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time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
analysis, the lag operator (L) or back
shift operator In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator. Shift o ...
(B) operates on an element of a time series to produce the previous element. For example, given some time series :X= \ then : L X_t = X_ for all t > 1 or similarly in terms of the backshift operator ''B'': B X_t = X_ for all t > 1. Equivalently, this definition can be represented as : X_t = L X_ for all t \geq 1 The lag operator (as well as backshift operator) can be raised to arbitrary integer powers so that : L^ X_ = X_ and : L^k X_ = X_.


Lag polynomials

Polynomials of the lag operator can be used, and this is a common notation for ARMA (autoregressive moving average) models. For example, : \varepsilon_t = X_t - \sum_^p \varphi_i X_ = \left(1 - \sum_^p \varphi_i L^i\right) X_t specifies an AR(''p'') model. A
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
of lag operators is called a lag polynomial so that, for example, the ARMA model can be concisely specified as : \varphi (L) X_t = \theta (L) \varepsilon_t where \varphi (L) and \theta (L) respectively represent the lag polynomials : \varphi (L) = 1 - \sum_^p \varphi_i L^i and : \theta (L)= 1 + \sum_^q \theta_i L^i.\, Polynomials of lag operators follow similar rules of multiplication and division as do numbers and polynomials of variables. For example, : X_t = \frac\varepsilon_t, means the same thing as :\varphi (L) X_t = \theta (L) \varepsilon_t . As with polynomials of variables, a polynomial in the lag operator can be divided by another one using
polynomial long division In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version of the familiar arithmetic technique called long division. It can be done easily by hand, becaus ...
. 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). Note that \varphi \left( 1 \right) denotes the sum of coefficients: : \varphi \left( 1 \right) = 1 - \sum_^p \varphi_i


Difference operator

In time series analysis, the first difference operator : \Delta : \begin \Delta X_t & = X_t - X_ \\ \Delta X_t & = (1-L)X_t ~. \end Similarly, the second difference operator works as follows: : \begin \Delta ( \Delta X_t ) & = \Delta X_t - \Delta X_ \\ \Delta^2 X_t & = (1-L)\Delta X_t \\ \Delta^2 X_t & = (1-L)(1-L)X_t \\ \Delta^2 X_t & = (1-L)^2 X_t ~. \end The above approach generalises to the ''i''-th difference operator \Delta ^i X_t = (1-L)^i X_t \ .


Conditional expectation

It is common in stochastic processes to care about the expected value of a variable given a previous information set. Let \Omega_t 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: :E \Omega_t= E_t
X_ X, or x, is the twenty-fourth and third-to-last letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''"ex"'' (pronounced ), ...
. 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: :L^n E_t
X_ X, or x, is the twenty-fourth and third-to-last letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''"ex"'' (pronounced ), ...
= E_
X_ X, or x, is the twenty-fourth and third-to-last letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''"ex"'' (pronounced ), ...
, :B^n E_t
X_ X, or x, is the twenty-fourth and third-to-last letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''"ex"'' (pronounced ), ...
= E_t
X_ X, or x, is the twenty-fourth and third-to-last letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''"ex"'' (pronounced ), ...
.


See also

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Autoregressive model In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...
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Autoregressive moving average model In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...
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Moving average model In time series analysis, the moving-average model (MA model), also known as moving-average process, is a common approach for modeling univariate time series. The moving-average model specifies that the output variable is cross-correlated with a ...
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Shift operator In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator. Shift o ...
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Z-transform In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation. It can be considered as a discrete-tim ...


References

* * * * {{DEFAULTSORT:Lag Operator Time series