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

:$,\; L\; X\_t\; =\; X\_\{t-1\}$ 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

:$,\; L^k\; X\_\{t\}\; =\; X\_\{t-k\}.,$

Lag polynomials

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

:$varepsilon\_t\; =\; X\_t\; -\; sum\_\{i=1\}^p\; varphi\_i\; X\_\{t-i\}\; =\; left(1\; -\; sum\_\{i=1\}^p\; varphi\_i\; L^i\; ight)\; X\_t,$

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

:$varphi\; X\_t\; =\; heta\; varepsilon\_t,$

where φ and θ respectively represent the lag polynomials,

:$varphi\; =\; 1\; -\; sum\_\{i=1\}^p\; varphi\_i\; L^i,$

and

:$heta=\; 1\; +\; sum\_\{i=1\}^q\; heta\_i\; L^i.,$

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.

:

Similarly, the second difference operator

:

The above approach generalises to the "i" 'th difference operator$Delta\; ^i\; X\_t\; =\; (1-L)^i\; X\_t$

See also

* Autoregressive moving average model
* Shift operator
* Z-transform