Lag operator

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 arbitrary 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 \varphi_i X_{t-i} = \left(1 – \sum_{i=1}^p \varphi_i L^i\right) X_t\,<math>

specifies an AR(p) model.

A polynomial of lag operators is called a lag polynomial so that, for example, the ARMA model can be concisely specified as

<math> \varphi X_t = \theta \varepsilon_t\,<math>

where φ and θ respectively represent the lag polynomials,

<math> \varphi = 1 – \sum_{i=1}^p \varphi_i L^i\,<math>

and

<math> \theta= 1 + \sum_{i=1}^q \theta_i L^i.\,<math>

An annihilator operator, denoted <math>[\ ]_+<math>, removes the entries of the polynomial with negative power (future values).