Auto-Regressive (AR) models for time-series prediction
A popular model for the prediction of time series is based on the so-called auto-regressive model
where 
‘s are constant coefficients, and 
is the ‘‘memory length’’ of the model. The interpretation of the model is that the next output is a linear function of the past. Elaborate variants of auto-regressive models are widely used for the prediction of time series arising in finance and economics.
To find the coefficient vector theta in 
, we collect observations 
(with 
) of the time series, and try to minimize the total squared error in the above equation:
![\[\min _\theta: \sum_{t=m}^T\left(y_t-\theta_1 y_{t-1}-\ldots-\theta_m y_{t-m}\right)^2 .\]](https://ecampusontario.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-bb2eb731b9ccf2c2298339bbdea50f85_l3.png "Rendered by QuickLaTeX.com")
This can be expressed as a linear least-squares problem, with appropriate data 
.
See also: Linear regression via Least-Squares.