Lag operator

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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

X= \{X_1, X_2, \dots \}\,

then

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

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

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

and

\, L^k X_{t} = X_{t-k}.\,

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

\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\,

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

\varphi X_t = \theta \varepsilon_t\,

where φ and θ respectively represent the lag polynomials,

\varphi = 1 - \sum_{i=1}^p \varphi_i L^i\,

and

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

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

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