Lag operator

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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

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

then

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

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

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

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

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

and

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