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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Lag operator</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">"Backshift" redirects here. For the linguistic sense, see <a href="Sequence_of_tenses" title="Sequence of tenses">Sequence of tenses</a>.</div>
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<p>In <a href="Time_series" title="Time series">time series</a> analysis, the <b>lag operator</b> (L) or <b>back<a href="Shift_operator" title="Shift operator">shift operator</a></b> (B) operates on an element of a time series to produce the previous element. For example, given some time series
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\{X_{1},X_{2},\dots \}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>=</mo>
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<mi>X</mi>
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<annotation encoding="application/x-tex">{\displaystyle X=\{X_{1},X_{2},\dots \}}</annotation>
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</math></span><img src="./051e8fc3a6fca766a13f9c72ec61ffad411655b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.152ex; height:2.843ex;" alt="{\displaystyle X=\{X_{1},X_{2},\dots \}}" loading="lazy"></span></dd></dl>
<p>then
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle LX_{t}=X_{t-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<msub>
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<mi>t</mi>
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<mi>t</mi>
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<annotation encoding="application/x-tex">{\displaystyle LX_{t}=X_{t-1}}</annotation>
</semantics>
</math></span><img src="./431ba8829207d547cc3af39838bf7006198bfc8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.282ex; height:2.509ex;" alt="{\displaystyle LX_{t}=X_{t-1}}" loading="lazy"></span> for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t>1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
<mo>&gt;</mo>
<mn>1</mn>
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<annotation encoding="application/x-tex">{\displaystyle t&gt;1}</annotation>
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</math></span><img src="./730f3de856e6f8850f89a9b990cfc7f7ba7c28bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t>1}" loading="lazy"></span></dd></dl>
<p>or similarly in terms of the backshift operator <i>B</i>: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BX_{t}=X_{t-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
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</msub>
<mo>=</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle BX_{t}=X_{t-1}}</annotation>
</semantics>
</math></span><img src="./206fcc16d79854ac00d784463d52282016a28763.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.463ex; height:2.509ex;" alt="{\displaystyle BX_{t}=X_{t-1}}" loading="lazy"></span> for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t>1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
<mo>&gt;</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t&gt;1}</annotation>
</semantics>
</math></span><img src="./730f3de856e6f8850f89a9b990cfc7f7ba7c28bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t>1}" loading="lazy"></span>. Equivalently, this definition can be represented as
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}=LX_{t+1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
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<mo>=</mo>
<mi>L</mi>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle X_{t}=LX_{t+1}}</annotation>
</semantics>
</math></span><img src="./204cfcca68df9a1e8a888c862e09e5661a648714.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.282ex; height:2.509ex;" alt="{\displaystyle X_{t}=LX_{t+1}}" loading="lazy"></span> for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\geq 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
<mo>≥<!-- ≥ --></mo>
<mn>1</mn>
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<annotation encoding="application/x-tex">{\displaystyle t\geq 1}</annotation>
</semantics>
</math></span><img src="./6946f8f50822de9e8e5e4c460ecbd4c05c644e15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.101ex; height:2.343ex;" alt="{\displaystyle t\geq 1}" loading="lazy"></span></dd></dl>
<p>The lag operator (as well as backshift operator) can be raised to arbitrary integer powers so that
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{-1}X_{t}=X_{t+1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
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</msub>
<mo>=</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
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</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L^{-1}X_{t}=X_{t+1}}</annotation>
</semantics>
</math></span><img src="./2634131346f3a30c210ce39bb1d4c0bdb2709687.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.615ex; height:3.009ex;" alt="{\displaystyle L^{-1}X_{t}=X_{t+1}}" loading="lazy"></span></dd></dl>
<p>and
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{k}X_{t}=X_{t-k}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
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</msup>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
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<mi>t</mi>
<mo>−<!-- − --></mo>
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<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L^{k}X_{t}=X_{t-k}.}</annotation>
</semantics>
</math></span><img src="./85b3a2673f4e82ea72b37a43e0cfebb67a106f23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.052ex; height:3.009ex;" alt="{\displaystyle L^{k}X_{t}=X_{t-k}.}" loading="lazy"></span></dd></dl>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Lag_polynomials">Lag polynomials</h2></div>
<p>Polynomials of the lag operator can be used, and this is a common notation for <a href="Autoregressive_moving_average" class="mw-redirect" title="Autoregressive moving average">ARMA</a> (autoregressive moving average) models. For example,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ε<!-- ε --></mi>
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<mi>t</mi>
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<mo>=</mo>
<msub>
<mi>X</mi>
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<mi>t</mi>
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<mo>−<!-- − --></mo>
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<mo>∑<!-- ∑ --></mo>
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<mi>i</mi>
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<mi>φ<!-- φ --></mi>
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<mo>−<!-- − --></mo>
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<mo>∑<!-- ∑ --></mo>
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<msub>
<mi>φ<!-- φ --></mi>
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<mi>i</mi>
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<msup>
<mi>L</mi>
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<mo>)</mo>
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<annotation encoding="application/x-tex">{\displaystyle \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}}</annotation>
</semantics>
</math></span><img src="./447d382a94a7cb7f6cec2cc6237753b29494f8bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.621ex; height:7.509ex;" alt="{\displaystyle \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}}" loading="lazy"></span></dd></dl>
<p>specifies an AR(<i>p</i>) model.
</p><p>A <a href="Polynomial" title="Polynomial">polynomial</a> of lag operators is called a <b>lag polynomial</b> so that, for example, the ARMA model can be concisely specified as
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (L)X_{t}=\theta (L)\varepsilon _{t}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>φ<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mi>L</mi>
<mo stretchy="false">)</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
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<mi>L</mi>
<mo stretchy="false">)</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
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<annotation encoding="application/x-tex">{\displaystyle \varphi (L)X_{t}=\theta (L)\varepsilon _{t}}</annotation>
</semantics>
</math></span><img src="./89ba93f98a80c919095357ff6ee23d807c9acd87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.153ex; height:2.843ex;" alt="{\displaystyle \varphi (L)X_{t}=\theta (L)\varepsilon _{t}}" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (L)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>φ<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mi>L</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi (L)}</annotation>
</semantics>
</math></span><img src="./ee0d9eb7efb10eec56d8cf1fb2d04462f5b0f70c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.912ex; height:2.843ex;" alt="{\displaystyle \varphi (L)}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta (L)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mo stretchy="false">(</mo>
<mi>L</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta (L)}</annotation>
</semantics>
</math></span><img src="./332b2411a4e06d1fa0a9e03e99c50d414720b688.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle \theta (L)}" loading="lazy"></span> respectively represent the lag polynomials
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (L)=1-\sum _{i=1}^{p}\varphi _{i}L^{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>φ<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mi>L</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</munderover>
<msub>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi (L)=1-\sum _{i=1}^{p}\varphi _{i}L^{i}}</annotation>
</semantics>
</math></span><img src="./b2bd58b37ebd3e0e1d697b60c14a6fd2d6c0231c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.458ex; height:7.009ex;" alt="{\displaystyle \varphi (L)=1-\sum _{i=1}^{p}\varphi _{i}L^{i}}" loading="lazy"></span></dd></dl>
<p>and
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta (L)=1+\sum _{i=1}^{q}\theta _{i}L^{i}.\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mo stretchy="false">(</mo>
<mi>L</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>q</mi>
</mrow>
</munderover>
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<mo>.</mo>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta (L)=1+\sum _{i=1}^{q}\theta _{i}L^{i}.\,}</annotation>
</semantics>
</math></span><img src="./2bcb382df680002b7ef1aedd733ca5a9923ac7fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.633ex; height:7.009ex;" alt="{\displaystyle \theta (L)=1+\sum _{i=1}^{q}\theta _{i}L^{i}.\,}" loading="lazy"></span></dd></dl>
<p>Polynomials of lag operators follow similar rules of multiplication and division as do numbers and polynomials of variables. For example,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}={\frac {\theta (L)}{\varphi (L)}}\varepsilon _{t},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">(</mo>
<mi>L</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mi>L</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{t}={\frac {\theta (L)}{\varphi (L)}}\varepsilon _{t},}</annotation>
</semantics>
</math></span><img src="./6ac58914b88604d2373a5d2aecd72ef98fa0fe45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.153ex; height:6.509ex;" alt="{\displaystyle X_{t}={\frac {\theta (L)}{\varphi (L)}}\varepsilon _{t},}" loading="lazy"></span></dd></dl>
<p>means the same thing as
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (L)X_{t}=\theta (L)\varepsilon _{t}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>φ<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mi>L</mi>
<mo stretchy="false">)</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">(</mo>
<mi>L</mi>
<mo stretchy="false">)</mo>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi (L)X_{t}=\theta (L)\varepsilon _{t}.}</annotation>
</semantics>
</math></span><img src="./24d4f188142640a2d3991eab2c8b9d2c1fbd0f3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.8ex; height:2.843ex;" alt="{\displaystyle \varphi (L)X_{t}=\theta (L)\varepsilon _{t}.}" loading="lazy"></span></dd></dl>
<p>As with polynomials of variables, a polynomial in the lag operator can be divided by another one using <a href="Polynomial_long_division" title="Polynomial long division">polynomial long division</a>. In general dividing one such polynomial by another, when each has a finite order (highest exponent), results in an infinite-order polynomial.
</p><p>An <b>annihilator operator</b>, denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\ ]_{+}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mtext>&nbsp;</mtext>
<msub>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [\ ]_{+}}</annotation>
</semantics>
</math></span><img src="./992881bf3cb310791e15cbc7acba599bda5e34f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.385ex; height:2.843ex;" alt="{\displaystyle [\ ]_{+}}" loading="lazy"></span>, removes the entries of the polynomial with negative power (future values).
</p><p>Note that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \left(1\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>φ<!-- φ --></mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi \left(1\right)}</annotation>
</semantics>
</math></span><img src="./399d3d8d617d87fae8dcf2670c15677c49f830a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.879ex; height:2.843ex;" alt="{\displaystyle \varphi \left(1\right)}" loading="lazy"></span> denotes the sum of coefficients:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \left(1\right)=1-\sum _{i=1}^{p}\varphi _{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>φ<!-- φ --></mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</munderover>
<msub>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi \left(1\right)=1-\sum _{i=1}^{p}\varphi _{i}}</annotation>
</semantics>
</math></span><img src="./3b82dea00167ee6a6e28134a8c616820149aa7d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.042ex; height:7.009ex;" alt="{\displaystyle \varphi \left(1\right)=1-\sum _{i=1}^{p}\varphi _{i}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Difference_operator">Difference operator</h2></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Finite_difference" title="Finite difference">Finite difference</a></div>
<p>In time series analysis, the first difference operator &nbsp;:<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>
</semantics>
</math></span><img src="./32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"></span>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Delta X_{t}&amp;=X_{t}-X_{t-1}\\\Delta X_{t}&amp;=(1-L)X_{t}~.\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>L</mi>
<mo stretchy="false">)</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mtext>&nbsp;</mtext>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Delta X_{t}&amp;=X_{t}-X_{t-1}\\\Delta X_{t}&amp;=(1-L)X_{t}~.\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./396f480bbe96c9b32cf026b7dcd0f1e17709f1e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.909ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\Delta X_{t}&amp;=X_{t}-X_{t-1}\\\Delta X_{t}&amp;=(1-L)X_{t}~.\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>Similarly, the second difference operator works as follows:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Delta (\Delta X_{t})&amp;=\Delta X_{t}-\Delta X_{t-1}\\\Delta ^{2}X_{t}&amp;=(1-L)\Delta X_{t}\\\Delta ^{2}X_{t}&amp;=(1-L)(1-L)X_{t}\\\Delta ^{2}X_{t}&amp;=(1-L)^{2}X_{t}~.\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mo stretchy="false">(</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msup>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>L</mi>
<mo stretchy="false">)</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msup>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>L</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>L</mi>
<mo stretchy="false">)</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msup>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>L</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mtext>&nbsp;</mtext>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Delta (\Delta X_{t})&amp;=\Delta X_{t}-\Delta X_{t-1}\\\Delta ^{2}X_{t}&amp;=(1-L)\Delta X_{t}\\\Delta ^{2}X_{t}&amp;=(1-L)(1-L)X_{t}\\\Delta ^{2}X_{t}&amp;=(1-L)^{2}X_{t}~.\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./f84b720ed67709076e99c55cd5db83e7819c3547.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:29.821ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}\Delta (\Delta X_{t})&amp;=\Delta X_{t}-\Delta X_{t-1}\\\Delta ^{2}X_{t}&amp;=(1-L)\Delta X_{t}\\\Delta ^{2}X_{t}&amp;=(1-L)(1-L)X_{t}\\\Delta ^{2}X_{t}&amp;=(1-L)^{2}X_{t}~.\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>The above approach generalises to the <i>i</i>-th difference operator
<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ^{i}X_{t}=(1-L)^{i}X_{t}\ .}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>L</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mtext>&nbsp;</mtext>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta ^{i}X_{t}=(1-L)^{i}X_{t}\ .}</annotation>
</semantics>
</math></span><img src="./0475710857916cdefc18c5084b43d9ca2bc08a6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.757ex; height:3.176ex;" alt="{\displaystyle \Delta ^{i}X_{t}=(1-L)^{i}X_{t}\ .}" loading="lazy"></span>
</p>
<div class="mw-heading mw-heading2"><h2 id="Conditional_expectation">Conditional expectation</h2></div>
<p>It is common in stochastic processes to care about the expected value of a variable given a previous information set. Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{t}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega _{t}}</annotation>
</semantics>
</math></span><img src="./635203506e879a91aadea447cbca82bd3c265364.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.504ex; height:2.509ex;" alt="{\displaystyle \Omega _{t}}" loading="lazy"></span> be all information that is common knowledge at time <i>t</i> (this is often subscripted below the expectation operator); then the expected value of the realisation of <i>X</i>, <i>j</i> time-steps in the future, can be written equivalently as:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[X_{t+j}|\Omega _{t}]=E_{t}[X_{t+j}].}">
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<annotation encoding="application/x-tex">{\displaystyle E[X_{t+j}|\Omega _{t}]=E_{t}[X_{t+j}].}</annotation>
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</math></span><img src="./0834937cce8e77b24deab11db341f444c9d5efda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.213ex; height:3.009ex;" alt="{\displaystyle E[X_{t+j}|\Omega _{t}]=E_{t}[X_{t+j}].}" loading="lazy"></span></dd></dl>
<p>With these time-dependent conditional expectations, there is the need to distinguish between the backshift operator (<i>B</i>) that only adjusts the date of the forecasted variable and the Lag operator (<i>L</i>) that adjusts equally the date of the forecasted variable and the information set:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{n}E_{t}[X_{t+j}]=E_{t-n}[X_{t+j-n}],}">
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<mrow class="MJX-TeXAtom-ORD">
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<msup>
<mi>L</mi>
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<mo stretchy="false">[</mo>
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<mi>X</mi>
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<mo stretchy="false">]</mo>
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<annotation encoding="application/x-tex">{\displaystyle L^{n}E_{t}[X_{t+j}]=E_{t-n}[X_{t+j-n}],}</annotation>
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</math></span><img src="./d41871114290a3444e022ae552afe8847b54d006.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.158ex; height:3.009ex;" alt="{\displaystyle L^{n}E_{t}[X_{t+j}]=E_{t-n}[X_{t+j-n}],}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B^{n}E_{t}[X_{t+j}]=E_{t}[X_{t+j-n}].}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>B</mi>
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</msup>
<msub>
<mi>E</mi>
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<annotation encoding="application/x-tex">{\displaystyle B^{n}E_{t}[X_{t+j}]=E_{t}[X_{t+j-n}].}</annotation>
</semantics>
</math></span><img src="./d8f16a7034f484634558e8e5c6cc00ebe901f6dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.075ex; height:3.009ex;" alt="{\displaystyle B^{n}E_{t}[X_{t+j}]=E_{t}[X_{t+j-n}].}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Autoregressive_model" title="Autoregressive model">Autoregressive model</a></li>
<li><a href="Autoregressive_moving_average_model" class="mw-redirect" title="Autoregressive moving average model">Autoregressive moving average model</a></li>
<li><a href="Moving_average_model" class="mw-redirect" title="Moving average model">Moving average model</a></li>
<li><a href="Shift_operator" title="Shift operator">Shift operator</a></li>
<li><a href="Z-transform" title="Z-transform">Z-transform</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFHamilton1994" class="citation book cs1">Hamilton, James Douglas (1994). <i>Time Series Analysis</i>. Princeton University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>0-691-04289-6</bdi>.</cite></li>
<li><cite id="CITEREFVerbeek2008" class="citation book cs1"><a href="Marno_Verbeek" title="Marno Verbeek">Verbeek, Marno</a> (2008). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/guidetomoderneco0003verb"><i>A Guide to Modern Econometrics</i></a></span>. John Wiley and Sons. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>0-470-51769-7</bdi>.</cite></li>
<li><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/DifferenceOperator.html">"Wolfram MathWorld"</a>. <i>WolframMathworld: Difference Operator</i>. Wolfram Research<span class="reference-accessdate">. Retrieved <span class="nowrap">10 November</span> 2017</span>.</cite></li>
<li><cite id="CITEREFBoxJenkinsReinselLjung2016" class="citation book cs1">Box, George E. P.; Jenkins, Gwilym M.; Reinsel, Gregory C.; Ljung, Greta M. (2016). <i>Time Series Analysis: Forecasting and Control</i> (5th&nbsp;ed.). New Jersey: Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-1-118-67502-1</bdi>.</cite></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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