For example, given some time series then or similarly in terms of the backshift operator B:
Equivalently, this definition can be represented as The lag operator (as well as backshift operator) can be raised to arbitrary integer powers so that and Polynomials of the lag operator can be used, and this is a common notation for ARMA (autoregressive moving average) models.
A polynomial of lag operators is called a lag polynomial so that, for example, the ARMA model can be concisely specified as where
In general dividing one such polynomial by another, when each has a finite order (highest exponent), results in an infinite-order polynomial.
, removes the entries of the polynomial with negative power (future values).
denotes the sum of coefficients: In time series analysis, the first difference operator :
Similarly, the second difference operator works as follows: The above approach generalises to the i-th difference operator
It is common in stochastic processes to care about the expected value of a variable given a previous information set.
be all information that is common knowledge at time t (this is often subscripted below the expectation operator); then the expected value of the realisation of X, j time-steps in the future, can be written equivalently as: With these time-dependent conditional expectations, there is the need to distinguish between the backshift operator (B) that only adjusts the date of the forecasted variable and the Lag operator (L) that adjusts equally the date of the forecasted variable and the information set: