Simultaneous equations model

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class="vector-toc-list"> <li id="toc-Using_cross-equation_restrictions_to_achieve_identification" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_cross-equation_restrictions_to_achieve_identification"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Using cross-equation restrictions to achieve identification</span> </div> </a> <ul id="toc-Using_cross-equation_restrictions_to_achieve_identification-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Estimation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Estimation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Estimation</span> </div> </a> <button aria-controls="toc-Estimation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Estimation subsection</span> </button> <ul id="toc-Estimation-sublist" class="vector-toc-list"> <li id="toc-Two-stage_least_squares_(2SLS)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Two-stage_least_squares_(2SLS)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Two-stage least squares (2SLS)</span> </div> </a> <ul id="toc-Two-stage_least_squares_(2SLS)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Indirect_least_squares" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Indirect_least_squares"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Indirect least squares</span> </div> </a> <ul id="toc-Indirect_least_squares-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limited_information_maximum_likelihood_(LIML)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Limited_information_maximum_likelihood_(LIML)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Limited information maximum likelihood (LIML)</span> </div> </a> <ul id="toc-Limited_information_maximum_likelihood_(LIML)-sublist" class="vector-toc-list"> <li id="toc-K_class_estimators" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#K_class_estimators"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.1</span> <span>K class estimators</span> </div> </a> <ul id="toc-K_class_estimators-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Three-stage_least_squares_(3SLS)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Three-stage_least_squares_(3SLS)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Three-stage least squares (3SLS)</span> </div> </a> <ul id="toc-Three-stage_least_squares_(3SLS)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications_in_social_science" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications_in_social_science"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applications in social science</span> </div> </a> <ul id="toc-Applications_in_social_science-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" 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mw-first-heading"><span class="mw-page-title-main">Simultaneous equations model</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Type of statistical model</div> <p><b>Simultaneous equations models</b> are a type of <a href="/wiki/Statistical_model" title="Statistical model">statistical model</a> in which the <a href="/wiki/Dependent_and_independent_variables" title="Dependent and independent variables">dependent variables</a> are functions of other dependent variables, rather than just independent variables.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> This means some of the explanatory variables are <a href="/wiki/Endogeneity_(econometrics)" title="Endogeneity (econometrics)">jointly determined</a> with the dependent variable, which in <a href="/wiki/Economics" title="Economics">economics</a> usually is the consequence of some underlying <a href="/wiki/Economic_equilibrium" title="Economic equilibrium">equilibrium mechanism</a>. Take the typical <a href="/wiki/Supply_and_demand" title="Supply and demand">supply and demand</a> model: whilst typically one would determine the quantity supplied and demanded to be a function of the price set by the market, it is also possible for the reverse to be true, where producers observe the quantity that consumers demand <i>and then</i> set the price.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Simultaneity poses challenges for the <a href="/wiki/Point_estimation" title="Point estimation">estimation</a> of the statistical parameters of interest, because the <a href="/wiki/Gauss%E2%80%93Markov_theorem" title="Gauss–Markov theorem">Gauss–Markov assumption</a> of <a href="/wiki/Gauss%E2%80%93Markov_theorem#Strict_exogeneity" title="Gauss–Markov theorem">strict exogeneity</a> of the regressors is violated. And while it would be natural to estimate all simultaneous equations at once, this often leads to a <a href="/wiki/Computational_complexity" title="Computational complexity">computationally costly</a> non-linear optimization problem even for the simplest <a href="/wiki/System_of_linear_equations" title="System of linear equations">system of linear equations</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> This situation prompted the development, spearheaded by the <a href="/wiki/Cowles_Commission" class="mw-redirect" title="Cowles Commission">Cowles Commission</a> in the 1940s and 1950s,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> of various techniques that estimate each equation in the model seriatim, most notably <a href="/wiki/Limited_information_maximum_likelihood" class="mw-redirect" title="Limited information maximum likelihood">limited information maximum likelihood</a> and <a href="/wiki/Two-stage_least_squares" class="mw-redirect" title="Two-stage least squares">two-stage least squares</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Structural_and_reduced_form">Structural and reduced form</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simultaneous_equations_model&action=edit&section=1" title="Edit section: Structural and reduced form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose there are <i>m</i> regression equations of the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{it}=y_{-i,t}'\gamma _{i}+x_{it}'\;\!\beta _{i}+u_{it},\quad i=1,\ldots ,m,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mo>,</mo> <mi>t</mi> </mrow> <mo>′</mo> </msubsup> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> <mo>′</mo> </msubsup> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{it}=y_{-i,t}'\gamma _{i}+x_{it}'\;\!\beta _{i}+u_{it},\quad i=1,\ldots ,m,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f95f94a8d6f478111a228324866b79e58c150201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:41.689ex; height:3.176ex;" alt="{\displaystyle y_{it}=y_{-i,t}'\gamma _{i}+x_{it}'\;\!\beta _{i}+u_{it},\quad i=1,\ldots ,m,}"></span></dd></dl> <p>where <i>i</i> is the equation number, and <span class="nowrap"><i>t</i> = 1, ..., <i>T</i></span> is the observation index. In these equations <i>x<sub>it</sub></i> is the <i>k<sub>i</sub>×</i>1 vector of exogenous variables, <i>y<sub>it</sub></i> is the dependent variable, <i>y<sub>−i,t</sub></i> is the <i>n<sub>i</sub>×</i>1 vector of all other endogenous variables which enter the <i>i</i><sup>th</sup> equation on the right-hand side, and <i>u<sub>it</sub></i> are the error terms. The “−<i>i</i>” notation indicates that the vector <i>y<sub>−i,t</sub></i> may contain any of the <i>y</i>’s except for <i>y<sub>it</sub></i> (since it is already present on the left-hand side). The regression coefficients <i>β<sub>i</sub></i> and <i>γ<sub>i</sub></i> are of dimensions <i>k<sub>i</sub>×</i>1 and <i>n<sub>i</sub>×</i>1 correspondingly. Vertically stacking the <i>T</i> observations corresponding to the <i>i</i><sup>th</sup> equation, we can write each equation in vector form as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}=Y_{-i}\gamma _{i}+X_{i}\beta _{i}+u_{i},\quad i=1,\ldots ,m,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}=Y_{-i}\gamma _{i}+X_{i}\beta _{i}+u_{i},\quad i=1,\ldots ,m,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d7108bd64327616645c72d9d187b3c3f00d0343" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.405ex; height:2.676ex;" alt="{\displaystyle y_{i}=Y_{-i}\gamma _{i}+X_{i}\beta _{i}+u_{i},\quad i=1,\ldots ,m,}"></span></dd></dl> <p>where <i>y<sub>i</sub></i> and <i>u<sub>i</sub></i> are <i>T×</i>1 vectors, <i>X<sub>i</sub></i> is a <i>T×k<sub>i</sub></i> matrix of exogenous regressors, and <i>Y<sub>−i</sub></i> is a <i>T×n<sub>i</sub></i> matrix of endogenous regressors on the right-hand side of the <i>i</i><sup>th</sup> equation. Finally, we can move all endogenous variables to the left-hand side and write the <i>m</i> equations jointly in vector form as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\Gamma =X\mathrm {B} +U.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo>+</mo> <mi>U</mi> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\Gamma =X\mathrm {B} +U.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3e0f1443582c8c2041c1b1142f14b0dedb20b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.607ex; height:2.343ex;" alt="{\displaystyle Y\Gamma =X\mathrm {B} +U.\,}"></span></dd></dl> <p>This representation is known as the <b>structural form</b>. In this equation <span class="nowrap"><i>Y</i> = [<i>y</i><sub>1</sub> <i>y</i><sub>2</sub> ... <i>y<sub>m</sub></i>]</span> is the <i>T×m</i> matrix of dependent variables. Each of the matrices <i>Y<sub>−i</sub></i> is in fact an <i>n<sub>i</sub></i>-columned submatrix of this <i>Y</i>. The <i>m×m</i> matrix Γ, which describes the relation between the dependent variables, has a complicated structure. It has ones on the diagonal, and all other elements of each column <i>i</i> are either the components of the vector <i>−γ<sub>i</sub></i> or zeros, depending on which columns of <i>Y</i> were included in the matrix <i>Y<sub>−i</sub></i>. The <i>T×k</i> matrix <i>X</i> contains all exogenous regressors from all equations, but without repetitions (that is, matrix <i>X</i> should be of full rank). Thus, each <i>X<sub>i</sub></i> is a <i>k<sub>i</sub></i>-columned submatrix of <i>X</i>. Matrix Β has size <i>k×m</i>, and each of its columns consists of the components of vectors <i>β<sub>i</sub></i> and zeros, depending on which of the regressors from <i>X</i> were included or excluded from <i>X<sub>i</sub></i>. Finally, <span class="nowrap"><i>U</i> = [<i>u</i><sub>1</sub> <i>u</i><sub>2</sub> ... <i>u<sub>m</sub></i>]</span> is a <i>T×m</i> matrix of the error terms. </p><p>Postmultiplying the structural equation by <span class="nowrap">Γ<sup> −1</sup></span>, the system can be written in the <b><a href="/wiki/Reduced_form" title="Reduced form">reduced form</a></b> as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=X\mathrm {B} \Gamma ^{-1}+U\Gamma ^{-1}=X\Pi +V.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <msup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mi>U</mi> <msup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>X</mi> <mi mathvariant="normal">Π</mi> <mo>+</mo> <mi>V</mi> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=X\mathrm {B} \Gamma ^{-1}+U\Gamma ^{-1}=X\Pi +V.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/478192a50e31df5a25b05b27676990cf5da12717" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:33.175ex; height:2.843ex;" alt="{\displaystyle Y=X\mathrm {B} \Gamma ^{-1}+U\Gamma ^{-1}=X\Pi +V.\,}"></span></dd></dl> <p>This is already a simple <a href="/wiki/General_linear_model" title="General linear model">general linear model</a>, and it can be estimated for example by <a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">ordinary least squares</a>. Unfortunately, the task of decomposing the estimated matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\hat {\Pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Π</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\hat {\Pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72243ca9100a026c2b225ef09bbe22293791454e" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.338ex; width:1.233ex; height:2.176ex;" aria-hidden="true" alt="{\displaystyle \scriptstyle {\hat {\Pi }}}"></span> into the individual factors Β and <span class="nowrap">Γ<sup> −1</sup></span> is quite complicated, and therefore the reduced form is more suitable for prediction but not inference. </p> <div class="mw-heading mw-heading3"><h3 id="Assumptions">Assumptions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simultaneous_equations_model&action=edit&section=2" title="Edit section: Assumptions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Firstly, the rank of the matrix <i>X</i> of exogenous regressors must be equal to <i>k</i>, both in finite samples and in the limit as <span class="nowrap"><i>T</i> → ∞</span> (this later requirement means that in the limit the expression <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\frac {1}{T}}X'\!X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msup> <mi>X</mi> <mo>′</mo> </msup> <mspace width="negativethinmathspace" /> <mi>X</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\frac {1}{T}}X'\!X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/796740c73f7d145e81b29b8eba25cce8855a0364" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -1.171ex; width:4.732ex; height:3.176ex;" aria-hidden="true" alt="{\displaystyle \scriptstyle {\frac {1}{T}}X'\!X}"></span> should converge to a nondegenerate <i>k×k</i> matrix). Matrix Γ is also assumed to be non-degenerate. </p><p>Secondly, error terms are assumed to be serially <a href="/wiki/Independent_and_identically_distributed" class="mw-redirect" title="Independent and identically distributed">independent and identically distributed</a>. That is, if the <i>t</i><sup>th</sup> row of matrix <i>U</i> is denoted by <i>u</i><sub>(<i>t</i>)</sub>, then the sequence of vectors {<i>u</i><sub>(<i>t</i>)</sub>} should be iid, with zero mean and some covariance matrix Σ (which is unknown). In particular, this implies that <span class="nowrap">E[<i>U</i>] = 0</span>, and <span class="nowrap">E[<i>U′U</i>] = <i>T</i> Σ</span>. </p><p>Lastly, assumptions are required for identification. </p> <div class="mw-heading mw-heading2"><h2 id="Identification">Identification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simultaneous_equations_model&action=edit&section=3" title="Edit section: Identification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Identification_condition" class="mw-redirect" title="Identification condition">identification conditions</a> require that the <a href="/wiki/System_of_linear_equations" title="System of linear equations">system of linear equations</a> be solvable for the unknown parameters. </p><p>More specifically, the <i>order condition</i>, a necessary condition for identification, is that for each equation <span class="texhtml"><i>k<sub>i</sub> + n<sub>i</sub> ≤ k</i></span>, which can be phrased as “the number of excluded exogenous variables is greater or equal to the number of included endogenous variables”. </p><p>The <i>rank condition</i>, a stronger condition which is necessary and sufficient, is that the <a href="/wiki/Rank_(linear_algebra)" title="Rank (linear algebra)">rank</a> of <span class="texhtml">Π<sub><i>i</i>0</sub></span> equals <span class="texhtml"><i>n<sub>i</sub></i></span>, where <span class="texhtml">Π<sub><i>i</i>0</sub></span> is a <span class="texhtml">(<i>k − k<sub>i</sub></i>)×<i>n<sub>i</sub></i></span> matrix which is obtained from <span class="texhtml"><i>Π</i></span> by crossing out those columns which correspond to the excluded endogenous variables, and those rows which correspond to the included exogenous variables. </p> <div class="mw-heading mw-heading3"><h3 id="Using_cross-equation_restrictions_to_achieve_identification">Using cross-equation restrictions to achieve identification</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simultaneous_equations_model&action=edit&section=4" title="Edit section: Using cross-equation restrictions to achieve identification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In simultaneous equations models, the most common method to achieve <a href="/wiki/Parameter_identification_problem" title="Parameter identification problem">identification</a> is by imposing within-equation parameter restrictions.<sup id="cite_ref-Woolridge_6-0" class="reference"><a href="#cite_note-Woolridge-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Yet, identification is also possible using cross equation restrictions. </p><p>To illustrate how cross equation restrictions can be used for identification, consider the following example from Wooldridge<sup id="cite_ref-Woolridge_6-1" class="reference"><a href="#cite_note-Woolridge-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><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}y_{1}&=\gamma _{12}y_{2}+\delta _{11}z_{1}+\delta _{12}z_{2}+\delta _{13}z_{3}+u_{1}\\y_{2}&=\gamma _{21}y_{1}+\delta _{21}z_{1}+\delta _{22}z_{2}+u_{2}\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> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y_{1}&=\gamma _{12}y_{2}+\delta _{11}z_{1}+\delta _{12}z_{2}+\delta _{13}z_{3}+u_{1}\\y_{2}&=\gamma _{21}y_{1}+\delta _{21}z_{1}+\delta _{22}z_{2}+u_{2}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7287c92bd205aa44c3aa3a543f40e893e7fe57fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.195ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}y_{1}&=\gamma _{12}y_{2}+\delta _{11}z_{1}+\delta _{12}z_{2}+\delta _{13}z_{3}+u_{1}\\y_{2}&=\gamma _{21}y_{1}+\delta _{21}z_{1}+\delta _{22}z_{2}+u_{2}\end{aligned}}}"></span></dd></dl> <p>where z's are uncorrelated with u's and y's are <a href="/wiki/Endogenous_variable" class="mw-redirect" title="Endogenous variable">endogenous</a> variables. Without further restrictions, the first equation is not identified because there is no excluded exogenous variable. The second equation is just identified if <span class="texhtml"><i>δ</i><sub>13</sub>≠0</span>, which is assumed to be true for the rest of discussion. </p><p>Now we impose the cross equation restriction of <span class="texhtml"><i>δ</i><sub>12</sub>=<i>δ</i><sub>22</sub></span>. Since the second equation is identified, we can treat <span class="texhtml"><i>δ</i><sub>12</sub></span> as known for the purpose of identification. Then, the first equation becomes: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{1}-\delta _{12}z_{2}=\gamma _{12}y_{2}+\delta _{11}z_{1}+\delta _{13}z_{3}+u_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{1}-\delta _{12}z_{2}=\gamma _{12}y_{2}+\delta _{11}z_{1}+\delta _{13}z_{3}+u_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56a924d28f95e12a292df618843ef9c09aebef6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.443ex; height:2.843ex;" alt="{\displaystyle y_{1}-\delta _{12}z_{2}=\gamma _{12}y_{2}+\delta _{11}z_{1}+\delta _{13}z_{3}+u_{1}}"></span></dd></dl> <p>Then, we can use <span class="texhtml">(<i>z</i><sub>1</sub>, <i>z</i><sub>2</sub>, <i>z</i><sub>3</sub>)</span> as <a href="/wiki/Instrumental_variable" class="mw-redirect" title="Instrumental variable">instruments</a> to estimate the coefficients in the above equation since there are one endogenous variable (<span class="texhtml"><i>y</i><sub>2</sub></span>) and one excluded exogenous variable (<span class="texhtml"><i>z</i><sub>2</sub></span>) on the right hand side. Therefore, cross equation restrictions in place of within-equation restrictions can achieve identification. </p> <div class="mw-heading mw-heading2"><h2 id="Estimation">Estimation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simultaneous_equations_model&action=edit&section=5" title="Edit section: Estimation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Two-stage_least_squares_(2SLS)"><span id="Two-stage_least_squares_.282SLS.29"></span>Two-stage least squares (2SLS)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simultaneous_equations_model&action=edit&section=6" title="Edit section: Two-stage least squares (2SLS)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The simplest and the most common estimation method for the simultaneous equations model is the so-called <a href="/wiki/Two-stage_least_squares" class="mw-redirect" title="Two-stage least squares">two-stage least squares</a> method,<sup id="cite_ref-Greene_2003_loc=p._399_7-0" class="reference"><a href="#cite_note-Greene_2003_loc=p._399-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> developed independently by <a href="#CITEREFTheil1953">Theil (1953)</a> and <a href="#CITEREFBasmann1957">Basmann (1957)</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> It is an equation-by-equation technique, where the endogenous regressors on the right-hand side of each equation are being instrumented with the regressors <i>X</i> from all other equations. The method is called “two-stage” because it conducts estimation in two steps:<sup id="cite_ref-Greene_2003_loc=p._399_7-1" class="reference"><a href="#cite_note-Greene_2003_loc=p._399-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><i>Step 1</i>: Regress <i>Y<sub>−i</sub></i> on <i>X</i> and obtain the predicted values <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\hat {Y}}_{\!-i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace" /> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\hat {Y}}_{\!-i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b6d49fcf4badb3033058da5198491e578c41883" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.671ex; width:2.53ex; height:2.509ex;" aria-hidden="true" alt="{\displaystyle \scriptstyle {\hat {Y}}_{\!-i}}"></span>;</dd> <dd><i>Step 2</i>: Estimate <i>γ<sub>i</sub></i>, <i>β<sub>i</sub></i> by the <a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">ordinary least squares</a> regression of <i>y<sub>i</sub></i> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\hat {Y}}_{\!-i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace" /> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\hat {Y}}_{\!-i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b6d49fcf4badb3033058da5198491e578c41883" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.671ex; width:2.53ex; height:2.509ex;" aria-hidden="true" alt="{\displaystyle \scriptstyle {\hat {Y}}_{\!-i}}"></span> and <i>X<sub>i</sub></i>.</dd></dl> <p>If the <i>i</i><sup>th</sup> equation in the model is written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}={\begin{pmatrix}Y_{-i}&X_{i}\end{pmatrix}}{\begin{pmatrix}\gamma _{i}\\\beta _{i}\end{pmatrix}}+u_{i}\equiv Z_{i}\delta _{i}+u_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≡</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}={\begin{pmatrix}Y_{-i}&X_{i}\end{pmatrix}}{\begin{pmatrix}\gamma _{i}\\\beta _{i}\end{pmatrix}}+u_{i}\equiv Z_{i}\delta _{i}+u_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e4d75ea9db46b7b0ecf16bea2035c779b71e153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.266ex; height:6.176ex;" alt="{\displaystyle y_{i}={\begin{pmatrix}Y_{-i}&X_{i}\end{pmatrix}}{\begin{pmatrix}\gamma _{i}\\\beta _{i}\end{pmatrix}}+u_{i}\equiv Z_{i}\delta _{i}+u_{i},}"></span></dd></dl> <p>where <i>Z<sub>i</sub></i> is a <i>T×</i>(<i>n<sub>i</sub> + k<sub>i</sub></i>) matrix of both endogenous and exogenous regressors in the <i>i</i><sup>th</sup> equation, and <i>δ<sub>i</sub></i> is an (<i>n<sub>i</sub> + k<sub>i</sub></i>)-dimensional vector of regression coefficients, then the 2SLS estimator of <i>δ<sub>i</sub></i> will be given by<sup id="cite_ref-Greene_2003_loc=p._399_7-2" class="reference"><a href="#cite_note-Greene_2003_loc=p._399-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\delta }}_{i}={\big (}{\hat {Z}}'_{i}{\hat {Z}}_{i}{\big )}^{-1}{\hat {Z}}'_{i}y_{i}={\big (}Z'_{i}PZ_{i}{\big )}^{-1}Z'_{i}Py_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>δ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msubsup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <mi>P</mi> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msubsup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <mi>P</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\delta }}_{i}={\big (}{\hat {Z}}'_{i}{\hat {Z}}_{i}{\big )}^{-1}{\hat {Z}}'_{i}y_{i}={\big (}Z'_{i}PZ_{i}{\big )}^{-1}Z'_{i}Py_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa3dbe84c8387336df71fdab4e189e13dea29073" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.853ex; height:3.843ex;" alt="{\displaystyle {\hat {\delta }}_{i}={\big (}{\hat {Z}}'_{i}{\hat {Z}}_{i}{\big )}^{-1}{\hat {Z}}'_{i}y_{i}={\big (}Z'_{i}PZ_{i}{\big )}^{-1}Z'_{i}Py_{i},}"></span></dd></dl> <p>where <span class="nowrap"><i>P</i> = <i>X</i> (<i>X</i> ′<i>X</i>)<sup>−1</sup><i>X</i> ′</span> is the projection matrix onto the linear space spanned by the exogenous regressors <i>X</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Indirect_least_squares">Indirect least squares</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simultaneous_equations_model&action=edit&section=7" title="Edit section: Indirect least squares"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Indirect least squares is an approach in <a href="/wiki/Econometrics" title="Econometrics">econometrics</a> where the <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> in a simultaneous equations model are estimated from the <a href="/wiki/Reduced_form" title="Reduced form">reduced form</a> model using <a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">ordinary least squares</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> For this, the structural system of equations is transformed into the reduced form first. Once the coefficients are estimated the model is put back into the structural form. </p> <div class="mw-heading mw-heading3"><h3 id="Limited_information_maximum_likelihood_(LIML)"><span id="Limited_information_maximum_likelihood_.28LIML.29"></span>Limited information maximum likelihood (LIML)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simultaneous_equations_model&action=edit&section=8" title="Edit section: Limited information maximum likelihood (LIML)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The “limited information” maximum likelihood method was suggested by <a href="/wiki/Meyer_Abraham_Girshick" title="Meyer Abraham Girshick">M. A. Girshick</a> in 1947,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> and formalized by <a href="/wiki/Theodore_Wilbur_Anderson" title="Theodore Wilbur Anderson">T. W. Anderson</a> and <a href="/w/index.php?title=Herman_Rubin&action=edit&redlink=1" class="new" title="Herman Rubin (page does not exist)">H. Rubin</a> in 1949.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> It is used when one is interested in estimating a single structural equation at a time (hence its name of limited information), say for observation i: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}=Y_{-i}\gamma _{i}+X_{i}\beta _{i}+u_{i}\equiv Z_{i}\delta _{i}+u_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≡</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}=Y_{-i}\gamma _{i}+X_{i}\beta _{i}+u_{i}\equiv Z_{i}\delta _{i}+u_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e0a966ce7b0c131a15cb7577eda136e94dbecc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.407ex; height:2.843ex;" alt="{\displaystyle y_{i}=Y_{-i}\gamma _{i}+X_{i}\beta _{i}+u_{i}\equiv Z_{i}\delta _{i}+u_{i}}"></span></dd></dl> <p>The structural equations for the remaining endogenous variables Y<sub>−i</sub> are not specified, and they are given in their reduced form: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{-i}=X\Pi +U_{-i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>X</mi> <mi mathvariant="normal">Π</mi> <mo>+</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{-i}=X\Pi +U_{-i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18aa49402fa9c16b82835c099d86496162da99f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.756ex; height:2.509ex;" alt="{\displaystyle Y_{-i}=X\Pi +U_{-i}}"></span></dd></dl> <p>Notation in this context is different than for the simple <a href="/wiki/Instrumental_variable" class="mw-redirect" title="Instrumental variable">IV</a> case. One has: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{-i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{-i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4883fa33be5c232e0abaeab4a7f43ed710794410" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.429ex; height:2.509ex;" alt="{\displaystyle Y_{-i}}"></span>: The endogenous variable(s).</li> <li><span class="mwe-math-element"><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_{-i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{-i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e426c1d5e159d30dc69e2e260018f9bf099fb67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.002ex; height:2.509ex;" alt="{\displaystyle X_{-i}}"></span>: The exogenous variable(s)</li> <li><span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>: The instrument(s) (often denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span>)</li></ul> <p>The explicit formula for the LIML is:<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\delta }}_{i}={\Big (}Z'_{i}(I-\lambda M)Z_{i}{\Big )}^{\!-1}Z'_{i}(I-\lambda M)y_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>δ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <msubsup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>I</mi> <mo>−</mo> <mi>λ</mi> <mi>M</mi> <mo stretchy="false">)</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace" /> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msubsup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>I</mi> <mo>−</mo> <mi>λ</mi> <mi>M</mi> <mo stretchy="false">)</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\delta }}_{i}={\Big (}Z'_{i}(I-\lambda M)Z_{i}{\Big )}^{\!-1}Z'_{i}(I-\lambda M)y_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01656dd69172935f949dc07da3c580b3c4c6418b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.918ex; height:5.176ex;" alt="{\displaystyle {\hat {\delta }}_{i}={\Big (}Z'_{i}(I-\lambda M)Z_{i}{\Big )}^{\!-1}Z'_{i}(I-\lambda M)y_{i},}"></span></dd></dl> <p>where <span class="nowrap"><i>M</i> = <i>I − X</i> (<i>X</i> ′<i>X</i>)<sup>−1</sup><i>X</i> ′</span>, and <i>λ</i> is the smallest characteristic root of the matrix: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Big (}{\begin{bmatrix}y_{i}\\Y_{-i}\end{bmatrix}}M_{i}{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}{\Big )}{\Big (}{\begin{bmatrix}y_{i}\\Y_{-i}\end{bmatrix}}M{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}{\Big )}^{\!-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace" /> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Big (}{\begin{bmatrix}y_{i}\\Y_{-i}\end{bmatrix}}M_{i}{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}{\Big )}{\Big (}{\begin{bmatrix}y_{i}\\Y_{-i}\end{bmatrix}}M{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}{\Big )}^{\!-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fbff566aacf3c74578934c3c65bc52b4690be3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.224ex; margin-bottom: -0.281ex; width:45.734ex; height:6.176ex;" alt="{\displaystyle {\Big (}{\begin{bmatrix}y_{i}\\Y_{-i}\end{bmatrix}}M_{i}{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}{\Big )}{\Big (}{\begin{bmatrix}y_{i}\\Y_{-i}\end{bmatrix}}M{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}{\Big )}^{\!-1}}"></span></dd></dl> <p>where, in a similar way, <span class="nowrap"><i>M<sub>i</sub></i> = <i>I − X<sub>i</sub></i> (<i>X<sub>i</sub></i>′<i>X<sub>i</sub></i>)<sup>−1</sup><i>X<sub>i</sub></i>′</span>. </p><p>In other words, <i>λ</i> is the smallest solution of the <a href="/wiki/Generalized_eigenvalue_problem#Generalized_eigenvalue_problem" class="mw-redirect" title="Generalized eigenvalue problem">generalized eigenvalue problem</a>, see <a href="#CITEREFTheil1971">Theil (1971</a>, p. 503): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Big |}{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}'M_{i}{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}-\lambda {\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}'M{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}{\Big |}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>′</mo> </msup> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>−</mo> <mi>λ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>′</mo> </msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Big |}{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}'M_{i}{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}-\lambda {\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}'M{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}{\Big |}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2d9bca90d63d9dcddf87452f52c16093ffb41a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:55.557ex; height:4.176ex;" alt="{\displaystyle {\Big |}{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}'M_{i}{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}-\lambda {\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}'M{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}{\Big |}=0}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="K_class_estimators">K class estimators</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simultaneous_equations_model&action=edit&section=9" title="Edit section: K class estimators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The LIML is a special case of the K-class estimators:<sup id="cite_ref-DavidsonMacKinnon649_16-0" class="reference"><a href="#cite_note-DavidsonMacKinnon649-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\delta }}={\Big (}Z'(I-\kappa M)Z{\Big )}^{\!-1}Z'(I-\kappa M)y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>δ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <msup> <mi>Z</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>I</mi> <mo>−</mo> <mi>κ</mi> <mi>M</mi> <mo stretchy="false">)</mo> <mi>Z</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace" /> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>Z</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>I</mi> <mo>−</mo> <mi>κ</mi> <mi>M</mi> <mo stretchy="false">)</mo> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\delta }}={\Big (}Z'(I-\kappa M)Z{\Big )}^{\!-1}Z'(I-\kappa M)y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82753c97ae528ea11622dfd6ff006742e6e2f891" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.596ex; height:5.176ex;" alt="{\displaystyle {\hat {\delta }}={\Big (}Z'(I-\kappa M)Z{\Big )}^{\!-1}Z'(I-\kappa M)y,}"></span></dd></dl> <p>with: </p> <ul><li><span class="mwe-math-element"><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 ={\begin{bmatrix}\beta _{i}&\gamma _{i}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta ={\begin{bmatrix}\beta _{i}&\gamma _{i}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95db08ff60bed1e423e8bf92c6436b1a5e55f24a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.634ex; height:2.843ex;" alt="{\displaystyle \delta ={\begin{bmatrix}\beta _{i}&\gamma _{i}\end{bmatrix}}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z={\begin{bmatrix}X_{i}&Y_{-i}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z={\begin{bmatrix}X_{i}&Y_{-i}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd4d3fc65660d8492c77a2774cfa671e844bf25e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.591ex; margin-bottom: -0.247ex; width:15.299ex; height:2.843ex;" alt="{\displaystyle Z={\begin{bmatrix}X_{i}&Y_{-i}\end{bmatrix}}}"></span></li></ul> <p>Several estimators belong to this class: </p> <ul><li>κ=0: <a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">OLS</a></li> <li>κ=1: 2SLS. Note indeed that in this case, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I-\kappa M=I-M=P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>−</mo> <mi>κ</mi> <mi>M</mi> <mo>=</mo> <mi>I</mi> <mo>−</mo> <mi>M</mi> <mo>=</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I-\kappa M=I-M=P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06c956fb3cc60eb95a673737131e93eb4fe1ab68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:22.19ex; height:2.343ex;" alt="{\displaystyle I-\kappa M=I-M=P}"></span> the usual projection matrix of the 2SLS</li> <li>κ=λ: LIML</li> <li>κ=λ - α / (n-K): <a href="#CITEREFFuller1977">Fuller (1977)</a> estimator.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Here K represents the number of instruments, n the sample size, and α a positive constant to specify. A value of α=1 will yield an estimator that is approximately unbiased.<sup id="cite_ref-DavidsonMacKinnon649_16-1" class="reference"><a href="#cite_note-DavidsonMacKinnon649-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Three-stage_least_squares_(3SLS)"><span id="Three-stage_least_squares_.283SLS.29"></span>Three-stage least squares (3SLS)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simultaneous_equations_model&action=edit&section=10" title="Edit section: Three-stage least squares (3SLS)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The three-stage least squares estimator was introduced by <a href="#CITEREFZellnerTheil1962">Zellner & Theil (1962)</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> It can be seen as a special case of multi-equation <a href="/wiki/Generalized_method_of_moments" title="Generalized method of moments">GMM</a> where the set of <a href="/wiki/Instrumental_variable" class="mw-redirect" title="Instrumental variable">instrumental variables</a> is common to all equations.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> If all regressors are in fact predetermined, then 3SLS reduces to <a href="/wiki/Seemingly_unrelated_regressions" title="Seemingly unrelated regressions">seemingly unrelated regressions</a> (SUR). Thus it may also be seen as a combination of <a href="/wiki/2SLS" class="mw-redirect" title="2SLS">two-stage least squares</a> (2SLS) with SUR. </p> <div class="mw-heading mw-heading2"><h2 id="Applications_in_social_science">Applications in social science</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simultaneous_equations_model&action=edit&section=11" title="Edit section: Applications in social science"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Across fields and disciplines simultaneous equation models are applied to various observational phenomena. These equations are applied when phenomena are assumed to be reciprocally causal. The classic example is supply and demand in <a href="/wiki/Economics" title="Economics">economics</a>. In other disciplines there are examples such as candidate evaluations and party identification<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> or public opinion and social policy in <a href="/wiki/Political_science" title="Political science">political science</a>;<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> road investment and travel demand in geography;<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> and educational attainment and parenthood entry in <a href="/wiki/Sociology" title="Sociology">sociology</a> or <a href="/wiki/Demography" title="Demography">demography</a>.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> The simultaneous equation model requires a theory of reciprocal causality that includes special features if the causal effects are to be estimated as simultaneous feedback as opposed to one-sided 'blocks' of an equation where a researcher is interested in the causal effect of X on Y while holding the causal effect of Y on X constant, or when the researcher knows the exact amount of time it takes for each causal effect to take place, i.e., the length of the causal lags. Instead of lagged effects, simultaneous feedback means estimating the simultaneous and perpetual impact of X and Y on each other. This requires a theory that causal effects are simultaneous in time, or so complex that they appear to behave simultaneously; a common example are the moods of roommates.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> To estimate simultaneous feedback models a theory of equilibrium is also necessary – that X and Y are in relatively steady states or are part of a system (society, market, classroom) that is in a relatively stable state.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simultaneous_equations_model&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li> <li><a href="/wiki/Seemingly_unrelated_regressions" title="Seemingly unrelated regressions">Seemingly unrelated regressions</a></li> <li><a href="/wiki/Reduced_form" title="Reduced form">Reduced form</a></li> <li><a href="/wiki/Parameter_identification_problem" title="Parameter identification problem">Parameter identification problem</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simultaneous_equations_model&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMartinHurnHarris2013" class="citation book cs1">Martin, Vance; Hurn, Stan; Harris, David (2013). <i>Econometric Modelling with Time Series</i>. Cambridge University Press. p. 159. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-19660-4" title="Special:BookSources/978-0-521-19660-4"><bdi>978-0-521-19660-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Econometric+Modelling+with+Time+Series&rft.pages=159&rft.pub=Cambridge+University+Press&rft.date=2013&rft.isbn=978-0-521-19660-4&rft.aulast=Martin&rft.aufirst=Vance&rft.au=Hurn%2C+Stan&rft.au=Harris%2C+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaddalaLahiri2009" class="citation book cs1">Maddala, G. S.; Lahiri, Kajal (2009). <i>Introduction to Econometrics</i> (Fourth ed.). Wiley. pp. 355–357. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-01512-4" title="Special:BookSources/978-0-470-01512-4"><bdi>978-0-470-01512-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Econometrics&rft.pages=355-357&rft.edition=Fourth&rft.pub=Wiley&rft.date=2009&rft.isbn=978-0-470-01512-4&rft.aulast=Maddala&rft.aufirst=G.+S.&rft.au=Lahiri%2C+Kajal&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQuandt1983" class="citation book cs1">Quandt, Richard E. (1983). "Computational Problems and Methods". In Griliches, Z.; Intriligator, M. D. (eds.). <i>Handbook of Econometrics</i>. Vol. I. North-Holland. pp. 699–764. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-86185-8" title="Special:BookSources/0-444-86185-8"><bdi>0-444-86185-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Computational+Problems+and+Methods&rft.btitle=Handbook+of+Econometrics&rft.pages=699-764&rft.pub=North-Holland&rft.date=1983&rft.isbn=0-444-86185-8&rft.aulast=Quandt&rft.aufirst=Richard+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChrist1994" class="citation journal cs1">Christ, Carl F. (1994). "The Cowles Commission's Contributions to Econometrics at Chicago, 1939–1955". <i><a href="/wiki/Journal_of_Economic_Literature" title="Journal of Economic Literature">Journal of Economic Literature</a></i>. <b>32</b> (1): 30–59. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2728422">2728422</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Economic+Literature&rft.atitle=The+Cowles+Commission%27s+Contributions+to+Econometrics+at+Chicago%2C+1939%E2%80%931955&rft.volume=32&rft.issue=1&rft.pages=30-59&rft.date=1994&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2728422%23id-name%3DJSTOR&rft.aulast=Christ&rft.aufirst=Carl+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnston1971" class="citation book cs1"><a href="/wiki/John_Johnston_(econometrician)" title="John Johnston (econometrician)">Johnston, J.</a> (1971). "Simultaneous-equation Methods: Estimation". <i>Econometric Methods</i> (Second ed.). New York: McGraw-Hill. pp. 376–423. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-032679-7" title="Special:BookSources/0-07-032679-7"><bdi>0-07-032679-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Simultaneous-equation+Methods%3A+Estimation&rft.btitle=Econometric+Methods&rft.place=New+York&rft.pages=376-423&rft.edition=Second&rft.pub=McGraw-Hill&rft.date=1971&rft.isbn=0-07-032679-7&rft.aulast=Johnston&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-Woolridge-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Woolridge_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Woolridge_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Wooldridge, J.M., Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass.</span> </li> <li id="cite_note-Greene_2003_loc=p._399-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Greene_2003_loc=p._399_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Greene_2003_loc=p._399_7-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Greene_2003_loc=p._399_7-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreene2002" class="citation book cs1">Greene, William H. (2002). <i>Econometric analysis</i> (5th ed.). Prentice Hall. pp. 398–99. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-066189-9" title="Special:BookSources/0-13-066189-9"><bdi>0-13-066189-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Econometric+analysis&rft.pages=398-99&rft.edition=5th&rft.pub=Prentice+Hall&rft.date=2002&rft.isbn=0-13-066189-9&rft.aulast=Greene&rft.aufirst=William+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTheil1953" class="citation report cs1">Theil, H. (1953). Estimation and Simultaneous Correlation in Complete Equation Systems (Memorandum). Central Planning Bureau.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=Estimation+and+Simultaneous+Correlation+in+Complete+Equation+Systems&rft.pub=Central+Planning+Bureau&rft.date=1953&rft.aulast=Theil&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span> Reprinted in <i>Henri Theil’s Contributions to Economics and Econometrics</i> (Springer, 1992), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-94-011-2546-8_6">10.1007/978-94-011-2546-8_6</a>.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBasmann1957" class="citation journal cs1"><a href="/wiki/Robert_Basmann" title="Robert Basmann">Basmann, R. L.</a> (1957). "A generalized classical method of linear estimation of coefficients in a structural equation". <i><a href="/wiki/Econometrica" title="Econometrica">Econometrica</a></i>. <b>25</b> (1): 77–83. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1907743">10.2307/1907743</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1907743">1907743</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Econometrica&rft.atitle=A+generalized+classical+method+of+linear+estimation+of+coefficients+in+a+structural+equation&rft.volume=25&rft.issue=1&rft.pages=77-83&rft.date=1957&rft_id=info%3Adoi%2F10.2307%2F1907743&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1907743%23id-name%3DJSTOR&rft.aulast=Basmann&rft.aufirst=R.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTheil1971" class="citation book cs1"><a href="/wiki/Henri_Theil" title="Henri Theil">Theil, Henri</a> (1971). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/principlesofecon0000thei"><i>Principles of Econometrics</i></a></span>. New York: John Wiley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Econometrics&rft.place=New+York&rft.pub=John+Wiley&rft.date=1971&rft.aulast=Theil&rft.aufirst=Henri&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprinciplesofecon0000thei&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Park, S-B. (1974) "On Indirect Least Squares Estimation of a Simultaneous Equation System", <i>The Canadian Journal of Statistics / La Revue Canadienne de Statistique</i>, 2 (1), 75–82 <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3314964">3314964</a></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVajdaValkoGodfrey1987" class="citation journal cs1">Vajda, S.; Valko, P.; Godfrey, K.R. (1987). "Direct and indirect least squares methods in continuous-time parameter estimation". <i>Automatica</i>. <b>23</b> (6): 707–718. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0005-1098%2887%2990027-6">10.1016/0005-1098(87)90027-6</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Automatica&rft.atitle=Direct+and+indirect+least+squares+methods+in+continuous-time+parameter+estimation&rft.volume=23&rft.issue=6&rft.pages=707-718&rft.date=1987&rft_id=info%3Adoi%2F10.1016%2F0005-1098%2887%2990027-6&rft.aulast=Vajda&rft.aufirst=S.&rft.au=Valko%2C+P.&rft.au=Godfrey%2C+K.R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">First application by <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGirshickHaavelmo1947" class="citation journal cs1">Girshick, M. A.; Haavelmo, Trygve (1947). "Statistical Analysis of the Demand for Food: Examples of Simultaneous Estimation of Structural Equations". <i><a href="/wiki/Econometrica" title="Econometrica">Econometrica</a></i>. <b>15</b> (2): 79–110. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1907066">10.2307/1907066</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1907066">1907066</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Econometrica&rft.atitle=Statistical+Analysis+of+the+Demand+for+Food%3A+Examples+of+Simultaneous+Estimation+of+Structural+Equations&rft.volume=15&rft.issue=2&rft.pages=79-110&rft.date=1947&rft_id=info%3Adoi%2F10.2307%2F1907066&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1907066%23id-name%3DJSTOR&rft.aulast=Girshick&rft.aufirst=M.+A.&rft.au=Haavelmo%2C+Trygve&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAndersonRubin1949" class="citation journal cs1">Anderson, T.W.; Rubin, H. (1949). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177730090">"Estimator of the parameters of a single equation in a complete system of stochastic equations"</a>. <i><a href="/wiki/Annals_of_Mathematical_Statistics" title="Annals of Mathematical Statistics">Annals of Mathematical Statistics</a></i>. <b>20</b> (1): 46–63. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177730090">10.1214/aoms/1177730090</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2236803">2236803</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematical+Statistics&rft.atitle=Estimator+of+the+parameters+of+a+single+equation+in+a+complete+system+of+stochastic+equations&rft.volume=20&rft.issue=1&rft.pages=46-63&rft.date=1949&rft_id=info%3Adoi%2F10.1214%2Faoms%2F1177730090&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2236803%23id-name%3DJSTOR&rft.aulast=Anderson&rft.aufirst=T.W.&rft.au=Rubin%2C+H.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faoms%252F1177730090&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAmemiya1985" class="citation book cs1">Amemiya, Takeshi (1985). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/advancedeconomet00amem"><i>Advanced Econometrics</i></a></span>. Cambridge, Massachusetts: Harvard University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/advancedeconomet00amem/page/235">235</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-674-00560-0" title="Special:BookSources/0-674-00560-0"><bdi>0-674-00560-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Econometrics&rft.place=Cambridge%2C+Massachusetts&rft.pages=235&rft.pub=Harvard+University+Press&rft.date=1985&rft.isbn=0-674-00560-0&rft.aulast=Amemiya&rft.aufirst=Takeshi&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fadvancedeconomet00amem&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-DavidsonMacKinnon649-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-DavidsonMacKinnon649_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-DavidsonMacKinnon649_16-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavidsonMacKinnon1993" class="citation book cs1">Davidson, Russell; MacKinnon, James G. (1993). <i>Estimation and inference in econometrics</i>. Oxford University Press. p. 649. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-506011-3" title="Special:BookSources/0-19-506011-3"><bdi>0-19-506011-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Estimation+and+inference+in+econometrics&rft.pages=649&rft.pub=Oxford+University+Press&rft.date=1993&rft.isbn=0-19-506011-3&rft.aulast=Davidson&rft.aufirst=Russell&rft.au=MacKinnon%2C+James+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFuller1977" class="citation journal cs1"><a href="/wiki/Wayne_Fuller" title="Wayne Fuller">Fuller, Wayne</a> (1977). "Some Properties of a Modification of the Limited Information Estimator". <i>Econometrica</i>. <b>45</b> (4): 939–953. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1912683">10.2307/1912683</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1912683">1912683</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Econometrica&rft.atitle=Some+Properties+of+a+Modification+of+the+Limited+Information+Estimator&rft.volume=45&rft.issue=4&rft.pages=939-953&rft.date=1977&rft_id=info%3Adoi%2F10.2307%2F1912683&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1912683%23id-name%3DJSTOR&rft.aulast=Fuller&rft.aufirst=Wayne&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZellnerTheil1962" class="citation journal cs1"><a href="/wiki/Arnold_Zellner" title="Arnold Zellner">Zellner, Arnold</a>; <a href="/wiki/Henri_Theil" title="Henri Theil">Theil, Henri</a> (1962). "Three-stage least squares: simultaneous estimation of simultaneous equations". <i>Econometrica</i>. <b>30</b> (1): 54–78. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1911287">10.2307/1911287</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1911287">1911287</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Econometrica&rft.atitle=Three-stage+least+squares%3A+simultaneous+estimation+of+simultaneous+equations&rft.volume=30&rft.issue=1&rft.pages=54-78&rft.date=1962&rft_id=info%3Adoi%2F10.2307%2F1911287&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1911287%23id-name%3DJSTOR&rft.aulast=Zellner&rft.aufirst=Arnold&rft.au=Theil%2C+Henri&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKmenta1986" class="citation book cs1">Kmenta, Jan (1986). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Bxq7AAAAIAAJ&pg=PA695">"System Methods of Estimation"</a>. <i>Elements of Econometrics</i> (Second ed.). New York: Macmillan. pp. 695–701. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780023650703" title="Special:BookSources/9780023650703"><bdi>9780023650703</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=System+Methods+of+Estimation&rft.btitle=Elements+of+Econometrics&rft.place=New+York&rft.pages=695-701&rft.edition=Second&rft.pub=Macmillan&rft.date=1986&rft.isbn=9780023650703&rft.aulast=Kmenta&rft.aufirst=Jan&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBxq7AAAAIAAJ%26pg%3DPA695&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHayashi2000" class="citation book cs1">Hayashi, Fumio (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QyIW8WUIyzcC&pg=PA276">"Multiple-Equation GMM"</a>. <i>Econometrics</i>. Princeton University Press. pp. 276–279. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1400823838" title="Special:BookSources/1400823838"><bdi>1400823838</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Multiple-Equation+GMM&rft.btitle=Econometrics&rft.pages=276-279&rft.pub=Princeton+University+Press&rft.date=2000&rft.isbn=1400823838&rft.aulast=Hayashi&rft.aufirst=Fumio&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQyIW8WUIyzcC%26pg%3DPA276&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPageJones1979" class="citation journal cs1">Page, Benjamin I.; Jones, Calvin C. (1979-12-01). "Reciprocal Effects of Policy Preferences, Party Loyalties and the Vote". <i>American Political Science Review</i>. <b>73</b> (4): 1071–1089. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1953990">10.2307/1953990</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-0554">0003-0554</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1953990">1953990</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:144984371">144984371</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Political+Science+Review&rft.atitle=Reciprocal+Effects+of+Policy+Preferences%2C+Party+Loyalties+and+the+Vote&rft.volume=73&rft.issue=4&rft.pages=1071-1089&rft.date=1979-12-01&rft.issn=0003-0554&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A144984371%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1953990%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F1953990&rft.aulast=Page&rft.aufirst=Benjamin+I.&rft.au=Jones%2C+Calvin+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWlezien1995" class="citation journal cs1">Wlezien, Christopher (1995-01-01). "The Public as Thermostat: Dynamics of Preferences for Spending". <i>American Journal of Political Science</i>. <b>39</b> (4): 981–1000. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2111666">10.2307/2111666</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2111666">2111666</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Political+Science&rft.atitle=The+Public+as+Thermostat%3A+Dynamics+of+Preferences+for+Spending&rft.volume=39&rft.issue=4&rft.pages=981-1000&rft.date=1995-01-01&rft_id=info%3Adoi%2F10.2307%2F2111666&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2111666%23id-name%3DJSTOR&rft.aulast=Wlezien&rft.aufirst=Christopher&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBreznau2016" class="citation journal cs1">Breznau, Nate (2016-07-01). <a rel="nofollow" class="external text" href="http://osf.io/wt376/">"Positive Returns and Equilibrium: Simultaneous Feedback Between Public Opinion and Social Policy"</a>. <i>Policy Studies Journal</i>. <b>45</b> (4): 583–612. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fpsj.12171">10.1111/psj.12171</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1541-0072">1541-0072</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Policy+Studies+Journal&rft.atitle=Positive+Returns+and+Equilibrium%3A+Simultaneous+Feedback+Between+Public+Opinion+and+Social+Policy&rft.volume=45&rft.issue=4&rft.pages=583-612&rft.date=2016-07-01&rft_id=info%3Adoi%2F10.1111%2Fpsj.12171&rft.issn=1541-0072&rft.aulast=Breznau&rft.aufirst=Nate&rft_id=http%3A%2F%2Fosf.io%2Fwt376%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFXieLevinson2010" class="citation journal cs1">Xie, F.; Levinson, D. (2010-05-01). "How streetcars shaped suburbanization: a Granger causality analysis of land use and transit in the Twin Cities". <i>Journal of Economic Geography</i>. <b>10</b> (3): 453–470. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fjeg%2Flbp031">10.1093/jeg/lbp031</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/11299%2F179996">11299/179996</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1468-2702">1468-2702</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Economic+Geography&rft.atitle=How+streetcars+shaped+suburbanization%3A+a+Granger+causality+analysis+of+land+use+and+transit+in+the+Twin+Cities&rft.volume=10&rft.issue=3&rft.pages=453-470&rft.date=2010-05-01&rft_id=info%3Ahdl%2F11299%2F179996&rft.issn=1468-2702&rft_id=info%3Adoi%2F10.1093%2Fjeg%2Flbp031&rft.aulast=Xie&rft.aufirst=F.&rft.au=Levinson%2C+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarini1984" class="citation journal cs1">Marini, Margaret Mooney (1984-01-01). "Women's Educational Attainment and the Timing of Entry into Parenthood". <i>American Sociological Review</i>. <b>49</b> (4): 491–511. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2095464">10.2307/2095464</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2095464">2095464</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Sociological+Review&rft.atitle=Women%27s+Educational+Attainment+and+the+Timing+of+Entry+into+Parenthood&rft.volume=49&rft.issue=4&rft.pages=491-511&rft.date=1984-01-01&rft_id=info%3Adoi%2F10.2307%2F2095464&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2095464%23id-name%3DJSTOR&rft.aulast=Marini&rft.aufirst=Margaret+Mooney&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWongLaw1999" class="citation journal cs1">Wong, Chi-Sum; Law, Kenneth S. (1999-01-01). "Testing Reciprocal Relations by Nonrecursive Structuralequation Models Using Cross-Sectional Data". <i>Organizational Research Methods</i>. <b>2</b> (1): 69–87. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F109442819921005">10.1177/109442819921005</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1094-4281">1094-4281</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122284566">122284566</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Organizational+Research+Methods&rft.atitle=Testing+Reciprocal+Relations+by+Nonrecursive+Structuralequation+Models+Using+Cross-Sectional+Data&rft.volume=2&rft.issue=1&rft.pages=69-87&rft.date=1999-01-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122284566%23id-name%3DS2CID&rft.issn=1094-4281&rft_id=info%3Adoi%2F10.1177%2F109442819921005&rft.aulast=Wong&rft.aufirst=Chi-Sum&rft.au=Law%2C+Kenneth+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">2013. “Reverse Arrow Dynamics: Feedback Loops and Formative Measurement.” In <i>Structural Equation Modeling: A Second Course</i>, edited by <a href="/wiki/Gregory_R._Hancock" title="Gregory R. Hancock">Gregory R. Hancock</a> and Ralph O. Mueller, 2nd ed., 41–79. Charlotte, NC: Information Age Publishing</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simultaneous_equations_model&action=edit&section=14" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAsteriouHall2011" class="citation book cs1">Asteriou, Dimitrios; Hall, Stephen G. (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6qYcBQAAQBAJ&pg=PA395"><i>Applied Econometrics</i></a> (Second ed.). Basingstoke: Palgrave Macmillan. p. 395. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-230-27182-1" title="Special:BookSources/978-0-230-27182-1"><bdi>978-0-230-27182-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Econometrics&rft.place=Basingstoke&rft.pages=395&rft.edition=Second&rft.pub=Palgrave+Macmillan&rft.date=2011&rft.isbn=978-0-230-27182-1&rft.aulast=Asteriou&rft.aufirst=Dimitrios&rft.au=Hall%2C+Stephen+G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6qYcBQAAQBAJ%26pg%3DPA395&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChow1983" class="citation book cs1"><a href="/wiki/Gregory_Chow" title="Gregory Chow">Chow, Gregory C.</a> (1983). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/econometrics0000chow/page/117"><i>Econometrics</i></a></span>. New York: McGraw-Hill. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/econometrics0000chow/page/117">117–121</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-010847-1" title="Special:BookSources/0-07-010847-1"><bdi>0-07-010847-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Econometrics&rft.place=New+York&rft.pages=117-121&rft.pub=McGraw-Hill&rft.date=1983&rft.isbn=0-07-010847-1&rft.aulast=Chow&rft.aufirst=Gregory+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Feconometrics0000chow%2Fpage%2F117&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFombyHillJohnson1984" class="citation book cs1">Fomby, Thomas B.; Hill, R. Carter; Johnson, Stanley R. (1984). "Simultaneous Equations Models". <i>Advanced Econometric Methods</i>. New York: Springer. pp. 437–552. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90908-7" title="Special:BookSources/0-387-90908-7"><bdi>0-387-90908-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Simultaneous+Equations+Models&rft.btitle=Advanced+Econometric+Methods&rft.place=New+York&rft.pages=437-552&rft.pub=Springer&rft.date=1984&rft.isbn=0-387-90908-7&rft.aulast=Fomby&rft.aufirst=Thomas+B.&rft.au=Hill%2C+R.+Carter&rft.au=Johnson%2C+Stanley+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaddalaLahiri2009" class="citation book cs1"><a href="/wiki/G._S._Maddala" title="G. S. Maddala">Maddala, G. S.</a>; Lahiri, Kajal (2009). "Simultaneous Equations Models". <i>Introduction to Econometrics</i> (Fourth ed.). New York: Wiley. pp. 355–400. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-01512-4" title="Special:BookSources/978-0-470-01512-4"><bdi>978-0-470-01512-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Simultaneous+Equations+Models&rft.btitle=Introduction+to+Econometrics&rft.place=New+York&rft.pages=355-400&rft.edition=Fourth&rft.pub=Wiley&rft.date=2009&rft.isbn=978-0-470-01512-4&rft.aulast=Maddala&rft.aufirst=G.+S.&rft.au=Lahiri%2C+Kajal&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRuud2000" class="citation book cs1">Ruud, Paul A. (2000). "Simultaneous Equations". <i>An Introduction to Classical Econometric Theory</i>. Oxford University Press. pp. 697–746. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-511164-8" title="Special:BookSources/0-19-511164-8"><bdi>0-19-511164-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Simultaneous+Equations&rft.btitle=An+Introduction+to+Classical+Econometric+Theory&rft.pages=697-746&rft.pub=Oxford+University+Press&rft.date=2000&rft.isbn=0-19-511164-8&rft.aulast=Ruud&rft.aufirst=Paul+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSargan1988" class="citation book cs1"><a href="/wiki/Denis_Sargan" title="Denis Sargan">Sargan, Denis</a> (1988). <i>Lectures on Advanced Econometric Theory</i>. Oxford: Basil Blackwell. pp. 68–89. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-631-14956-2" title="Special:BookSources/0-631-14956-2"><bdi>0-631-14956-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Advanced+Econometric+Theory&rft.place=Oxford&rft.pages=68-89&rft.pub=Basil+Blackwell&rft.date=1988&rft.isbn=0-631-14956-2&rft.aulast=Sargan&rft.aufirst=Denis&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWooldridge2013" class="citation book cs1"><a href="/wiki/Jeffrey_Wooldridge" title="Jeffrey Wooldridge">Wooldridge, Jeffrey M.</a> (2013). "Simultaneous Equations Models". <i>Introductory Econometrics</i> (Fifth ed.). South-Western. pp. 554–582. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-111-53104-1" title="Special:BookSources/978-1-111-53104-1"><bdi>978-1-111-53104-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Simultaneous+Equations+Models&rft.btitle=Introductory+Econometrics&rft.pages=554-582&rft.edition=Fifth&rft.pub=South-Western&rft.date=2013&rft.isbn=978-1-111-53104-1&rft.aulast=Wooldridge&rft.aufirst=Jeffrey+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimultaneous+equations+model" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simultaneous_equations_model&action=edit&section=15" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=D5lt9bhOshc&list=PLD15D38DC7AA3B737&index=15"><span class="plainlinks">Lecture on the Identification Problem in 2SLS, and Estimation</span></a> on <a href="/wiki/YouTube_video_(identifier)" class="mw-redirect" title="YouTube video (identifier)">YouTube</a> by <a href="/wiki/Mark_Thoma" title="Mark Thoma">Mark Thoma</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Simultaneous_equations_model&oldid=1231034568">https://en.wikipedia.org/w/index.php?title=Simultaneous_equations_model&oldid=1231034568</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Simultaneous_equation_methods_(econometrics)" title="Category:Simultaneous equation methods (econometrics)">Simultaneous equation methods (econometrics)</a></li><li><a href="/wiki/Category:Regression_models" title="Category:Regression models">Regression models</a></li><li><a href="/wiki/Category:Mathematical_and_quantitative_methods_(economics)" title="Category:Mathematical and quantitative methods (economics)">Mathematical and quantitative methods (economics)</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 26 June 2024, at 02:35<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Simultaneous equations model - Wikipedia