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 then or similarly in terms of the backshift operator B: for all. Equivalently, this definition can be represented as The lag operator can be raised to arbitrary integer powers so that and
Lag polynomials
Polynomials of the lag operator can be used, and this is a common notation for ARMA models. For example, specifies an AR 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 and 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, results in an infinite-order polynomial. An annihilator operator, denoted, removes the entries of the polynomial with negative power. Note that denotes the sum of coefficients:
Difference operator
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
Conditional expectation
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 ; 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 that only adjusts the date of the forecasted variable and the Lag operator that adjusts equally the date of the forecasted variable and the information set:
