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 dth-order difference can be written as (1 - B)^{d} y_{t}.

Backshift notation is particularly 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.