Autocorrelation is the correlation between a time series and a lagged version of itself. It measures whether today's value tends to move together with a past value. Because observations are ordered in time, neighbouring values are often dependent rather than independent.
Autocorrelation is the correlation between a time series and a lagged version of itself. It measures whether today's value tends to move together with a past value. Because observations are ordered in time, neighbouring values are often dependent rather than independent.
If this month's sales are high, next month's sales are also often fairly high. That lag-1 pattern is a form of autocorrelation.
A lag is the number of time steps between two observations being compared.
For monthly data, lag 1 is last month and lag 12 is the same month last year.
Positive autocorrelation means a high value tends to be followed by another high value (and a low by a low). Negative autocorrelation means a high value tends to be followed by a low value, or the reverse.
Slow decay of autocorrelation across many lags often suggests persistence or a wandering level. Near-zero autocorrelation after lag 0 is closer to unstructured noise. These are guides for inspection, not automatic model names.
Partial autocorrelation at lag k is the correlation between Yt and Yt−k after the linear effect of the intermediate lags 1, 2, …, k−1 has been accounted for. It asks: is there a direct link at lag k, once shorter lags are controlled for?
Sales may correlate with lag 2 partly because both lag 2 and today are linked through lag 1. Partial autocorrelation at lag 2 removes that shorter-lag path and looks at what remains.
| Concept | Meaning | What it helps identify |
|---|---|---|
| Autocorrelation | Correlation of the series with its lagged values, including indirect paths through shorter lags | Overall dependence on the past; persistence; possible seasonal lag spikes |
| Partial Autocorrelation | Correlation at a chosen lag after accounting for intermediate lags | More direct lag-k dependence; often used when thinking about autoregressive order |
The ACF plot shows sample autocorrelation at lags 1, 2, 3, … . The PACF plot shows sample partial autocorrelation at those lags. Bars that stand well away from zero (relative to a simple confidence band on the plot) suggest that the lag is worth noticing. Finite samples are noisy, so one extra bar should not be over-interpreted.
At an introductory level:
Teaching rules of thumb (for later AR/MA discussion) say that a pure autoregressive pattern often shows ACF tailing off and PACF cutting off, while a pure moving-average pattern often shows the reverse. Mixed series and short samples do not follow textbooks perfectly.
ACF and PACF are identification tools. They do not automatically determine the final ARMA or ARIMA model. Always read them with a time plot and, after fitting, with residual checks.