Editing Lag operator (section)

{{Short description|Operator for offsetting time series elements}}{{Redirect|Backshift|the linguistic sense|Sequence of tenses}}
{{More footnotes|date=January 2011}}

In [[time series]] analysis, the '''lag operator''' (L) or '''back[[shift operator]]''' (B) operates on an element of a time series to produce the previous element.  For example, given some time series

:<math>X= \{X_1, X_2, \dots \}</math>

then

:<math> L X_t = X_{t-1} </math> for all <math> t > 1</math>

or similarly in terms of the backshift operator ''B'': <math> B X_t = X_{t-1} </math> for all <math> t > 1</math>. Equivalently, this definition can be represented as

:<math> X_t = L X_{t+1}</math> for all <math> t \geq 1</math>

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

:<math> L^{-1} X_{t} = X_{t+1}</math>

and

:<math> L^k X_{t} = X_{t-k}.</math>