Lag operator
In time series analysis, the lag operator (L) or backshift operator (B) operates on an element of a time series to produce the previous element. For example, given some time series then for all or similarly in terms of the backshift operator B: for all . Equivalently, this definition can be represented as for all The lag operator (as well as backshift operator) can be raised to arbitrary integer powers so that and Lag polynomials Polynomials of the lag operator can be used, and this is a common notation for ARMA (autoregressive moving average) models. For example, 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 where and respectively represent the lag polynomials and Polynomials of lag operators follow similar rules of multiplication and division as do numbers and polynomials of variables. For example, means the same thing as 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