Autoregressive Model

An Autoregressive Model (AR Model) is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. In this model, the output variable relies on its own previous values and a stochastic term. It is usually employed for forecasting in time series analysis, as it takes a specific period of historical data and uses it to predict future values.

How Autoregressive Model works

An autoregressive model works by using a linear combination of past values. The idea is that past values have an effect on present values. For instance, the value of a stock, a product's sales, or the weather conditions of a particular location can be influenced by their respective past values.

In an autoregressive model, the forecasted output equals a constant plus a weighted sum of previous values. If we denote the output at time 't' as 'y(t)', then the equation for an AR model of order 'p' can be expressed as:

y(t) = c + φ1y(t-1) + φ2y(t-2) + ... + φp*y(t-p) + e(t)

This is read as: the value at time 't' equals a constant 'c' plus the sum of the effects of the 'p' previous values multiplied by their respective weights. 'e(t)' is the error term, representing the difference between the actual value and the predicted value at time 't'.

The weights (φ1, φ2,..., φp) are parameters of the model that are typically estimated from the data. These weights show how much of an impact the previous values have on the current value. An important aspect of this model is that it is a stochastic process, meaning it includes a random error term. This term represents the unpredictable part of the process that can't be estimated from past values.

In summary, an autoregressive model operates under the assumption that the current value of the time series can be predicted from its own past values.