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

or similarly in terms of the backshift operator B: ${\displaystyle BX_{t}=X_{t-1}}$ for all ${\displaystyle t>1}$. Equivalently, this definition can be represented as

The lag operator (as well as backshift operator) can be raised to arbitrary integer powers so that

and

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