In time series analysis, the lag operator or backshift operator operates on an element of a time series to produce the previous element. For example, given some time series
- for all
- for all
where L is the lag operator. Sometimes the symbol B for backshift is used instead. Note that the lag operator can be raised to arbitrary integer powers so that
Also polynomials of the lag operator can be used, and this is a common notation for ARMA (autoregressive moving average) models. For example,
specifies an AR(p) model.
A polynomial of lag operators is called a lag polynomial so that, for example, the ARMA model can be concisely specified as
where and respectively represent the lag polynomials
Polynomials of lag operators follow similar rules of multiplication and division as do numbers and polynomials of variables. For example,
means the same thing as
As with polynomials of variables, a polynomial in the lag operator can be divided by another one using polynomial long division. In general dividing one such polynomial by another, when each has a finite order (highest exponent), results in an infinite-order polynomial.
An annihilator operator, denoted , removes the entries of the polynomial with negative power (future values).
In time series analysis, the first difference operator Δ is a special case of lag polynomial.
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. Let 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: