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.