## Note on a differencing and lag operators in time series modeling

Differencing operator $\Delta$ is defined by
$\Delta X_t := X_t - X_{t-1}$
Lag operator $B$ is defined by:
$B X_t := X_{t-1}$
It is common to see in literature the following:
$\Delta X_t = X_t - X_{t-1} = X_t - B X_t = (1 - B) X_t$
Authors often omit to say that $(1 - B)$ in the last equality is an operator whose definition is the preceding equation; the last equality in above should better read as:
$X_t - B X_t := (1 - B) X_t$
Otherwise, one can ask, how the hell subtraction of operator from a real number is supposed to be carried out ?! 🙂
If we want to be more formal in definition of operators, which are maps, we should specify domains and ranges of maps:
$\Delta : \mathcal{R}^w \mapsto \mathcal{R}^w$
$B : \mathcal{R}^w \mapsto \mathcal{R}^w$
This is a formal way to say, that the domain and range is a set of all sequences of real numbers.