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

then

for all

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

and

Also polynomials of the lag operator can be used, and this is a common notation for ARMA 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 φ and θ respectively represent the lag polynomials,

and

An annihilator operator, denoted , removes the entries of the polynomial with negative power (future values).