101 Concepts for the Level I Exam

Essential Concept 7: Autoregressive (AR) Models

An autoregressive time series model is a linear model that predicts its current value using its most recent past value as the independent variable. An AR model of order p, denoted by AR(p) uses p lags of a time series to predict its current value.

$\mathrm{x}}_{\mathrm{t}~}\mathrm{=}~{\mathrm{b}}_{0~}\mathrm{+}~{\mathrm{b}}_{\mathrm{1}}{\mathrm{x}}_{\left(\mathrm{t}-\mathrm{1}\right)~}\mathrm{+}~{\mathrm{b}}_{\mathrm{2}}{\mathrm{x}}_{\left(\mathrm{t}-\mathrm{2}\right)~}\mathrm{+\ \dots \ +}~{\mathrm{b}}_{\mathrm{p}}\mathrm{x}\left(\mathrm{t}-\mathrm{p}\right)~\mathrm{+\ }{\mathrm{\epsilon}}_{\mathrm{t}~}$

The chain rule of forecasting is used to predict successive forecasts.
The one-period ahead forecast of $x${}_{t}$$ from an AR(1) model is ${\widehat{\mathrm{x}}}_{\mathrm{t+1}}\mathrm{=\ }{\widehat{\mathrm{b}}}_0\mathrm{+\ }{\widehat{\mathrm{b}}}_{\mathrm{1}}{\mathrm{x}}_{\mathrm{t}}\$

${\mathrm{x}}_{\mathrm{t+1}}$ can be used to forecast the two-period ahead value : ${\widehat{\mathrm{x}}}_{\mathrm{t+2}}\mathrm{=\ }{\widehat{\mathrm{b}}}_0\mathrm{+\ }{\widehat{\mathrm{b}}}_{\mathrm{1}}{\mathrm{x}}_{\mathrm{t+1}}$$