AR models are a very useful tool to tackle the prediction problem in relation to a time series. A strong correlation between consecutive values of a series is often observed. In this case, we speak of autocorrelation of the first order when we consider adjacent values, of the second order if we refer to the relation between the values of the series after two periods, and in general of the p^th order if the values considered have p periods between them. AR models allow exploiting these bonds to obtain useful forecasts of the future behavior of the series.

AR is a linear predictive modeling technique. This model tries to predict the time series based on the previous values assumed using the AR parameters as coefficients. The number of samples used for the forecast determines the order of the model (p). As the name indicates, it is a regression of the variable against itself; that is, a linear combination of past values of the variables is used to forecast the future value. The AR model of p order is defined as:

In the previous formula, the terms are defined as follows:

• Y_t is the actual value at time period t
• c is a constant
• ϕ_i (i = 1,2,..., p) are model parameters
• Y_t-i is the past value at time period t-i
• ε_t is the random error at time period t (white noise)

It may happen that the constant term is omitted; this is done to make the model as simple as possible.