An AR model is a part of the stochastic process, wherein specific lagged values of yt are used as predictor variables and regressed on yt in order to estimate its values. Lagged values are values of the series of the previous period that tend to have an impact on the current value of the series. Let's look at an example. Say we have to assess and predict tomorrow's weather. We would start by thinking of what today's weather is and what yesterday's weather was, as this will help us in predicting whether it will be rainy, bright and sunny, or cloudy. Subconsciously, we are also cognizant of the fact that the weather of the previous day might have an association with today's weather. This is what we call an AR model.
This has a degree of uncertainty that results in less accuracy in the prediction of future values. The formula is the same as the formula for a series with p lag, as follows:
In the previous equation, ω is the white noise term and α is the coefficient, which can't be zero. The aggregated equation appears as follows:
Occasionally, we might talk about the order of a model. For example, we might describe an AR model as being of order p. In this case, the p represents the number of lagged variables used within the model. For example, an AR(2) model or second-order AR model looks like the following:
