The backshift operator B (aka the lag operator L)

The lagged value can be expressed by means of backshift operator B (sometimes it is called the lag operator L). Having time series yt=1,2…={y1,y2,…} the operator B ’s function is to refer to value lagged n times backwards:

It has standard multiplication and division properties:

It can be also used as a polynomial:

The advantage of B operator is that time series’ differencing Δ can be calculated easily:

The first order difference is for example:

And the (rarely used) second order difference (kind of “derivative”) is:

Not to be confused with seasonal differencing:

For example, having monthly time series, the annual seasonality can be expressed through seasonal differencing:

To continue the example, in combination with first differencing: