Editing Lag operator (section)

==Lag polynomials==
Polynomials of the lag operator can be used, and this is a common notation for [[autoregressive moving average|ARMA]] (autoregressive moving average) 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 (L) X_t = \theta (L) \varepsilon_t</math>

where <math> \varphi (L)</math> and <math>\theta (L)</math> respectively represent the lag polynomials

:<math> \varphi (L) = 1 - \sum_{i=1}^p \varphi_i L^i</math>

and

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

Polynomials of lag operators follow similar rules of multiplication and division as do numbers and polynomials of variables. For example,

:<math> X_t = \frac{\theta (L) }{\varphi (L)}\varepsilon_t,</math>

means the same thing as

:<math>\varphi (L) X_t = \theta (L) \varepsilon_t .</math>

As with polynomials of variables, a polynomial in the lag operator can be divided by another one using [[polynomial long division]]. In general dividing one such polynomial by another, when each has a finite order (highest exponent), results in an infinite-order polynomial.

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

Note that <math>\varphi \left( 1 \right)</math>  denotes the sum of coefficients:
:<math> \varphi \left( 1 \right) = 1 - \sum_{i=1}^p \varphi_i </math>