# Lag operator

## Operator for offsetting time series elements / From Wikipedia, the free encyclopedia

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"Backshift" redirects here. For the linguistic sense, see Sequence of tenses.

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

- $X=\{X_{1},X_{2},\dots \}$

This article includes a list of general references, but it lacks sufficient corresponding inline citations. (January 2011) |

then

- $LX_{t}=X_{t-1}$ for all $t>1$

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

- $X_{t}=LX_{t+1}$ for all $t\geq 1$

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

- $L^{-1}X_{t}=X_{t+1}$

and

- $L^{k}X_{t}=X_{t-k}.$