Autocorrelation (ACF) measures how a time series relates to a copy of itself shifted by a chosen number of time steps. Partial autocorrelation (PACF) measures the lag-specific relationship left after accounting for the shorter lags. Their plots can help suggest autoregressive (AR) and moving-average (MA) model orders, but they are diagnostic clues—not automatic model selectors.
What does autocorrelation measure?
A lag is the number of time steps between observations. At lag 1, for example, the ACF compares each value with the value one time step earlier; at lag 2, it compares values two steps apart. This interpretation assumes equally spaced observations.
The autocorrelation function (ACF) summarizes how strongly values at each lag move together. A positive autocorrelation means values separated by that lag tend to be similar in direction; a negative value means they tend to move in opposite directions. The sample ACF is estimated from observed data by comparing deviations from the series’ sample mean across each lag, using a normalized sum of products (NIST’s description of autocorrelation).
What does partial autocorrelation add?
The partial autocorrelation function (PACF) asks whether there is a distinct relationship between values k steps apart once the intervening lags 1 through k−1 have been accounted for. NIST defines it as “the autocorrelation between X_t and X_{t-k} that is not accounted for by lags 1 through k-1” (NIST’s partial autocorrelation explanation).
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For an intuitive example, suppose today’s value resembles yesterday’s. The lag-1 ACF may be positive. If today also resembles the value from two days ago, that resemblance could arise because yesterday links the two observations. The lag-2 PACF asks whether a separate lag-2 association remains after accounting for lag 1. This is an illustration of the definitions, not a result from a measured dataset.
ACF vs. PACF: which should you use?
| Function | Question it answers | Useful textbook clue |
|---|---|---|
| ACF | How correlated are observations separated by each lag? | In a simple moving-average MA(q) process, the theoretical ACF cuts off beyond lag q. |
| PACF | What association remains at a lag after accounting for shorter lags? | In a simple autoregressive AR(p) process, the theoretical PACF becomes zero beyond lag p. |
These patterns make ACF especially informative when proposing an MA order and PACF especially informative when proposing an AR order. They describe theoretical behavior in simple model families; real data may not display a clean cutoff. For mixed or less straightforward patterns, the plots may not identify a model clearly (NIST on PACF use; NIST on model identification).
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Look for a pattern, not a single decisive bar
Each bar is an estimate from a finite sample. Sampling variation can produce visible spikes or obscure a theoretical cutoff. NIST cautions that sample autocorrelation and partial autocorrelation functions are random variables and may not reproduce their theoretical patterns (NIST’s model-identification guidance).
Treat confidence bands as approximate
NIST gives an approximate 95% PACF interval of ±2/√N, where N is the sample size. This is a commonly used approximation, not a universal pass/fail threshold. A bar outside a band is not, by itself, proof that a lag belongs in the final model.
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ACF/PACF intervals and standard-error calculations depend on assumptions and methods. The statsmodels ACF documentation describes confidence intervals and standard-error considerations; its PACF API offers several estimators, including Yule-Walker, OLS, Levinson-Durbin, and Burg. For a reproducible analysis, record the software version, estimator, number of lags, and interval settings.
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How to use ACF and PACF in model identification
- Inspect both plots. Use the ACF to notice overall lagged dependence and possible MA behavior; use the PACF to look for remaining lag-specific relationships and possible AR behavior.
- Propose candidate orders. Treat apparent cutoffs and spikes as hypotheses, especially when the pattern resembles a simple AR or MA case.
- Fit candidate models and compare them. ACF/PACF plots narrow the possibilities; they do not settle mixed or ambiguous cases. Information criteria such as AIC can also help compare candidate models.
- Check residuals and diagnostics. Assess whether the fitted model leaves meaningful structure unexplained rather than accepting an order solely because a plot looked tidy.
- Document the choices. Record estimator, lag count, confidence-interval method, and software version so another analyst can reproduce the plots.
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