## 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 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.