermediate values. For a stable AR(p) process, the PACF cuts off after lag p, while the ACF tails.

MA(q) Process

For an MA(1) process the autocovariances are easy to compute, and the ACF cuts off after lag 1, indicating no correlation beyond that lag. The process can be written as a linear combination of an infinite past of white‑noise terms; its PACF does not cut off but decays toward zero.

The invertibility condition requires the MA polynomial roots to lie outside the unit circle; otherwise the influence of distant lags would increase, which is unrealistic.

In general, an MA(q) process has autocovariances that vanish for lags greater than q, producing a truncated ACF. The PACF of an MA(q) is non‑truncated but approaches zero.

Identification rule: if the sample ACF cuts off after lag q while the PACF tails, the series is likely MA(q). In practice, sample ACF estimates fluctuate around zero; statistical tests based on asymptotic normality can be used to assess truncation.

ARMA(p,q) Process

The ACF of an ARMA(p,q) process is a mixture of the ACFs of an AR(p) and an MA(q) component. When the MA part is zero, the ACF tails; when the AR part is zero, the ACF cuts off. The PACF shows a few significant spikes up to lag p and then decays, while the ACF shows spikes up to lag q before decaying.

Typical identification pattern:

ACF tails, PACF cuts off after lag p → AR(p) model.

ACF cuts off after lag q, PACF tails → MA(q) model.

Both ACF and PACF tail → ARMA(p,q) model.

Examples (illustrated below) demonstrate these patterns.