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

$X=\{X_{1},X_{2},\dots \}\,$

then

$\,LX_{t}=X_{t-1}$ 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

Tarik RULES

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

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).

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

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

The above approach generalises to the i 'th difference operator $\Delta ^{i}X_{t}=(1-L)^{i}X_{t}\$

See also

• Shift operator
• Autoregressive moving average model