Editing Autoregressive model (section)

==Intertemporal effect of shocks==

In an AR process, a one-time shock affects values of the evolving variable infinitely far into the future. For example, consider the AR(1) model <math> X_t = \varphi_1 X_{t-1} + \varepsilon_t</math>. A non-zero value for <math>\varepsilon_t</math> at say time ''t''=1 affects <math>X_1</math> by the amount  <math>\varepsilon_1</math>. Then by the AR equation for <math>X_2</math> in terms of <math>X_1</math>, this affects <math>X_2</math> by the amount <math>\varphi_1 \varepsilon_1</math>. Then by the AR equation for <math>X_3</math> in terms of <math>X_2</math>, this affects <math>X_3</math> by the amount <math>\varphi_1^2 \varepsilon_1</math>. Continuing this process shows that the effect of <math>\varepsilon_1</math> never ends, although if the process is [[stationary process|stationary]] then the effect diminishes toward zero in the limit.

Because each shock affects ''X'' values infinitely far into the future from when they occur, any given value ''X''<sub>''t''</sub> is affected by shocks occurring infinitely far into the past. This can also be seen by rewriting the autoregression

:<math>\phi (B)X_t=  \varepsilon_t \,</math>

(where the constant term has been suppressed by assuming that the variable has been measured as deviations from its mean) as

:<math>X_t= \frac{1}{\phi (B)}\varepsilon_t \, .</math>

When the [[polynomial long division|polynomial division]] on the right side is carried out, the polynomial in the backshift operator applied to <math>\varepsilon_t</math> has an infinite order—that is, an infinite number of lagged values of <math>\varepsilon_t</math> appear on the right side of the equation.