Autoregressive models predict a series from its own recent past, which is where much of its information lies.
They formalise the idea that today's value is a guide to tomorrow's.
They are a building block of many larger forecasting models.
An AR(1) model says the next value equals a constant plus a weight times the current value, plus noise.
A weight between minus 1 and 1 gives a stationary series that keeps returning to its average.
A weight of 1 gives a random walk, which never settles.
Higher orders, such as AR(3), use more past values and can capture more complex patterns.
The PACF plot suggests the order, since its bars tend to cut off sharply after the right number of lags.
The weights are estimated from the data, usually by regressing the series on its own lags.
Forecasts more than a few steps ahead drift back towards the series average.
Extra columns, such as price or holidays, can be added alongside the lags.
AR models assume the relationship between lags stays constant, which breaks after structural changes.
Including too many lags fits noise and makes forecasts worse.
In practice, comparing AR models of several orders on past data picks a sensible number of lags.
The usual mistake is fitting an AR model to a trending series without making it stationary first.