# 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

Note that a first difference is represented by $(1 - B)$. Similarly, if second-order differences have to be computed, then:

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

the same result we obtained earlier.