Lag operator
Encyclopedia
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
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
and
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 φ and θ respectively represent the lag polynomials,
and
An annihilator operator, denoted
, removes the entries of the polynomial with negative power (future values).
is a special case of lag polynomial.
Similarly, the second difference operator
The above approach generalises to the i 'th difference operator
be all information that is common knowledge at time t (this is often subscripted below the expectation operator), then the expected value of X that is some 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:
Time series
In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...
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
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
and
Lag polynomials
Also polynomials of the lag operator can be used, and this is a common notation for ARMA models. For example,specifies an AR(p) model.
A polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
of lag operators is called a lag polynomial so that, for example, the ARMA model can be concisely specified as
where φ and θ respectively represent the lag polynomials,
and
An annihilator operator, denoted
, removes the entries of the polynomial with negative power (future values).Difference operator
In time series analysis, the first difference operator is a special case of lag polynomial.Similarly, the second difference operator
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 (this is often subscripted below the expectation operator), then the expected value of X that is some 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:
See also
- Autoregressive modelAutoregressive modelIn statistics and signal processing, an autoregressive model is a type of random process which is often used to model and predict various types of natural phenomena...
- Autoregressive moving average modelAutoregressive moving average modelIn statistics and signal processing, autoregressive–moving-average models, sometimes called Box–Jenkins models after the iterative Box–Jenkins methodology usually used to estimate them, are typically applied to autocorrelated time series data.Given a time series of data Xt, the ARMA model is a...
- Shift operatorShift operatorIn mathematics, and in particular functional analysis, the shift operator or translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator....
- Z-transformZ-transformIn mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation....