8.2 Backshift notation

The backward shift operator \(B\) is a useful notational device when working with time series lags: \[ B y_{t} = y_{t - 1} \: . \] (Some references use \(L\) for “lag” instead of \(B\) for “backshift”.) In other words, \(B\), operating on \(y_{t}\), has the effect of shifting the data back one period. Two applications of \(B\) to \(y_{t}\) shifts the data back two periods: \[ B(By_{t}) = B^{2}y_{t} = y_{t-2}\: . \] For monthly data, if we wish to consider “the same month last year,” the notation is \(B^{12}y_{t}\) = \(y_{t-12}\).

The backward shift operator is convenient for describing the process of differencing. A first difference can be written as \[ y'_{t} = y_{t} - y_{t-1} = y_t - By_{t} = (1 - B)y_{t}\: . \] Note that a first difference is represented by \((1 - B)\). Similarly, if second-order differences have to be computed, then: \[ y''_{t} = y_{t} - 2y_{t - 1} + y_{t - 2} = (1-2B+B^2)y_t = (1 - B)^{2} y_{t}\: . \] In general, a \(d\)th-order difference can be written as \[ (1 - B)^{d} y_{t}. \]

Backshift notation is very useful when combining differences as the operator can be treated using ordinary algebraic rules. In particular, terms involving \(B\) can be multiplied together.

For example, a seasonal difference followed by a first difference can be written as \[\begin{align*} (1-B)(1-B^m)y_t &= (1 - B - B^m + B^{m+1})y_t \\ &= y_t-y_{t-1}-y_{t-m}+y_{t-m-1}, \end{align*}\] the same result we obtained earlier.