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| In [[time series]] analysis, the '''lag operator''' or '''back[[shift operator]]''' operates on an element of a time series to produce the previous element. For example, given some time series
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| :<math>X= \{X_1, X_2, \dots \}\,</math> | |
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| then
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| :<math>\, L X_t = X_{t-1} </math> for all <math>\; t > 1\,</math>
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| or equivalently
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| :<math>\, X_t = L X_{t+1}</math> for all <math>\; t \geq 1\,</math>
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| 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
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| :<math>\, L^{-1} X_{t} = X_{t+1}\,</math>
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| and
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| :<math>\, L^k X_{t} = X_{t-k}.\,</math>
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| ==Lag polynomials== | |
| Also polynomials of the lag operator can be used, and this is a common notation for [[autoregressive moving average|ARMA]] (autoregressive moving average) models. For example,
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| :<math> \varepsilon_t = X_t - \sum_{i=1}^p \varphi_i X_{t-i} = \left(1 - \sum_{i=1}^p \varphi_i L^i\right) X_t\,</math>
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| specifies an AR(''p'') model.
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| A [[polynomial]] of lag operators is called a '''lag polynomial''' so that, for example, the ARMA model can be concisely specified as
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| :<math> \varphi (L) X_t = \theta (L) \varepsilon_t\,</math> | |
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| where <math> \varphi (L)</math> and <math>\theta (L)</math> respectively represent the lag polynomials
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| :<math> \varphi (L) = 1 - \sum_{i=1}^p \varphi_i L^i\,</math>
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| and
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| :<math> \theta (L)= 1 + \sum_{i=1}^q \theta_i L^i.\,</math>
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| Polynomials of lag operators follow similar rules of multiplication and division as do numbers and polynomials of variables. For example,
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| :<math> X_t = \frac{\theta (L) }{\varphi (L)}\varepsilon_t,</math> | |
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| means the same thing as
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| :<math>\varphi (L) X_t = \theta (L) \varepsilon_t\, .</math>
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| 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.
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| An '''annihilator operator''', denoted <math>[\ ]_+</math>, removes the entries of the polynomial with negative power (future values).
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| ==Difference operator== | |
| {{main|Finite difference}}
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| In time series analysis, the first difference operator ''Δ'' is a special case of lag polynomial.
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| :<math> | |
| \begin{array}{lcr}
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| \Delta X_t & = X_t - X_{t-1} \\
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| \Delta X_t & = (1-L)X_t ~.
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| \end{array}
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| </math>
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| Similarly, the second difference operator works as follows:
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| :<math>
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| \begin{align}
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| \Delta ( \Delta X_t ) & = \Delta X_t - \Delta X_{t-1} \\
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| \Delta^2 X_t & = (1-L)\Delta X_t \\
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| \Delta^2 X_t & = (1-L)(1-L)X_t \\
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| \Delta^2 X_t & = (1-L)^2 X_t ~.
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| \end{align}
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| </math>
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| The above approach generalises to the ''i''th difference operator
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| <math> \Delta ^i X_t = (1-L)^i X_t \ .</math>
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| ==Conditional expectation==
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| It is common in stochastic processes to care about the expected value of a variable given a previous information set. Let <math>\Omega_t</math> 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:
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| :<math>E [ X_{t+j} | \Omega_t] = E_t [ X_{t+j} ] \,.</math>
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| 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:
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| :<math>L^n E_t [ X_{t+j} ] = E_{t-n} [ X_{t+j-n} ] \, ,</math>
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| :<math>B^n E_t [ X_{t+j} ] = E_t [ X_{t+j-n} ] \, .</math> | |
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| ==References==
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| *{{cite book | first = James Douglas| last = Hamilton| year = 1994| title = Time series analysis | publisher = Princeton University Press | id = ISBN 0-691-04289-6 }}
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| *{{cite book | first = Marno| last = Verbeek| year = 2008| title = A guide to modern econometrics | publisher = John Wiley and Sons | id = ISBN 0-470-51769-7 }}
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| {{More footnotes|date=January 2011}}
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| ==See also==
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| * [[Autoregressive model]]
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| * [[Autoregressive moving average model]]
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| * [[Moving average model]]
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| * [[Shift operator]]
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| * [[Z-transform]]
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| {{DEFAULTSORT:Lag Operator}}
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| [[Category:Stochastic processes]]
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