Lag operator

Polynomials of the lag operator can be used, and this is a common notation for ARMA (autoregressive moving average) models. For example,

:$\backslash varepsilon\_t\; =\; X\_t\; -\; \backslash sum\_\{i=1\}^p\; \backslash varphi\_i\; X\_\{t-i\}\; =\; \backslash left(1\; -\; \backslash sum\_\{i=1\}^p\; \backslash varphi\_i\; L^i\backslash right)\; X\_t$

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

:$\backslash varphi\; (L)\; X\_t\; =\; \backslash theta\; (L)\; \backslash varepsilon\_t$

where $\backslash varphi\; (L)$ and $\backslash theta\; (L)$ respectively represent the lag polynomials

:$\backslash varphi\; (L)\; =\; 1\; -\; \backslash sum\_\{i=1\}^p\; \backslash varphi\_i\; L^i$

and

:$\backslash theta\; (L)=\; 1\; +\; \backslash sum\_\{i=1\}^q\; \backslash theta\_i\; L^i.\backslash ,$

Polynomials of lag operators follow similar rules of multiplication and division as do numbers and polynomials of variables. For example,

:$X\_t\; =\; \backslash frac\{\backslash theta\; (L)\; \}\{\backslash varphi\; (L)\}\backslash varepsilon\_t,$

means the same thing as

:$\backslash varphi\; (L)\; X\_t\; =\; \backslash theta\; (L)\; \backslash varepsilon\_t\; .$

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 $[\backslash \; ]\_+$, removes the entries of the polynomial with negative power (future values).

Note that $\backslash varphi\; \backslash left(\; 1\; \backslash right)$ denotes the sum of coefficients:

:$\backslash varphi\; \backslash left(\; 1\; \backslash right)\; =\; 1\; -\; \backslash sum\_\{i=1\}^p\; \backslash varphi\_i$

{{main|Finite difference}}

In time series analysis, the first difference operator :$\backslash Delta$

:$$

\begin{align}

\Delta X_t & = X_t - X_{t-1} \\

\Delta X_t & = (1-L)X_t ~.

\end{align}

Similarly, the second difference operator works as follows:

:$$

\begin{align}

\Delta ( \Delta X_t ) & = \Delta X_t - \Delta X_{t-1} \\

\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{align}

The above approach generalises to the i-th difference operator

$\backslash Delta\; ^i\; X\_t\; =\; (1-L)^i\; X\_t\; \backslash \; .$

It is common in stochastic processes to care about the expected value of a variable given a previous information set. Let $\backslash Omega\_t$ 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:

:$E\; [\; X\_\{t+j\}\; |\; \backslash Omega\_t]\; =\; E\_t\; [\; X\_\{t+j\}\; ]\; .$

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:

:$L^n\; E\_t\; [\; X\_\{t+j\}\; ]\; =\; E\_\{t-n\}\; [\; X\_\{t+j-n\}\; ]\; ,$

:$B^n\; E\_t\; [\; X\_\{t+j\}\; ]\; =\; E\_t\; [\; X\_\{t+j-n\}\; ]\; .$

• Autoregressive model
• Autoregressive moving average model
• Moving average model
• Shift operator
• Z-transform

{{Reflist}}

• {{cite book | first = James Douglas| last = Hamilton| year = 1994| title = Time Series Analysis | publisher = Princeton University Press | isbn = 0-691-04289-6 }}
• {{cite book | first = Marno | last = Verbeek | author-link = Marno Verbeek | year = 2008 | title = A Guide to Modern Econometrics | publisher = John Wiley and Sons | isbn = 0-470-51769-7 | url-access = registration | url = https://archive.org/details/guidetomoderneco0003verb }}
• {{cite web|last1=Weisstein|first1=Eric|title=Wolfram MathWorld|url=http://mathworld.wolfram.com/DifferenceOperator.html|website=WolframMathworld: Difference Operator|publisher=Wolfram Research|accessdate=10 November 2017}}
• {{cite book|last1=Box|first1=George E. P.|last2=Jenkins|first2=Gwilym M.|last3=Reinsel|first3=Gregory C.|last4=Ljung|first4=Greta M.|title=Time Series Analysis: Forecasting and Control|date=2016|publisher=Wiley|location=New Jersey|isbn=978-1-118-67502-1|edition=5th}}

{{DEFAULTSORT:Lag Operator}}

Category:Time series