Autoregressive Processes (AR)

Autoregression (AR) is probably one of the most intuitive ways to think about recurrence in univariate time series. We’ll define AR as the method to predict future values of a time series, $y_t$, using its past realizations. In this lesson, we will study the most famous type of AR model and the keystone of all the rest: the AR($p$).

Definition

The AR($p$) is a linear model that assumes that $y_t$ is a weighted sum of its past values. Perhaps not surprisingly, the acronym AR stands for autoregressive. We can express it mathematically as:

Or, to put it more succinctly:

In the equation above, $p$ is the order of the AR. It denotes the number of lags of $y_t$ that make up the model. Or, in other words, how many past realizations of $y_t$ have an impact on its present value. Next, $\phi_i$ is the coefficient of $y_{t-i}$, and it determines the weight that $y_{t-i}$ ...