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

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

[edit] Lag Polynomials

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).

[edit] Difference Operator

In time series analysis, the first difference operator $\Delta \$ is a special case of lag polynomial.

$\begin{array}{lcr} \Delta X_t & = X_t - X_{t-1} \\ \Delta X_t & = (1-L)X_t \end{array}$

Similarly, the 2nd Difference Operator

$\begin{align} \Delta ( \Delta X_t ) & = \Delta X_t - \Delta X_{t-1} \\ \Delta^2 X_t & = (1-L)\Delta X_t \\ \Delta^2 X_t & = (1-L)(1-L)X_t \\ \Delta^2 X_t & = (1-L)^2 X_t \end{align}$