Fixed effects model

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href="#Fixed_effects_estimator"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Fixed effects estimator</span> </div> </a> <ul id="toc-Fixed_effects_estimator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-First_difference_estimator" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#First_difference_estimator"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>First difference estimator</span> </div> </a> <ul id="toc-First_difference_estimator-sublist" class="vector-toc-list"> <li id="toc-Equality_of_fixed_effects_and_first_difference_estimators_when_T=2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Equality_of_fixed_effects_and_first_difference_estimators_when_T=2"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Equality of fixed effects and first difference estimators when T=2</span> </div> </a> <ul id="toc-Equality_of_fixed_effects_and_first_difference_estimators_when_T=2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Chamberlain_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Chamberlain_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Chamberlain method</span> </div> </a> <ul id="toc-Chamberlain_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hausman–Taylor_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hausman–Taylor_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Hausman–Taylor method</span> </div> </a> <ul id="toc-Hausman–Taylor_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalization_with_input_uncertainty" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalization_with_input_uncertainty"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Generalization with input uncertainty</span> </div> </a> <ul id="toc-Generalization_with_input_uncertainty-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Use_to_test_for_consistency" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Use_to_test_for_consistency"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Use to test for consistency</span> </div> </a> <ul id="toc-Use_to_test_for_consistency-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-Notes" class="vector-toc-list-item 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.sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></th></tr><tr><th class="sidebar-heading"> Models</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></li> <li><a href="/wiki/Simple_linear_regression" title="Simple linear regression">Simple regression</a></li> <li><a href="/wiki/Polynomial_regression" title="Polynomial regression">Polynomial regression</a></li> <li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></li> <li><a href="/wiki/Vector_generalized_linear_model" title="Vector generalized linear model">Vector generalized linear model</a></li> <li><a href="/wiki/Discrete_choice" title="Discrete choice">Discrete choice</a></li> <li><a href="/wiki/Binomial_regression" title="Binomial regression">Binomial regression</a></li> <li><a href="/wiki/Binary_regression" title="Binary regression">Binary regression</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic regression</a></li> <li><a href="/wiki/Multinomial_logistic_regression" title="Multinomial logistic regression">Multinomial logistic regression</a></li> <li><a href="/wiki/Mixed_logit" title="Mixed logit">Mixed logit</a></li> <li><a href="/wiki/Probit_model" title="Probit model">Probit</a></li> <li><a href="/wiki/Multinomial_probit" title="Multinomial probit">Multinomial probit</a></li> <li><a href="/wiki/Ordered_logit" title="Ordered logit">Ordered logit</a></li> <li><a href="/wiki/Ordered_probit" class="mw-redirect" title="Ordered probit">Ordered probit</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Poisson</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Multilevel_model" title="Multilevel model">Multilevel model</a></li> <li><a class="mw-selflink selflink">Fixed effects</a></li> <li><a href="/wiki/Random_effects_model" title="Random effects model">Random effects</a></li> <li><a href="/wiki/Mixed_model" title="Mixed model">Linear mixed-effects model</a></li> <li><a href="/wiki/Nonlinear_mixed-effects_model" title="Nonlinear mixed-effects model">Nonlinear mixed-effects model</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li> <li><a href="/wiki/Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li> <li><a href="/wiki/Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li> <li><a href="/wiki/Robust_regression" title="Robust regression">Robust</a></li> <li><a href="/wiki/Quantile_regression" title="Quantile regression">Quantile</a></li> <li><a href="/wiki/Isotonic_regression" title="Isotonic regression">Isotonic</a></li> <li><a href="/wiki/Principal_component_regression" title="Principal component regression">Principal components</a></li> <li><a href="/wiki/Least-angle_regression" title="Least-angle regression">Least angle</a></li> <li><a href="/wiki/Local_regression" title="Local regression">Local</a></li> <li><a href="/wiki/Segmented_regression" title="Segmented regression">Segmented</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Errors-in-variables_models" title="Errors-in-variables models">Errors-in-variables</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Estimation</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Least_squares" title="Least squares">Least squares</a></li> <li><a href="/wiki/Linear_least_squares" title="Linear least squares">Linear</a></li> <li><a href="/wiki/Non-linear_least_squares" title="Non-linear least squares">Non-linear</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary</a></li> <li><a href="/wiki/Weighted_least_squares" title="Weighted least squares">Weighted</a></li> <li><a href="/wiki/Generalized_least_squares" title="Generalized least squares">Generalized</a></li> <li><a href="/wiki/Generalized_estimating_equation" title="Generalized estimating equation">Generalized estimating equation</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Partial_least_squares_regression" title="Partial least squares regression">Partial</a></li> <li><a href="/wiki/Total_least_squares" title="Total least squares">Total</a></li> <li><a href="/wiki/Non-negative_least_squares" title="Non-negative least squares">Non-negative</a></li> <li><a href="/wiki/Tikhonov_regularization" class="mw-redirect" title="Tikhonov regularization">Ridge regression</a></li> <li><a href="/wiki/Regularized_least_squares" title="Regularized least squares">Regularized</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Least_absolute_deviations" title="Least absolute deviations">Least absolute deviations</a></li> <li><a href="/wiki/Iteratively_reweighted_least_squares" title="Iteratively reweighted least squares">Iteratively reweighted</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian</a></li> <li><a href="/wiki/Bayesian_multivariate_linear_regression" title="Bayesian multivariate linear regression">Bayesian multivariate</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Background</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Regression_validation" title="Regression validation">Regression validation</a></li> <li><a href="/wiki/Mean_and_predicted_response" class="mw-redirect" title="Mean and predicted response">Mean and predicted response</a></li> <li><a href="/wiki/Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li> <li><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></li> <li><a href="/wiki/Studentized_residual" title="Studentized residual">Studentized residual</a></li> <li><a href="/wiki/Gauss%E2%80%93Markov_theorem" title="Gauss–Markov theorem">Gauss–Markov theorem</a></li></ul></td> </tr><tr><td class="sidebar-below"> <ul><li><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, 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a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Regression_bar" title="Template:Regression bar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Regression_bar" title="Template talk:Regression bar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Regression_bar" title="Special:EditPage/Template:Regression bar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, a <b>fixed effects model</b> is a <a href="/wiki/Statistical_model" title="Statistical model">statistical model</a> in which the model <a href="/wiki/Parameter" title="Parameter">parameters</a> are fixed or non-random quantities. This is in contrast to <a href="/wiki/Random_effects_model" title="Random effects model">random effects models</a> and <a href="/wiki/Mixed_model" title="Mixed model">mixed models</a> in which all or some of the model parameters are random variables. In many applications including <a href="/wiki/Econometrics" title="Econometrics">econometrics</a><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> and <a href="/wiki/Biostatistics" title="Biostatistics">biostatistics</a><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><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><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><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><sup id="cite_ref-Gomes2022_6-0" class="reference"><a href="#cite_note-Gomes2022-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> a fixed effects model refers to a <a href="/wiki/Regression_model" class="mw-redirect" title="Regression model">regression model</a> in which the group means are fixed (non-random) as opposed to a random effects model in which the group means are a random sample from a population.<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><sup id="cite_ref-Gomes2022_6-1" class="reference"><a href="#cite_note-Gomes2022-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Generally, data can be grouped according to several observed factors. The group means could be modeled as fixed or random effects for each grouping. In a fixed effects model each group mean is a group-specific fixed quantity. </p><p>In <a href="/wiki/Panel_data" title="Panel data">panel data</a> where longitudinal observations exist for the same subject, fixed effects represent the subject-specific means. In <a href="/wiki/Panel_analysis" title="Panel analysis">panel data analysis</a> the term <b>fixed effects estimator</b> (also known as the <b>within estimator</b>) is used to refer to an <a href="/wiki/Estimator" title="Estimator">estimator</a> for the <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> in the regression model including those fixed effects (one time-invariant intercept for each subject). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Qualitative_description">Qualitative description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fixed_effects_model&action=edit&section=1" title="Edit section: Qualitative description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Such models assist in controlling for <a href="/wiki/Omitted_variable_bias" class="mw-redirect" title="Omitted variable bias">omitted variable bias</a> due to unobserved heterogeneity when this heterogeneity is constant over time. This heterogeneity can be removed from the data through differencing, for example by subtracting the group-level average over time, or by taking a <a href="/wiki/First_difference" class="mw-redirect" title="First difference">first difference</a> which will remove any time invariant components of the model. </p><p>There are two common assumptions made about the individual specific effect: the random effects assumption and the fixed effects assumption. The <a href="/wiki/Random_effects_model" title="Random effects model">random effects</a> assumption is that the individual-specific effects are uncorrelated with the independent variables. The fixed effect assumption is that the individual-specific effects are correlated with the independent variables. If the random effects assumption holds, the random effects estimator is more <a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">efficient</a> than the fixed effects estimator. However, if this assumption does not hold, the random effects estimator is not <a href="/wiki/Consistent_estimator" title="Consistent estimator">consistent</a>. The <a href="/wiki/Durbin%E2%80%93Wu%E2%80%93Hausman_test" title="Durbin–Wu–Hausman test">Durbin–Wu–Hausman test</a> is often used to discriminate between the fixed and the random effects models.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Formal_model_and_assumptions">Formal model and assumptions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fixed_effects_model&action=edit&section=2" title="Edit section: Formal model and assumptions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the linear unobserved effects model for <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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> observations and <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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> time periods: </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}=X_{it}\mathbf {\beta } +\alpha _{i}+u_{it}}"> <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> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mo>+</mo> <msub> <mi>α</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{it}=X_{it}\mathbf {\beta } +\alpha _{i}+u_{it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/801a97e7f39e6dfcd790c5290def77577cb43b04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.972ex; height:2.509ex;" alt="{\displaystyle y_{it}=X_{it}\mathbf {\beta } +\alpha _{i}+u_{it}}"></span> for <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 t=1,\dots ,T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=1,\dots ,T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fad83717deedfb250b91bbc133f7122dd63ab223" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.915ex; height:2.509ex;" alt="{\displaystyle t=1,\dots ,T}"></span> and <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=1,\dots ,N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,\dots ,N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2537dcf59e68022de4f253dbadbd152c9d3eec47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.305ex; height:2.509ex;" alt="{\displaystyle i=1,\dots ,N}"></span></dd></dl> <p>Where: </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_{it}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a78584966237d2707cee7f5492d7376cb51e884" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.533ex; height:2.009ex;" alt="{\displaystyle y_{it}}"></span> is the dependent variable observed for individual <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> at time <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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></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 X_{it}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/165eecf71464eaa5224e695c2b8fd57e39d9cbca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.318ex; height:2.509ex;" alt="{\displaystyle X_{it}}"></span> is the time-variant <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 1\times k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5db4902f7c824acf82ab19c3029fc919d25b15a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.214ex; height:2.176ex;" alt="{\displaystyle 1\times k}"></span> (the number of independent variables) regressor vector.</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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> is the <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 k\times 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>×</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\times 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97b29553642874fa905288da16160d18eb62be8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.214ex; height:2.176ex;" alt="{\displaystyle k\times 1}"></span> matrix of parameters.</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 \alpha _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1fb627423abe4988b7ed88d4920bf1ec074790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.287ex; height:2.009ex;" alt="{\displaystyle \alpha _{i}}"></span> is the unobserved time-invariant individual effect. For example, the innate ability for individuals or historical and institutional factors for countries.</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 u_{it}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f972110de72875c81eb192190ac92af49c74140b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.723ex; height:2.009ex;" alt="{\displaystyle u_{it}}"></span> is the <a href="/wiki/Errors_and_residuals_in_statistics" class="mw-redirect" title="Errors and residuals in statistics">error term</a>.</li></ul> <p>Unlike <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_{it}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/165eecf71464eaa5224e695c2b8fd57e39d9cbca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.318ex; height:2.509ex;" alt="{\displaystyle X_{it}}"></span>, <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 \alpha _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1fb627423abe4988b7ed88d4920bf1ec074790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.287ex; height:2.009ex;" alt="{\displaystyle \alpha _{i}}"></span> cannot be directly observed. </p><p>Unlike the <a href="/wiki/Random_effects_model" title="Random effects model">random effects model</a> where the unobserved <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 \alpha _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1fb627423abe4988b7ed88d4920bf1ec074790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.287ex; height:2.009ex;" alt="{\displaystyle \alpha _{i}}"></span> is independent of <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_{it}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/165eecf71464eaa5224e695c2b8fd57e39d9cbca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.318ex; height:2.509ex;" alt="{\displaystyle X_{it}}"></span> for all <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 t=1,...,T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=1,...,T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86b754d74421b40360b498cd86bcf69dba126fc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.906ex; height:2.509ex;" alt="{\displaystyle t=1,...,T}"></span>, the fixed effects (FE) model allows <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 \alpha _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1fb627423abe4988b7ed88d4920bf1ec074790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.287ex; height:2.009ex;" alt="{\displaystyle \alpha _{i}}"></span> to be correlated with the regressor 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 X_{it}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/165eecf71464eaa5224e695c2b8fd57e39d9cbca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.318ex; height:2.509ex;" alt="{\displaystyle X_{it}}"></span>. <a href="/wiki/Endogeneity_(econometrics)#Exogeneity_versus_endogeneity" title="Endogeneity (econometrics)">Strict exogeneity</a> with respect to the idiosyncratic error term <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 u_{it}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f972110de72875c81eb192190ac92af49c74140b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.723ex; height:2.009ex;" alt="{\displaystyle u_{it}}"></span> is still required. </p> <div class="mw-heading mw-heading2"><h2 id="Statistical_estimation">Statistical estimation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fixed_effects_model&action=edit&section=3" title="Edit section: Statistical estimation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Fixed_effects_estimator">Fixed effects estimator</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fixed_effects_model&action=edit&section=4" title="Edit section: Fixed effects estimator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since <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 \alpha _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1fb627423abe4988b7ed88d4920bf1ec074790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.287ex; height:2.009ex;" alt="{\displaystyle \alpha _{i}}"></span> is not observable, it cannot be directly <a href="/wiki/Controlling_for_a_variable" title="Controlling for a variable">controlled</a> for. The FE model eliminates <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 \alpha _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1fb627423abe4988b7ed88d4920bf1ec074790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.287ex; height:2.009ex;" alt="{\displaystyle \alpha _{i}}"></span> by de-meaning the variables using the <i>within</i> transformation: </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}-{\overline {y}}_{i}=\left(X_{it}-{\overline {X}}_{i}\right)\beta +\left(\alpha _{i}-{\overline {\alpha }}_{i}\right)+\left(u_{it}-{\overline {u}}_{i}\right)\implies {\ddot {y}}_{it}={\ddot {X}}_{it}\beta +{\ddot {u}}_{it}}"> <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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>β</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mi>β</mi> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{it}-{\overline {y}}_{i}=\left(X_{it}-{\overline {X}}_{i}\right)\beta +\left(\alpha _{i}-{\overline {\alpha }}_{i}\right)+\left(u_{it}-{\overline {u}}_{i}\right)\implies {\ddot {y}}_{it}={\ddot {X}}_{it}\beta +{\ddot {u}}_{it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93957968875e8dec110b6e14b7d4b2bcd2f6dea9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:71.222ex; height:4.843ex;" alt="{\displaystyle y_{it}-{\overline {y}}_{i}=\left(X_{it}-{\overline {X}}_{i}\right)\beta +\left(\alpha _{i}-{\overline {\alpha }}_{i}\right)+\left(u_{it}-{\overline {u}}_{i}\right)\implies {\ddot {y}}_{it}={\ddot {X}}_{it}\beta +{\ddot {u}}_{it}}"></span></dd></dl> <p>where <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 {\overline {y}}_{i}={\frac {1}{T}}\sum \limits _{t=1}^{T}y_{it}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </munderover> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {y}}_{i}={\frac {1}{T}}\sum \limits _{t=1}^{T}y_{it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb29841123d4b4512bdeec180d1fc75a372d2f2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.313ex; height:7.343ex;" alt="{\displaystyle {\overline {y}}_{i}={\frac {1}{T}}\sum \limits _{t=1}^{T}y_{it}}"></span>, <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 {\overline {X}}_{i}={\frac {1}{T}}\sum \limits _{t=1}^{T}X_{it}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}_{i}={\frac {1}{T}}\sum \limits _{t=1}^{T}X_{it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df581698a9f33e4a4840eaa85c32eaa5fdaca298" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.949ex; height:7.343ex;" alt="{\displaystyle {\overline {X}}_{i}={\frac {1}{T}}\sum \limits _{t=1}^{T}X_{it}}"></span>, and <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 {\overline {u}}_{i}={\frac {1}{T}}\sum \limits _{t=1}^{T}u_{it}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </munderover> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {u}}_{i}={\frac {1}{T}}\sum \limits _{t=1}^{T}u_{it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7835b313930ca1d180c70fbf0e5d1bbf039b7c2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.667ex; height:7.343ex;" alt="{\displaystyle {\overline {u}}_{i}={\frac {1}{T}}\sum \limits _{t=1}^{T}u_{it}}"></span>. </p><p>Since <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 \alpha _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1fb627423abe4988b7ed88d4920bf1ec074790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.287ex; height:2.009ex;" alt="{\displaystyle \alpha _{i}}"></span> is constant, <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 {\overline {\alpha _{i}}}=\alpha _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\alpha _{i}}}=\alpha _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/113cb8a3ac67046af77ed61d051ee61ebcaa8135" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.788ex; height:2.676ex;" alt="{\displaystyle {\overline {\alpha _{i}}}=\alpha _{i}}"></span> and hence the effect is eliminated. The FE estimator <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 {\beta }}_{FE}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\beta }}_{FE}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50fbd26a607ce41bc87f606c5aee7095db538db4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.17ex; height:3.343ex;" alt="{\displaystyle {\hat {\beta }}_{FE}}"></span> is then obtained by an OLS regression of <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 {\ddot {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>¨</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61a2fbf9e038c6f150fe23bfd2a6681c1f30120e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.509ex;" alt="{\displaystyle {\ddot {y}}}"></span> 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 {\ddot {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo>¨</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f9e88843729a40284a8353a3b6bb2b9dd2971f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.676ex;" alt="{\displaystyle {\ddot {X}}}"></span>. </p><p>At least three alternatives to the <i>within</i> transformation exist with variations. </p><p>One is to add a dummy variable for each individual <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>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dea233301b9ca8fe5dde94824f918c0ceaf7fd5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;" alt="{\displaystyle i>1}"></span> (omitting the first individual because of <a href="/wiki/Multicollinearity" title="Multicollinearity">multicollinearity</a>). This is numerically, but not computationally, equivalent to the fixed effect model and only works if the sum of the number of series and the number of global parameters is smaller than the number of observations.<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> The dummy variable approach is particularly demanding with respect to computer memory usage and it is not recommended for problems larger than the available RAM, and the applied program compilation, can accommodate. </p><p>Second alternative is to use consecutive reiterations approach to local and global estimations.<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> This approach is very suitable for low memory systems on which it is much more computationally efficient than the dummy variable approach. </p><p>The third approach is a nested estimation whereby the local estimation for individual series is programmed in as a part of the model definition.<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> This approach is the most computationally and memory efficient, but it requires proficient programming skills and access to the model programming code; although, it can be programmed including in SAS.<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><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> </p><p>Finally, each of the above alternatives can be improved if the series-specific estimation is linear (within a nonlinear model), in which case the direct linear solution for individual series can be programmed in as part of the nonlinear model definition.<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> <div class="mw-heading mw-heading3"><h3 id="First_difference_estimator">First difference estimator</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fixed_effects_model&action=edit&section=5" title="Edit section: First difference estimator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/First-difference_estimator" title="First-difference estimator">First-difference estimator</a></div> <p>An alternative to the within transformation is the <i>first difference</i> transformation, which produces a different estimator. For <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 t=2,\dots ,T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=2,\dots ,T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b970425ae5d7792ae83f5849a97eb8681a2960" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.915ex; height:2.509ex;" alt="{\displaystyle t=2,\dots ,T}"></span>: </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-1}=\left(X_{it}-X_{i,t-1}\right)\beta +\left(\alpha _{i}-\alpha _{i}\right)+\left(u_{it}-u_{i,t-1}\right)\implies \Delta y_{it}=\Delta X_{it}\beta +\Delta u_{it}.}"> <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> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>β</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">Δ</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mi>β</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{it}-y_{i,t-1}=\left(X_{it}-X_{i,t-1}\right)\beta +\left(\alpha _{i}-\alpha _{i}\right)+\left(u_{it}-u_{i,t-1}\right)\implies \Delta y_{it}=\Delta X_{it}\beta +\Delta u_{it}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd22ff9970921534f744a9152c959daba93961f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:85.368ex; height:3.009ex;" alt="{\displaystyle y_{it}-y_{i,t-1}=\left(X_{it}-X_{i,t-1}\right)\beta +\left(\alpha _{i}-\alpha _{i}\right)+\left(u_{it}-u_{i,t-1}\right)\implies \Delta y_{it}=\Delta X_{it}\beta +\Delta u_{it}.}"></span></dd></dl> <p>The FD estimator <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 {\beta }}_{FD}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mi>D</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\beta }}_{FD}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2411f9583ce0ca9ce0b92f20cb3fb9d5979e38f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.275ex; height:3.343ex;" alt="{\displaystyle {\hat {\beta }}_{FD}}"></span> is then obtained by an OLS regression of <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 y_{it}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta y_{it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46c503769873e454808fa56734cb390432559e9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.468ex; height:2.509ex;" alt="{\displaystyle \Delta y_{it}}"></span> 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 \Delta X_{it}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta X_{it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfe74ab9f96a3264d3fd0af28e89d867f09c612b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.254ex; height:2.509ex;" alt="{\displaystyle \Delta X_{it}}"></span>. </p><p>When <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 T=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99185bd5a8802847b853f8c862d25cd597061db7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.897ex; height:2.176ex;" alt="{\displaystyle T=2}"></span>, the first difference and fixed effects estimators are numerically equivalent. For <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 T>2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T>2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b919a459f5e02c3e28ccc65e9665139c27092027" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.897ex; height:2.176ex;" alt="{\displaystyle T>2}"></span>, they are not. If the error terms <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 u_{it}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f972110de72875c81eb192190ac92af49c74140b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.723ex; height:2.009ex;" alt="{\displaystyle u_{it}}"></span> are <a href="/wiki/Homoscedasticity" class="mw-redirect" title="Homoscedasticity">homoskedastic</a> with no <a href="/wiki/Serial_correlation" class="mw-redirect" title="Serial correlation">serial correlation</a>, the fixed effects estimator is more <a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">efficient</a> than the first difference estimator. If <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 u_{it}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f972110de72875c81eb192190ac92af49c74140b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.723ex; height:2.009ex;" alt="{\displaystyle u_{it}}"></span> follows a <a href="/wiki/Random_walk" title="Random walk">random walk</a>, however, the first difference estimator is more efficient.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Equality_of_fixed_effects_and_first_difference_estimators_when_T=2"><span id="Equality_of_fixed_effects_and_first_difference_estimators_when_T.3D2"></span>Equality of fixed effects and first difference estimators when T=2</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fixed_effects_model&action=edit&section=6" title="Edit section: Equality of fixed effects and first difference estimators when T=2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For the special two period 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 T=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99185bd5a8802847b853f8c862d25cd597061db7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.897ex; height:2.176ex;" alt="{\displaystyle T=2}"></span>), the fixed effects (FE) estimator and the first difference (FD) estimator are numerically equivalent. This is because the FE estimator effectively "doubles the data set" used in the FD estimator. To see this, establish that the fixed effects estimator is: <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 {FE}_{T=2}=\left[(x_{i1}-{\bar {x}}_{i})(x_{i1}-{\bar {x}}_{i})'+(x_{i2}-{\bar {x}}_{i})(x_{i2}-{\bar {x}}_{i})'\right]^{-1}\left[(x_{i1}-{\bar {x}}_{i})(y_{i1}-{\bar {y}}_{i})+(x_{i2}-{\bar {x}}_{i})(y_{i2}-{\bar {y}}_{i})\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mi>E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>=</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo>−</mo> <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"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <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"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {FE}_{T=2}=\left[(x_{i1}-{\bar {x}}_{i})(x_{i1}-{\bar {x}}_{i})'+(x_{i2}-{\bar {x}}_{i})(x_{i2}-{\bar {x}}_{i})'\right]^{-1}\left[(x_{i1}-{\bar {x}}_{i})(y_{i1}-{\bar {y}}_{i})+(x_{i2}-{\bar {x}}_{i})(y_{i2}-{\bar {y}}_{i})\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bec812495fc9c91ab3b4483db602006732de4e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:99.809ex; height:3.509ex;" alt="{\displaystyle {FE}_{T=2}=\left[(x_{i1}-{\bar {x}}_{i})(x_{i1}-{\bar {x}}_{i})'+(x_{i2}-{\bar {x}}_{i})(x_{i2}-{\bar {x}}_{i})'\right]^{-1}\left[(x_{i1}-{\bar {x}}_{i})(y_{i1}-{\bar {y}}_{i})+(x_{i2}-{\bar {x}}_{i})(y_{i2}-{\bar {y}}_{i})\right]}"></span> </p><p>Since each <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_{i1}-{\bar {x}}_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{i1}-{\bar {x}}_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/365f3d14425ae71233ac9e19a51fcb98301faeec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.73ex; height:2.843ex;" alt="{\displaystyle (x_{i1}-{\bar {x}}_{i})}"></span> can be re-written as <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_{i1}-{\dfrac {x_{i1}+x_{i2}}{2}})={\dfrac {x_{i1}-x_{i2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{i1}-{\dfrac {x_{i1}+x_{i2}}{2}})={\dfrac {x_{i1}-x_{i2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4af0f4b0dc937d214c5ff481d1cdb0506b1caad4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.858ex; height:5.009ex;" alt="{\displaystyle (x_{i1}-{\dfrac {x_{i1}+x_{i2}}{2}})={\dfrac {x_{i1}-x_{i2}}{2}}}"></span>, we'll re-write the line as: </p><p><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 {FE}_{T=2}=\left[\sum _{i=1}^{N}{\dfrac {x_{i1}-x_{i2}}{2}}{\dfrac {x_{i1}-x_{i2}}{2}}'+{\dfrac {x_{i2}-x_{i1}}{2}}{\dfrac {x_{i2}-x_{i1}}{2}}'\right]^{-1}\left[\sum _{i=1}^{N}{\dfrac {x_{i1}-x_{i2}}{2}}{\dfrac {y_{i1}-y_{i2}}{2}}+{\dfrac {x_{i2}-x_{i1}}{2}}{\dfrac {y_{i2}-y_{i1}}{2}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mi>E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>=</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>′</mo> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>′</mo> </msup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {FE}_{T=2}=\left[\sum _{i=1}^{N}{\dfrac {x_{i1}-x_{i2}}{2}}{\dfrac {x_{i1}-x_{i2}}{2}}'+{\dfrac {x_{i2}-x_{i1}}{2}}{\dfrac {x_{i2}-x_{i1}}{2}}'\right]^{-1}\left[\sum _{i=1}^{N}{\dfrac {x_{i1}-x_{i2}}{2}}{\dfrac {y_{i1}-y_{i2}}{2}}+{\dfrac {x_{i2}-x_{i1}}{2}}{\dfrac {y_{i2}-y_{i1}}{2}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ad1c97e73a14826b3cd0e14aec665df7916b2bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:108.591ex; height:8.009ex;" alt="{\displaystyle {FE}_{T=2}=\left[\sum _{i=1}^{N}{\dfrac {x_{i1}-x_{i2}}{2}}{\dfrac {x_{i1}-x_{i2}}{2}}'+{\dfrac {x_{i2}-x_{i1}}{2}}{\dfrac {x_{i2}-x_{i1}}{2}}'\right]^{-1}\left[\sum _{i=1}^{N}{\dfrac {x_{i1}-x_{i2}}{2}}{\dfrac {y_{i1}-y_{i2}}{2}}+{\dfrac {x_{i2}-x_{i1}}{2}}{\dfrac {y_{i2}-y_{i1}}{2}}\right]}"></span> </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 =\left[\sum _{i=1}^{N}2{\dfrac {x_{i2}-x_{i1}}{2}}{\dfrac {x_{i2}-x_{i1}}{2}}'\right]^{-1}\left[\sum _{i=1}^{N}2{\dfrac {x_{i2}-x_{i1}}{2}}{\dfrac {y_{i2}-y_{i1}}{2}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>′</mo> </msup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\left[\sum _{i=1}^{N}2{\dfrac {x_{i2}-x_{i1}}{2}}{\dfrac {x_{i2}-x_{i1}}{2}}'\right]^{-1}\left[\sum _{i=1}^{N}2{\dfrac {x_{i2}-x_{i1}}{2}}{\dfrac {y_{i2}-y_{i1}}{2}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e550eea4fb5bb3e05b5313a9ee93b4c27e481277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:59.024ex; height:8.009ex;" alt="{\displaystyle =\left[\sum _{i=1}^{N}2{\dfrac {x_{i2}-x_{i1}}{2}}{\dfrac {x_{i2}-x_{i1}}{2}}'\right]^{-1}\left[\sum _{i=1}^{N}2{\dfrac {x_{i2}-x_{i1}}{2}}{\dfrac {y_{i2}-y_{i1}}{2}}\right]}"></span></dd> <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 =2\left[\sum _{i=1}^{N}(x_{i2}-x_{i1})(x_{i2}-x_{i1})'\right]^{-1}\left[\sum _{i=1}^{N}{\frac {1}{2}}(x_{i2}-x_{i1})(y_{i2}-y_{i1})\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>2</mn> <msup> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =2\left[\sum _{i=1}^{N}(x_{i2}-x_{i1})(x_{i2}-x_{i1})'\right]^{-1}\left[\sum _{i=1}^{N}{\frac {1}{2}}(x_{i2}-x_{i1})(y_{i2}-y_{i1})\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cfb9fca2b4452e92c9bcb07e86dd495ef08573a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:63.366ex; height:8.009ex;" alt="{\displaystyle =2\left[\sum _{i=1}^{N}(x_{i2}-x_{i1})(x_{i2}-x_{i1})'\right]^{-1}\left[\sum _{i=1}^{N}{\frac {1}{2}}(x_{i2}-x_{i1})(y_{i2}-y_{i1})\right]}"></span></dd> <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 =\left[\sum _{i=1}^{N}(x_{i2}-x_{i1})(x_{i2}-x_{i1})'\right]^{-1}\sum _{i=1}^{N}(x_{i2}-x_{i1})(y_{i2}-y_{i1})={FD}_{T=2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mi>D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>=</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\left[\sum _{i=1}^{N}(x_{i2}-x_{i1})(x_{i2}-x_{i1})'\right]^{-1}\sum _{i=1}^{N}(x_{i2}-x_{i1})(y_{i2}-y_{i1})={FD}_{T=2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fdb23850f173a02c99cf064121292e00d89116b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:67.36ex; height:8.009ex;" alt="{\displaystyle =\left[\sum _{i=1}^{N}(x_{i2}-x_{i1})(x_{i2}-x_{i1})'\right]^{-1}\sum _{i=1}^{N}(x_{i2}-x_{i1})(y_{i2}-y_{i1})={FD}_{T=2}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Chamberlain_method">Chamberlain method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fixed_effects_model&action=edit&section=7" title="Edit section: Chamberlain method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Chamberlain%27s_approach_to_unobserved_effects_models" title="Chamberlain's approach to unobserved effects models">Chamberlain's approach to unobserved effects models</a></div> <p><a href="/w/index.php?title=Gary_Chamberlain&action=edit&redlink=1" class="new" title="Gary Chamberlain (page does not exist)">Gary Chamberlain</a>'s method, a generalization of the within estimator, replaces <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 \alpha _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1fb627423abe4988b7ed88d4920bf1ec074790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.287ex; height:2.009ex;" alt="{\displaystyle \alpha _{i}}"></span> with its <a href="/wiki/Linear_projection" class="mw-redirect" title="Linear projection">linear projection</a> onto the explanatory variables. Writing the linear projection 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 \alpha _{i}=\lambda _{0}+X_{i1}\lambda _{1}+X_{i2}\lambda _{2}+\dots +X_{iT}\lambda _{T}+e_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>T</mi> </mrow> </msub> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{i}=\lambda _{0}+X_{i1}\lambda _{1}+X_{i2}\lambda _{2}+\dots +X_{iT}\lambda _{T}+e_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/064894d11761da444c30f59711d296c1bdb82303" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:45.14ex; height:2.509ex;" alt="{\displaystyle \alpha _{i}=\lambda _{0}+X_{i1}\lambda _{1}+X_{i2}\lambda _{2}+\dots +X_{iT}\lambda _{T}+e_{i}}"></span></dd></dl> <p>this results in the following equation: </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}=\lambda _{0}+X_{i1}\lambda _{1}+X_{i2}\lambda _{2}+\dots +X_{it}(\lambda _{t}+\mathbf {\beta } )+\dots +X_{iT}\lambda _{T}+e_{i}+u_{it}}"> <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> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>T</mi> </mrow> </msub> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>e</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{it}=\lambda _{0}+X_{i1}\lambda _{1}+X_{i2}\lambda _{2}+\dots +X_{it}(\lambda _{t}+\mathbf {\beta } )+\dots +X_{iT}\lambda _{T}+e_{i}+u_{it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0160dcd575374920c3b0d2ac21da81007af75b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:70.833ex; height:2.843ex;" alt="{\displaystyle y_{it}=\lambda _{0}+X_{i1}\lambda _{1}+X_{i2}\lambda _{2}+\dots +X_{it}(\lambda _{t}+\mathbf {\beta } )+\dots +X_{iT}\lambda _{T}+e_{i}+u_{it}}"></span></dd></dl> <p>which can be estimated by <a href="/wiki/Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">minimum distance estimation</a>.<sup id="cite_ref-Chamberlain1984_17-0" class="reference"><a href="#cite_note-Chamberlain1984-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Hausman–Taylor_method"><span id="Hausman.E2.80.93Taylor_method"></span>Hausman–Taylor method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fixed_effects_model&action=edit&section=8" title="Edit section: Hausman–Taylor method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Need to have more than one time-variant regressor (<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>) and time-invariant regressor (<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>) and at least one <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> and one <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> that are uncorrelated with <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 \alpha _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1fb627423abe4988b7ed88d4920bf1ec074790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.287ex; height:2.009ex;" alt="{\displaystyle \alpha _{i}}"></span>. </p><p>Partition the <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> and <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> variables such that <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{array}{c}X=[{\underset {TN\times K1}{X_{1it}}}\vdots {\underset {TN\times K2}{X_{2it}}}]\\Z=[{\underset {TN\times G1}{Z_{1it}}}\vdots {\underset {TN\times G2}{Z_{2it}}}]\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>X</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mrow> <mi>T</mi> <mi>N</mi> <mo>×</mo> <mi>K</mi> <mn>1</mn> </mrow> </munder> </mrow> <mo>⋮</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mrow> <mi>T</mi> <mi>N</mi> <mo>×</mo> <mi>K</mi> <mn>2</mn> </mrow> </munder> </mrow> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <mi>Z</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mrow> <mi>T</mi> <mi>N</mi> <mo>×</mo> <mi>G</mi> <mn>1</mn> </mrow> </munder> </mrow> <mo>⋮</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mrow> <mi>T</mi> <mi>N</mi> <mo>×</mo> <mi>G</mi> <mn>2</mn> </mrow> </munder> </mrow> <mo stretchy="false">]</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}X=[{\underset {TN\times K1}{X_{1it}}}\vdots {\underset {TN\times K2}{X_{2it}}}]\\Z=[{\underset {TN\times G1}{Z_{1it}}}\vdots {\underset {TN\times G2}{Z_{2it}}}]\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb068178ecab189cd0508407871a109b928c171" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:20.126ex; height:11.843ex;" alt="{\displaystyle {\begin{array}{c}X=[{\underset {TN\times K1}{X_{1it}}}\vdots {\underset {TN\times K2}{X_{2it}}}]\\Z=[{\underset {TN\times G1}{Z_{1it}}}\vdots {\underset {TN\times G2}{Z_{2it}}}]\end{array}}}"></span> where <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70b2694445a5901b24338a2e7a7e58f02a72a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{1}}"></span> and <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cea9e950915c77b3dcf9d4d54101820f538bc077" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle Z_{1}}"></span> are uncorrelated with <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 \alpha _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1fb627423abe4988b7ed88d4920bf1ec074790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.287ex; height:2.009ex;" alt="{\displaystyle \alpha _{i}}"></span>. Need <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 K1>G2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mn>1</mn> <mo>></mo> <mi>G</mi> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K1>G2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26a9dc0a951b8a045ce3d2505a01cb1a1d67dc72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.316ex; height:2.176ex;" alt="{\displaystyle K1>G2}"></span>. </p><p>Estimating <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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> via OLS 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 {\widehat {di}}=Z_{i}\gamma +\varphi _{it}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>d</mi> <mi>i</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>γ</mi> <mo>+</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {di}}=Z_{i}\gamma +\varphi _{it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c29d8c652f284eebe34d021778ef7daaeec3ffc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.01ex; width:14.836ex; height:3.509ex;" alt="{\displaystyle {\widehat {di}}=Z_{i}\gamma +\varphi _{it}}"></span> using <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70b2694445a5901b24338a2e7a7e58f02a72a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{1}}"></span> and <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cea9e950915c77b3dcf9d4d54101820f538bc077" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle Z_{1}}"></span> as instruments yields a consistent estimate. </p> <div class="mw-heading mw-heading3"><h3 id="Generalization_with_input_uncertainty">Generalization with input uncertainty</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fixed_effects_model&action=edit&section=9" title="Edit section: Generalization with input uncertainty"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When there is input uncertainty for the <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> data, <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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc454b7f0e03259e9c5e8568fcef1c8e6b78230" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.204ex; height:2.676ex;" alt="{\displaystyle \delta y}"></span>, then the <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 \chi ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c0cc9237ec72a1da6d18bc8e7fb24cdda43a49a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:3.009ex;" alt="{\displaystyle \chi ^{2}}"></span> value, rather than the sum of squared residuals, should be minimized.<sup id="cite_ref-ren18_18-0" class="reference"><a href="#cite_note-ren18-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> This can be directly achieved from substitution rules: </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 {\frac {y_{it}}{\delta y_{it}}}=\mathbf {\beta } {\frac {X_{it}}{\delta y_{it}}}+\alpha _{i}{\frac {1}{\delta y_{it}}}+{\frac {u_{it}}{\delta y_{it}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mrow> <mi>δ</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mrow> <mi>δ</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>δ</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> <mrow> <mi>δ</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {y_{it}}{\delta y_{it}}}=\mathbf {\beta } {\frac {X_{it}}{\delta y_{it}}}+\alpha _{i}{\frac {1}{\delta y_{it}}}+{\frac {u_{it}}{\delta y_{it}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed055dd157c3e8262985eb24e2c377b10c38a749" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.068ex; height:5.843ex;" alt="{\displaystyle {\frac {y_{it}}{\delta y_{it}}}=\mathbf {\beta } {\frac {X_{it}}{\delta y_{it}}}+\alpha _{i}{\frac {1}{\delta y_{it}}}+{\frac {u_{it}}{\delta y_{it}}}}"></span>,</dd></dl> <p>then the values and standard deviations for <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 \mathbf {\beta } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\beta } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1aa99638e54aa8d94578d75aeb26380b9251e36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \mathbf {\beta } }"></span> and <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 \alpha _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1fb627423abe4988b7ed88d4920bf1ec074790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.287ex; height:2.009ex;" alt="{\displaystyle \alpha _{i}}"></span> can be determined via classical <a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">ordinary least squares</a> analysis and <a href="/wiki/Variance-covariance_matrix" class="mw-redirect" title="Variance-covariance matrix">variance-covariance matrix</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Use_to_test_for_consistency">Use to test for consistency</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fixed_effects_model&action=edit&section=10" title="Edit section: Use to test for consistency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Random effects estimators may be inconsistent sometimes in the long time series limit, if the random effects are misspecified (i.e. the model chosen for the random effects is incorrect). However, the fixed effects model may still be consistent in some situations. For example, if the time series being modeled is not stationary, random effects models assuming stationarity may not be consistent in the long-series limit. One example of this is if the time series has an upward trend. Then, as the series becomes longer, the model revises estimates for the mean of earlier periods upwards, giving increasingly biased predictions of coefficients. However, a model with fixed time effects does not pool information across time, and as a result earlier estimates will not be affected. </p><p>In situations like these where the fixed effects model is known to be consistent, the <a href="/wiki/Durbin-Wu-Hausman_test" class="mw-redirect" title="Durbin-Wu-Hausman test">Durbin-Wu-Hausman test</a> can be used to test whether the random effects model chosen is consistent. If <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 H_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43910602a221b7a4c373791f94793e3008622070" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{0}}"></span> is true, both <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 {\widehat {\beta }}_{RE}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}_{RE}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef259a229f8d49c065d1afd8e3a813791437b5e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.252ex; height:3.509ex;" alt="{\displaystyle {\widehat {\beta }}_{RE}}"></span> and <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 {\widehat {\beta }}_{FE}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}_{FE}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb2a92b3f965f8d56b1289472f5681ef8e7df0ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.235ex; height:3.509ex;" alt="{\displaystyle {\widehat {\beta }}_{FE}}"></span> are consistent, but only <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 {\widehat {\beta }}_{RE}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}_{RE}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef259a229f8d49c065d1afd8e3a813791437b5e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.252ex; height:3.509ex;" alt="{\displaystyle {\widehat {\beta }}_{RE}}"></span> is efficient. If <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 H_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/821c51b03158cde9bf208cc997e6484b9063b53a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.033ex; height:2.509ex;" alt="{\displaystyle H_{a}}"></span> is true the consistency of <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 {\widehat {\beta }}_{RE}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}_{RE}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef259a229f8d49c065d1afd8e3a813791437b5e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.252ex; height:3.509ex;" alt="{\displaystyle {\widehat {\beta }}_{RE}}"></span> cannot be guaranteed. </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=Fixed_effects_model&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Random_effects_model" title="Random effects model">Random effects model</a></li> <li><a href="/wiki/Mixed_model" title="Mixed model">Mixed model</a></li> <li><a href="/wiki/Dynamic_unobserved_effects_model" title="Dynamic unobserved effects model">Dynamic unobserved effects model</a></li> <li><a href="/wiki/Fixed-effect_Poisson_model" title="Fixed-effect Poisson model">Fixed-effect Poisson model</a></li> <li><a href="/wiki/Panel_analysis" title="Panel analysis">Panel analysis</a></li> <li><a href="/wiki/First-difference_estimator" title="First-difference estimator">First-difference estimator</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fixed_effects_model&action=edit&section=12" title="Edit section: Notes"><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">Greene, W.H., 2011. <i>Econometric Analysis</i>, 7th ed., Prentice Hall</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"><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="CITEREFDiggleHeagertyLiangZeger2002" class="citation book cs1">Diggle, Peter J.; Heagerty, Patrick; Liang, Kung-Yee; Zeger, Scott L. (2002). <i>Analysis of Longitudinal Data</i> (2nd ed.). 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Vol. 2. pp. 1247–1318. <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%2FS1573-4412%2884%2902014-6">10.1016/S1573-4412(84)02014-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780444861863" title="Special:BookSources/9780444861863"><bdi>9780444861863</bdi></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/1573-4412">1573-4412</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chapter+22+Panel+data&rft.series=Handbook+of+Econometrics&rft.pages=1247-1318&rft.date=1984&rft.issn=1573-4412&rft_id=info%3Adoi%2F10.1016%2FS1573-4412%2884%2902014-6&rft.isbn=9780444861863&rft.aulast=Chamberlain&rft.aufirst=Gary&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFixed+effects+model" class="Z3988"></span></span> </li> <li id="cite_note-ren18-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-ren18_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRenDongEspositoPueyo2018" class="citation journal cs1">Ren, Bin; Dong, Ruobing; Esposito, Thomas M.; Pueyo, Laurent; Debes, John H.; Poteet, Charles A.; Choquet, Élodie; Benisty, Myriam; Chiang, Eugene; Grady, Carol A.; Hines, Dean C.; Schneider, Glenn; Soummer, Rémi (2018). <a rel="nofollow" class="external text" href="https://doi.org/10.3847%2F2041-8213%2Faab7f5">"A Decade of MWC 758 Disk Images: Where Are the Spiral-Arm-Driving Planets?"</a>. <i>The Astrophysical Journal Letters</i>. <b>857</b> (1): L9. <a href="/wiki/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/1803.06776">1803.06776</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2018ApJ...857L...9R">2018ApJ...857L...9R</a>. <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/10.3847%2F2041-8213%2Faab7f5">10.3847/2041-8213/aab7f5</a></span>. <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:59427417">59427417</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Astrophysical+Journal+Letters&rft.atitle=A+Decade+of+MWC+758+Disk+Images%3A+Where+Are+the+Spiral-Arm-Driving+Planets%3F&rft.volume=857&rft.issue=1&rft.pages=L9&rft.date=2018&rft_id=info%3Aarxiv%2F1803.06776&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A59427417%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.3847%2F2041-8213%2Faab7f5&rft_id=info%3Abibcode%2F2018ApJ...857L...9R&rft.aulast=Ren&rft.aufirst=Bin&rft.au=Dong%2C+Ruobing&rft.au=Esposito%2C+Thomas+M.&rft.au=Pueyo%2C+Laurent&rft.au=Debes%2C+John+H.&rft.au=Poteet%2C+Charles+A.&rft.au=Choquet%2C+%C3%89lodie&rft.au=Benisty%2C+Myriam&rft.au=Chiang%2C+Eugene&rft.au=Grady%2C+Carol+A.&rft.au=Hines%2C+Dean+C.&rft.au=Schneider%2C+Glenn&rft.au=Soummer%2C+R%C3%A9mi&rft_id=https%3A%2F%2Fdoi.org%2F10.3847%252F2041-8213%252Faab7f5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFixed+effects+model" class="Z3988"></span></span> </li> </ol></div></div> <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=Fixed_effects_model&action=edit&section=13" title="Edit section: References"><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="CITEREFChristensen2002" class="citation book cs1">Christensen, Ronald (2002). <i>Plane Answers to Complex Questions: The Theory of Linear Models</i> (Third ed.). New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-95361-2" title="Special:BookSources/0-387-95361-2"><bdi>0-387-95361-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Plane+Answers+to+Complex+Questions%3A+The+Theory+of+Linear+Models&rft.place=New+York&rft.edition=Third&rft.pub=Springer&rft.date=2002&rft.isbn=0-387-95361-2&rft.aulast=Christensen&rft.aufirst=Ronald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFixed+effects+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGujaratiPorter2009" class="citation book cs1">Gujarati, Damodar N.; <a href="/wiki/Dawn_C._Porter" title="Dawn C. Porter">Porter, Dawn C.</a> (2009). "Panel Data Regression Models". <i>Basic Econometrics</i> (Fifth international ed.). Boston: McGraw-Hill. pp. 591–616. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-007-127625-2" title="Special:BookSources/978-007-127625-2"><bdi>978-007-127625-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Panel+Data+Regression+Models&rft.btitle=Basic+Econometrics&rft.place=Boston&rft.pages=591-616&rft.edition=Fifth+international&rft.pub=McGraw-Hill&rft.date=2009&rft.isbn=978-007-127625-2&rft.aulast=Gujarati&rft.aufirst=Damodar+N.&rft.au=Porter%2C+Dawn+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFixed+effects+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHsiao2003" class="citation book cs1">Hsiao, Cheng (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=i9iPG7C3EP4C&pg=PA95">"Fixed-effects models"</a>. <i>Analysis of Panel Data</i> (2nd ed.). New York: Cambridge University Press. pp. 95–103. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-52271-4" title="Special:BookSources/0-521-52271-4"><bdi>0-521-52271-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Fixed-effects+models&rft.btitle=Analysis+of+Panel+Data&rft.place=New+York&rft.pages=95-103&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=0-521-52271-4&rft.aulast=Hsiao&rft.aufirst=Cheng&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Di9iPG7C3EP4C%26pg%3DPA95&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFixed+effects+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWooldridge2013" class="citation book cs1">Wooldridge, Jeffrey M. (2013). "Fixed Effects Estimation". <i>Introductory Econometrics: A Modern Approach</i> (Fifth international ed.). Mason, OH: South-Western. pp. 466–474. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-111-53439-4" title="Special:BookSources/978-1-111-53439-4"><bdi>978-1-111-53439-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Fixed+Effects+Estimation&rft.btitle=Introductory+Econometrics%3A+A+Modern+Approach&rft.place=Mason%2C+OH&rft.pages=466-474&rft.edition=Fifth+international&rft.pub=South-Western&rft.date=2013&rft.isbn=978-1-111-53439-4&rft.aulast=Wooldridge&rft.aufirst=Jeffrey+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFixed+effects+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=Fixed_effects_model&action=edit&section=14" 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="http://teaching.sociology.ul.ie/DCW/confront/node45.html">Fixed and random effects models</a></li> <li><a rel="nofollow" class="external text" href="https://www.southampton.ac.uk/~cpd/anovas/datasets/index.htm">Examples of all ANOVA and ANCOVA models with up to three treatment factors, including randomized block, split plot, repeated measures, and Latin squares, and their analysis in R</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=Fixed_effects_model&oldid=1252212960">https://en.wikipedia.org/w/index.php?title=Fixed_effects_model&oldid=1252212960</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:Analysis_of_variance" title="Category:Analysis of variance">Analysis of variance</a></li><li><a href="/wiki/Category:Regression_models" title="Category:Regression models">Regression models</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 20 October 2024, at 10:05<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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Fixed effects model - Wikipedia