# Autoregressive Models

In time series analysis, Autoregressive (AR) Models are models that predict the next time-step of a quantity by utilizing the previous time-step values (lags) of that same quantity.

We can build an AR model by making use of PACF, analysis, determining which time lags have the strongest correlation for our signal.

Having observed the most strongly correlated lags, we can then write a simple function where the parameters are the lags, e.g.

\[ x_t = \beta_0 + \beta_1 x_{t-1} + \beta_3 x_{t-3} + \beta_4 x_{t-4} + \beta_12 x_{t-12} + \epsilon_t \]

For a monthly time series that is sensitive to monthly, quarterly, four-monthly and yearly effects. We are then left with the tasks of fitting the weights (\(\beta\)).

Note, an autoregressive model can only be used on stationary time series.