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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Control function (econometrics)</span></span>

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<p><b>Control functions</b> (also known as <b>two-stage residual inclusion</b>) are statistical methods to correct for <a href="Endogeneity_(econometrics)" title="Endogeneity (econometrics)">endogeneity</a> problems by modelling the endogeneity in the <a href="Errors_and_residuals" title="Errors and residuals">error term</a>. The approach thereby differs in important ways from other models that try to account for the same <a href="Econometric" class="mw-redirect" title="Econometric">econometric</a> problem. <a href="Instrumental_variable" class="mw-redirect" title="Instrumental variable">Instrumental variables</a>, for example, attempt to model the endogenous variable <i>X</i> as an often <a href="Invertible" class="mw-redirect" title="Invertible">invertible</a> model with respect to a relevant and <a href="Exogenous" class="mw-redirect" title="Exogenous">exogenous</a> instrument <i>Z</i>. <a href="Panel_analysis" title="Panel analysis">Panel analysis</a> uses special data properties to difference out unobserved heterogeneity that is assumed to be fixed over time.

</p><p>Control functions were introduced by <a href="James_Heckman" title="James Heckman">Heckman</a> and Robb<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> although the principle can be traced back to earlier papers.<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> A particular reason why they are popular is because they work for non-invertible models (such as <a href="Discrete_choice_model" class="mw-redirect" title="Discrete choice model">discrete choice models</a>) and allow for <a href="Heterogeneous" class="mw-redirect" title="Heterogeneous">heterogeneous</a> effects, where effects at the individual level can differ from effects at the aggregate.<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> A well-known example of the control function approach is the <a href="Heckman_correction" title="Heckman correction">Heckman correction</a>.

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<div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2></div>

<p>Assume we start from a standard endogenous variable setup with additive errors, where <i>X</i> is an endogenous variable, and <i>Z</i> is an exogenous variable that can serve as an instrument.

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</style><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=g(X)+U}">

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</math></span><img src="./52091e1b724569d319aac3df292a6271e755146c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.4ex; height:2.843ex;" alt="{\displaystyle Y=g(X)+U}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span></td></tr></tbody></table>

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<annotation encoding="application/x-tex">{\displaystyle X=\pi (Z)+V}</annotation>

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</math></span><img src="./0ecebfd4b53e39beb7e003b472f49c4e44a3fa20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.528ex; height:2.843ex;" alt="{\displaystyle X=\pi (Z)+V}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span></td></tr></tbody></table>

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<annotation encoding="application/x-tex">{\displaystyle E[U\mid Z,V]=E[U\mid V]}</annotation>

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</math></span><img src="./937f56c9ec07beacaee7e524ed734fe34db3e336.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.965ex; height:2.843ex;" alt="{\displaystyle E[U\mid Z,V]=E[U\mid V]}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span></td></tr></tbody></table>

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<annotation encoding="application/x-tex">{\displaystyle E[V\mid Z]=0}</annotation>

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</math></span><img src="./35969fe055934f12e1769cfb39a0a7bb2ff5322f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.735ex; height:2.843ex;" alt="{\displaystyle E[V\mid Z]=0}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_4" class="reference nourlexpansion" style="font-weight:bold;">4</span></td></tr></tbody></table>

<p>A popular instrumental variable approach is to use a two-step procedure and estimate equation (<b><a href="#math_2">2</a></b>) first and then use the estimates of this first step to estimate equation (<b><a href="#math_1">1</a></b>) in a second step. The control function, however, uses that this model implies

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<annotation encoding="application/x-tex">{\displaystyle E[Y\mid Z,V]=g(X)+E[U\mid Z,V]=g(X)+E[U\mid V]=g(X)+h(V)}</annotation>

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</math></span><img src="./bdf999b3a086f86f240188c5671e1b5c0de392b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:68.616ex; height:2.843ex;" alt="{\displaystyle E[Y\mid Z,V]=g(X)+E[U\mid Z,V]=g(X)+E[U\mid V]=g(X)+h(V)}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_5" class="reference nourlexpansion" style="font-weight:bold;">5</span></td></tr></tbody></table>

<p>The function <i>h</i>(<i>V</i>) is effectively the control function that models the endogeneity and where this econometric approach lends its name from.<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>

</p><p>In a <a href="Rubin_causal_model" title="Rubin causal model">Rubin causal model</a> potential outcomes framework, where <i>Y</i><sub>1</sub> is the outcome variable of people for who the participation indicator <i>D</i> equals 1, the control function approach leads to the following model

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<annotation encoding="application/x-tex">{\displaystyle E[Y_{1}\mid X,Z,D=1]=\mu _{1}(X)+E[U\mid D=1]}</annotation>

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</math></span><img src="./71b7ec26c52286b209abf302835f70337c8ac660.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.483ex; height:2.843ex;" alt="{\displaystyle E[Y_{1}\mid X,Z,D=1]=\mu _{1}(X)+E[U\mid D=1]}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_6" class="reference nourlexpansion" style="font-weight:bold;">6</span></td></tr></tbody></table>

<p>as long as the potential outcomes <i>Y</i><sub>0</sub> and <i>Y</i><sub>1</sub> are independent of <i>D</i> conditional on <i>X</i> and <i>Z</i>.<sup id="cite_ref-HeckmanVytlacil2007_5-0" class="reference"><a href="#cite_note-HeckmanVytlacil2007-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Variance_correction">Variance correction</h2></div>

<p>Since the second-stage regression includes <a href="Generated_regressor" title="Generated regressor">generated regressors</a>, its variance-covariance matrix needs to be adjusted.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Endogeneity_in_Poisson_regression">Endogeneity in Poisson regression</h3></div>

<p>Wooldridge and Terza provide a methodology to both deal with and test for endogeneity within the exponential regression framework, which the following discussion follows closely.<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> While the example focuses on a <a href="Poisson_regression" title="Poisson regression">Poisson regression</a> model, it is possible to generalize to other exponential regression models, although this may come at the cost of additional assumptions (e.g. for binary response or censored data models).

</p><p>Assume the following exponential regression model, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation>

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</math></span><img src="./0bc77764b2e74e64a63341054fa90f3e07db275f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}" loading="lazy"></span> is an unobserved term in the latent variable. We allow for correlation between <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation>

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</math></span><img src="./0bc77764b2e74e64a63341054fa90f3e07db275f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

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</math></span><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"></span> (implying <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

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</math></span><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"></span> is possibly endogenous), but allow for no such correlation between <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation>

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</math></span><img src="./0bc77764b2e74e64a63341054fa90f3e07db275f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle z_{i}}</annotation>

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</math></span><img src="./5c6e920bac39ad09fff4efef16254595091a1025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.881ex; height:2.009ex;" alt="{\displaystyle z_{i}}" loading="lazy"></span>.

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [y_{i}\mid x_{i},z_{i},a_{i}]=\exp(x_{i}b_{0}+z_{i}c_{0}+a_{i})}</annotation>

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</math></span><img src="./8ff01de7f917789d3637c2304b07103fea678810.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.154ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [y_{i}\mid x_{i},z_{i},a_{i}]=\exp(x_{i}b_{0}+z_{i}c_{0}+a_{i})}" loading="lazy"></span></dd></dl>

<p>The variables <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle z_{i}}</annotation>

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</math></span><img src="./5c6e920bac39ad09fff4efef16254595091a1025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.881ex; height:2.009ex;" alt="{\displaystyle z_{i}}" loading="lazy"></span> serve as instrumental variables for the potentially endogenous <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

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</math></span><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"></span>. One can assume a linear relationship between these two variables or alternatively project the endogenous variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

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</math></span><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"></span> onto the instruments to get the following reduced form equation:

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<annotation encoding="application/x-tex">{\displaystyle x_{i}=z_{i}\Pi +v_{i}}</annotation>

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</math></span><img src="./4b2cf27dd393cbd6fd4500fa20a0348c87157b8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.619ex; height:2.509ex;" alt="{\displaystyle x_{i}=z_{i}\Pi +v_{i}}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span></td></tr></tbody></table>

<p>The usual rank condition is needed to ensure identification. The endogeneity is then modeled in the following way, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math></span><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"></span> determines the severity of endogeneity and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{i}}</annotation>

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</math></span><img src="./7dffe5726650f6daac54829972a94f38eb8ec127.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i}}" loading="lazy"></span> is assumed to be independent of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{i}}</annotation>

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</math></span><img src="./ebdc3a9cb1583d3204eff8918b558c293e0d2cf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle e_{i}}" loading="lazy"></span>.

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<annotation encoding="application/x-tex">{\displaystyle a_{i}=v_{i}\rho +e_{i}}</annotation>

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</math></span><img src="./4df20e18273005399ff2377cc5eeb75cc0c50ec3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.981ex; height:2.509ex;" alt="{\displaystyle a_{i}=v_{i}\rho +e_{i}}" loading="lazy"></span></dd></dl>

<p>Imposing these assumptions, assuming the models are correctly specified, and normalizing <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [\exp(e_{i})]=1}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [\exp(e_{i})]=1}</annotation>

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</math></span><img src="./3912b3e85e13bb100eafd60edab35a891aeb502a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.382ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [\exp(e_{i})]=1}" loading="lazy"></span>, we can rewrite the conditional mean as follows:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [y_{i}\mid x_{i},z_{i},v_{i}]=\exp(x_{i}b_{0}+z_{i}c_{0}+v_{i}\rho )}</annotation>

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</math></span><img src="./431ba1ffd3b32996c73b63b194356ac439489e46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.151ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [y_{i}\mid x_{i},z_{i},v_{i}]=\exp(x_{i}b_{0}+z_{i}c_{0}+v_{i}\rho )}" loading="lazy"></span></td> <td></td> <td class="nowrap"><span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span></td></tr></tbody></table>

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<annotation encoding="application/x-tex">{\displaystyle v_{i}}</annotation>

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</math></span><img src="./7dffe5726650f6daac54829972a94f38eb8ec127.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i}}" loading="lazy"></span> were known at this point, it would be possible to estimate the relevant parameters by <a href="Quasi-maximum_likelihood_estimation" class="mw-redirect" title="Quasi-maximum likelihood estimation">quasi-maximum likelihood estimation</a> (QMLE). Following the two step procedure strategies, Wooldridge and Terza propose estimating equation (<b><a href="#math_1">1</a></b>) by <a href="Ordinary_least_squares" title="Ordinary least squares">ordinary least squares</a>. The fitted residuals from this regression can then be plugged into the estimating equation (<b><a href="#math_2">2</a></b>) and QMLE methods will lead to consistent estimators of the parameters of interest. Significance tests on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\rho }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\rho }}}</annotation>

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</math></span><img src="./a71ec0653c9ec1cad5e168085772c88e293fedef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.376ex; height:2.676ex;" alt="{\displaystyle {\hat {\rho }}}" loading="lazy"></span> can then be used to test for endogeneity within the model.

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<div class="mw-heading mw-heading2"><h2 id="Extensions">Extensions</h2></div>

<p>The original Heckit procedure makes <a href="Distributional_assumption" class="mw-redirect" title="Distributional assumption">distributional assumptions</a> about the error terms, however, more flexible estimation approaches with weaker distributional assumptions have been established.<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> Furthermore, Blundell and Powell show how the control function approach can be particularly helpful in models with nonadditive errors, such as discrete choice models.<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> This latter approach, however, does implicitly make strong distributional and functional form assumptions.<sup id="cite_ref-HeckmanVytlacil2007_5-1" class="reference"><a href="#cite_note-HeckmanVytlacil2007-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="Two-stage_least_squares" class="mw-redirect" title="Two-stage least squares">Two-stage least squares</a> – Technique in statistics<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>

<li><a href="Heckman_correction" title="Heckman correction">Heckman correction</a> – Statistical technique correcting sampling bias</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFHeckmanRobb1985" class="citation journal cs1"><a href="James_Heckman" title="James Heckman">Heckman, James J.</a>; Robb, Richard (1985). "Alternative methods for evaluating the impact of interventions". <i>Journal of Econometrics</i>. <b>30</b> (<span class="nowrap">1–</span>2). Elsevier BV: <span class="nowrap">239–</span>267. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0304-4076%2885%2990139-3">10.1016/0304-4076(85)90139-3</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0304-4076">0304-4076</a>.</cite></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"><cite id="CITEREFTelser1964" class="citation journal cs1"><a href="Lester_G._Telser" title="Lester G. Telser">Telser, L. G.</a> (1964). "Iterative Estimation of a Set of Linear Regression Equations". <i><a href="Journal_of_the_American_Statistical_Association" title="Journal of the American Statistical Association">Journal of the American Statistical Association</a></i>. <b>59</b> (307): <span class="nowrap">845–</span>862. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F01621459.1964.10480731">10.1080/01621459.1964.10480731</a>.</cite></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"><cite id="CITEREFArellano2008" class="citation web cs1"><a href="Manuel_Arellano" title="Manuel Arellano">Arellano, M.</a> (2008). <a rel="nofollow" class="external text" href="https://www.cemfi.es/~arellano/binary-endogeneity.pdf">"Binary Models with Endogenous Explanatory Variables"</a> <span class="cs1-format">(PDF)</span>. <i>Class notes</i>.</cite></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">Arellano, M. (2003): Endogeneity and Instruments in Nonparametric Models. Comments to papers by Darolles, Florens & Renault; and Blundell & Powell. Advances in Economics and Econometrics, Theory and Applications, Eight World Congress. Volume II, ed. by M. Dewatripont, L.P. Hansen, and S.J. Turnovsky. Cambridge University Press, Cambridge.</span>

</li>

<li id="cite_note-HeckmanVytlacil2007-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-HeckmanVytlacil2007_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-HeckmanVytlacil2007_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Heckman, J. J., and E. J. Vytlacil (2007): Econometric Evaluation of Social Programs, Part II: Using the Marginal Treatment Effect to Organize Alternative Econometric Estimators to Evaluate Social Programs, and to Forecast the Effects in New Environments. Handbook of Econometrics, Vol 6, ed. by J. J. Heckman and E. E. Leamer. North Holland.</span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFMurphyTopel1985" class="citation journal cs1"><a href="Kevin_M._Murphy" title="Kevin M. Murphy">Murphy, Kevin M.</a>; Topel, Robert H. (1985). "Estimation and Inference in Two-Step Econometric Models". <i>Journal of Business & Economic Statistics</i>. <b>3</b> (4): <span class="nowrap">370–</span>379. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F07350015.1985.10509471">10.1080/07350015.1985.10509471</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1391724">1391724</a>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFGauger1989" class="citation journal cs1">Gauger, Jean (1989). "The Generated Regressor Correction: Impacts Upon Inferences in Hypothesis Testing". <i><a href="Journal_of_Macroeconomics" title="Journal of Macroeconomics">Journal of Macroeconomics</a></i>. <b>11</b> (3): <span class="nowrap">383–</span>395. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0164-0704%2889%2990065-7">10.1016/0164-0704(89)90065-7</a>.</cite></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">Wooldridge 1997, pp. 382–383; Terza 1998</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"><cite id="CITEREFMatzkin2003" class="citation journal cs1"><a href="Rosa_Matzkin" title="Rosa Matzkin">Matzkin, R. L.</a> (2003). <a rel="nofollow" class="external text" href="https://webacademicos.udesa.edu.ar/pub/econ/doc38.pdf">"Nonparametric Estimation of Nonadditive Random Functions"</a> <span class="cs1-format">(PDF)</span>. <i><a href="Econometrica" title="Econometrica">Econometrica</a></i>. <b>71</b> (5): <span class="nowrap">1339–</span>1375. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2F1468-0262.00452">10.1111/1468-0262.00452</a>. <a href="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/10908%2F409">10908/409</a></span>.</cite></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">Blundell, R., and J. L. Powell (2003): Endogeneity in Nonparametric and Semiparametric Regression Models. Advances in Economics and Econometrics, Theory and Applications, Eight World Congress. Volume II, ed. by M. Dewatripont, L.P. Hansen, and S.J. Turnovsky. Cambridge University Press, Cambridge.</span>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFGuoSmall2016" class="citation journal cs1">Guo, Zijian; <a href="Dylan_S._Small" class="mw-redirect" title="Dylan S. Small">Small, Dylan S.</a> (2016). <a rel="nofollow" class="external text" href="http://www.jmlr.org/papers/v17/14-379.html">"Control Function Instrumental Variable Estimation of Nonlinear Causal Effect Models"</a>. <i><a href="Journal_of_Machine_Learning_Research" title="Journal of Machine Learning Research">Journal of Machine Learning Research</a></i>. <b>17</b> (100): <span class="nowrap">1–</span>35. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1602.01051">1602.01051</a></span>.</cite></li>

<li><cite id="CITEREFWooldridge2015" class="citation journal cs1"><a href="Jeffrey_Wooldridge" title="Jeffrey Wooldridge">Wooldridge, Jeffrey M.</a> (2015). "Control Function Methods in Applied Econometrics". <i><a href="Journal_of_Human_Resources" class="mw-redirect" title="Journal of Human Resources">Journal of Human Resources</a></i>. <b>50</b> (2): <span class="nowrap">420–</span>445. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3368%2Fjhr.50.2.420">10.3368/jhr.50.2.420</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119604644">119604644</a>.</cite></li></ul></div><div><div class="zim-footer">

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