Lag operator - Definition and Overview

In time series analysis the lag operator or backshift operator operates on an element of a time series to produce the previous element. For example, given some time series

<math>X= \{X_1, X_2, \dots \}<math>

then

<math>\, L X_t = X_{t-1} <math> for all <math>\; t > 1<math>

where L is the lag operator. Sometimes the symbol B for backshift is used instead. Note that the lag operator can be raised to aribtrary integer powers so that

<math>\, L^{-1} X_{t} = X_{t+1}<math>

and

<math>\, L^k X_{t} = X_{t-k}<math>

Also polynomials of the lag operator can be used, and this is a common notation for ARMA models. For example,

<math> \varepsilon_t = X_t - \sum_{i=1}^p \phi_i X_{t-i} = (1 - \sum_{i=1}^p \phi_i L^i) X_t<math>

specifies an AR(p) model.