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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Panel data</span></span>
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<p>In <a href="Statistics" title="Statistics">statistics</a> and <a href="Econometrics" title="Econometrics">econometrics</a>, <b>panel data</b> and <b>longitudinal data</b><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><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> are both multi-dimensional <a href="Data_set" title="Data set">data</a> involving measurements over time. Panel data is a subset of longitudinal data where observations are for the same subjects each time.
</p><p><a href="Time_series" title="Time series">Time series</a> and <a href="Cross-sectional_data" title="Cross-sectional data">cross-sectional data</a> can be thought of as special cases of panel data that are in one dimension only (one panel member or individual for the former, one time point for the latter). A literature search often involves time series, cross-sectional, or panel data.
</p><p>A study that uses panel data is called a <a href="Longitudinal_study" title="Longitudinal study">longitudinal study</a> or panel study.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>
<table class="wikitable sortable" style="display:inline-table">
<caption>MRPP balanced panel
</caption>
<tbody><tr>
<th scope="col">person
</th>
<th scope="col">year
</th>
<th scope="col">income
</th>
<th>
</th></tr>
<tr>
<td>1</td>
<td>2016</td>
<td>1300</td>
<td>27</td>
<td>1
</td></tr>
<tr>
<td>1</td>
<td>2017</td>
<td>1600</td>
<td>28</td>
<td>1
</td></tr>
<tr>
<td>1</td>
<td>2018</td>
<td>2000</td>
<td>29</td>
<td>1
</td></tr>
<tr>
<td>2</td>
<td>2016</td>
<td>2000</td>
<td>38</td>
<td>2
</td></tr>
<tr>
<td>2</td>
<td>2017</td>
<td>2300</td>
<td>39</td>
<td>2
</td></tr>
<tr>
<td>2</td>
<td>2018</td>
<td>2400</td>
<td>40</td>
<td>2
</td></tr></tbody></table>
<table class="wikitable sortable" style="display:inline-table">
<caption>MRPP unbalanced panel
</caption>
<tbody><tr>
<th scope="col">person
</th>
<th scope="col">year
</th>
<th scope="col">income
</th>
<th scope="col">age
</th>
<th scope="col">sex
</th></tr>
<tr>
<td>1</td>
<td>2016</td>
<td>1600</td>
<td>23</td>
<td>1
</td></tr>
<tr>
<td>1</td>
<td>2017</td>
<td>1500</td>
<td>24</td>
<td>1
</td></tr>
<tr>
<td>2</td>
<td>2016</td>
<td>1900</td>
<td>41</td>
<td>2
</td></tr>
<tr>
<td>2</td>
<td>2017</td>
<td>2000</td>
<td>42</td>
<td>2
</td></tr>
<tr>
<td>2</td>
<td>2018</td>
<td>2100</td>
<td>43</td>
<td>2
</td></tr>
<tr>
<td>3</td>
<td>2017</td>
<td>3300</td>
<td>34</td>
<td>1
</td></tr></tbody></table>
<p>In the <b>multiple response permutation procedure</b> (<b>MRPP</b>) example above, two datasets with a panel structure are shown and the objective is to test whether there's a significant difference between people in the sample data. Individual characteristics (income, age, sex) are collected for different persons and different years. In the first dataset, two persons (1, 2) are observed every year for three years (2016, 2017, 2018). In the second dataset, three persons (1, 2, 3) are observed two times (person 1), three times (person 2), and one time (person 3), respectively, over three years (2016, 2017, 2018); in particular, person 1 is not observed in year 2018 and person 3 is not observed in 2016 or 2018.
</p><p>A <b>balanced panel</b> (e.g., the first dataset above) is a dataset in which <i>each</i> panel member (i.e., person) is observed <i>every</i> year. Consequently, if a balanced panel contains <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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="./f5e3890c981ae85503089652feb48b191b57aae3.svg" 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}" loading="lazy"></span> panel members and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>
</semantics>
</math></span><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" 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}" loading="lazy"></span> periods, the number of observations (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>) in the dataset is necessarily <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=N\cdot T}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mi>N</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>T</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=N\cdot T}</annotation>
</semantics>
</math></span><img src="./533f1a02979cf1333f5cb3e33ba0ff7821afc5b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.872ex; height:2.176ex;" alt="{\displaystyle n=N\cdot T}" loading="lazy"></span>.
</p><p>An <b>unbalanced panel</b> (e.g., the second dataset above) is a dataset in which <i>at least one</i> panel member is not observed every period. Therefore, if an unbalanced panel contains <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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="./f5e3890c981ae85503089652feb48b191b57aae3.svg" 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}" loading="lazy"></span> panel members and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" 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}" loading="lazy"></span> periods, then the following strict inequality holds for the number of observations (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>) in the dataset: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n<N\cdot T}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo><</mo>
<mi>N</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>T</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n<N\cdot T}</annotation>
</semantics>
</math></span><img src="./c631bd437c0a4222278d6df40371a4a0359ecaeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.872ex; height:2.176ex;" alt="{\displaystyle n<N\cdot T}" loading="lazy"></span>.
</p><p>Both datasets above are structured in the <b>long format</b>, which is where one row holds one observation per time. Another way to structure panel data would be the <b>wide format</b> where one row represents one observational unit for <i>all</i> points in time (for the example, the wide format would have only two (first example) or three (second example) rows of data with additional columns for each time-varying variable (income, age).
</p>
<div class="mw-heading mw-heading2"><h2 id="Analysis">Analysis</h2></div>
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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Panel_analysis" title="Panel analysis">Panel analysis</a></div>
<p>A panel has the form
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{it},\quad i=1,\dots ,N,\quad t=1,\dots ,T,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
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<mi>t</mi>
</mrow>
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<mo>,</mo>
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<mo>,</mo>
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<mo>,</mo>
<mspace width="1em"></mspace>
<mi>t</mi>
<mo>=</mo>
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<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>T</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{it},\quad i=1,\dots ,N,\quad t=1,\dots ,T,}</annotation>
</semantics>
</math></span><img src="./d889c9b40518b497d78126903d2843e6a195c2cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.898ex; height:2.509ex;" alt="{\displaystyle X_{it},\quad i=1,\dots ,N,\quad t=1,\dots ,T,}" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
<semantics>
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<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
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</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" 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}" loading="lazy"></span> is the individual dimension and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" 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}" loading="lazy"></span> is the time dimension. A general panel data regression model is written as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{it}=\alpha +\beta 'X_{it}+u_{it}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
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<annotation encoding="application/x-tex">{\displaystyle y_{it}=\alpha +\beta 'X_{it}+u_{it}}</annotation>
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</math></span><img src="./7d926205e34c1643015eb55ac157b2a9a314ff4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.862ex; height:2.843ex;" alt="{\displaystyle y_{it}=\alpha +\beta 'X_{it}+u_{it}}" loading="lazy"></span>. Different assumptions can be made on the precise structure of this general model. Two important models are the <a href="Fixed_effects_model" title="Fixed effects model">fixed effects model</a> and the <a href="Random_effects_model" title="Random effects model">random effects model</a>.
</p><p>Consider a generic panel data model:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{it}=\alpha +\beta 'X_{it}+u_{it},}">
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<annotation encoding="application/x-tex">{\displaystyle y_{it}=\alpha +\beta 'X_{it}+u_{it},}</annotation>
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</math></span><img src="./87942b0a6c51f1cde0240cbd7f5e98a5247d2966.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.509ex; height:2.843ex;" alt="{\displaystyle y_{it}=\alpha +\beta 'X_{it}+u_{it},}" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{it}=\mu _{i}+v_{it}.}">
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<annotation encoding="application/x-tex">{\displaystyle u_{it}=\mu _{i}+v_{it}.}</annotation>
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</math></span><img src="./7e94e5ef1e237dd0c0e7bb06a7307c57df7bcd61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.031ex; height:2.509ex;" alt="{\displaystyle u_{it}=\mu _{i}+v_{it}.}" loading="lazy"></span></dd></dl>
<p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{i}}">
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<mi>μ<!-- μ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \mu _{i}}</annotation>
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</math></span><img src="./dea0a0293841cce9eef98b55e53a92b82ae59ee4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.201ex; height:2.176ex;" alt="{\displaystyle \mu _{i}}" loading="lazy"></span> are individual-specific, time-invariant effects (e.g., in a panel of countries this could include geography, climate, etc.) which are fixed over time, whereas <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{it}}">
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<annotation encoding="application/x-tex">{\displaystyle v_{it}}</annotation>
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</math></span><img src="./f10097478703659f6d88ae9e51441e5fb9c5fdd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.521ex; height:2.009ex;" alt="{\displaystyle v_{it}}" loading="lazy"></span> is a time-varying random component.
</p><p>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu _{i}}</annotation>
</semantics>
</math></span><img src="./dea0a0293841cce9eef98b55e53a92b82ae59ee4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.201ex; height:2.176ex;" alt="{\displaystyle \mu _{i}}" loading="lazy"></span> is unobserved, and correlated with at least one of the independent variables, then it will cause omitted variable bias in a standard <a href="Ordinary_least_squares" title="Ordinary least squares">OLS</a> regression. However, panel data methods, such as the fixed effects estimator or alternatively, the <a href="First-difference_estimator" title="First-difference estimator">first-difference estimator</a> can be used to control for it.
</p><p>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu _{i}}</annotation>
</semantics>
</math></span><img src="./dea0a0293841cce9eef98b55e53a92b82ae59ee4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.201ex; height:2.176ex;" alt="{\displaystyle \mu _{i}}" loading="lazy"></span> is not correlated with any of the independent variables, ordinary least squares linear regression methods can be used to yield unbiased and consistent estimates of the regression parameters. However, because <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu _{i}}</annotation>
</semantics>
</math></span><img src="./dea0a0293841cce9eef98b55e53a92b82ae59ee4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.201ex; height:2.176ex;" alt="{\displaystyle \mu _{i}}" loading="lazy"></span> is fixed over time, it will induce serial correlation in the error term of the regression. This means that more efficient estimation techniques are available. Random effects is one such method: it is a special case of feasible <a href="Generalized_least_squares" title="Generalized least squares">generalized least squares</a> which controls for the structure of the serial correlation induced by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu _{i}}</annotation>
</semantics>
</math></span><img src="./dea0a0293841cce9eef98b55e53a92b82ae59ee4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.201ex; height:2.176ex;" alt="{\displaystyle \mu _{i}}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Dynamic_panel_data">Dynamic panel data</h3></div>
<p>Dynamic panel data describes the case where a <a href="Lag_operator" title="Lag operator">lag</a> of the dependent variable is used as regressor:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{it}=\alpha +\beta 'X_{it}+\gamma y_{it-1}+u_{it}.}">
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<annotation encoding="application/x-tex">{\displaystyle y_{it}=\alpha +\beta 'X_{it}+\gamma y_{it-1}+u_{it}.}</annotation>
</semantics>
</math></span><img src="./69d621fe03bae20b915c1eeb238d518d5c2df9b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.245ex; height:3.009ex;" alt="{\displaystyle y_{it}=\alpha +\beta 'X_{it}+\gamma y_{it-1}+u_{it}.}" loading="lazy"></span></dd></dl>
<p>The presence of the lagged dependent variable violates strict exogeneity, that is, <a href="Endogeneity_(econometrics)" title="Endogeneity (econometrics)">endogeneity</a> may occur. The fixed effect estimator and the first differences estimator both rely on the assumption of strict exogeneity. Hence, if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u_{i}}</annotation>
</semantics>
</math></span><img src="./14f13cb025ff2e136dcbd2fc81ddf965b728e3d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle u_{i}}" loading="lazy"></span> is believed to be correlated with one of the independent variables, an alternative estimation technique must be used. Instrumental variables or GMM techniques are commonly used in this situation, such as the <a href="Arellano%E2%80%93Bond_estimator" title="Arellano–Bond estimator">Arellano–Bond estimator</a>.
While estimating this we should have the proper information about the instrumental variables.
</p>
<div class="mw-heading mw-heading2"><h2 id="Data_sets_which_have_a_panel_design">Data sets which have a panel design</h2></div>
<ul><li><i>German</i> <a href="Socio-Economic_Panel" title="Socio-Economic Panel">Socio-Economic Panel</a> (SOEP)</li>
<li><a href="Household%2C_Income_and_Labour_Dynamics_in_Australia_Survey" title="Household, Income and Labour Dynamics in Australia Survey">Household, Income and Labour Dynamics in Australia Survey</a> (HILDA)</li>
<li><a href="British_Household_Panel_Survey" title="British Household Panel Survey">British Household Panel Survey</a> (BHPS)</li>
<li><a href="Survey_of_Income_and_Program_Participation" title="Survey of Income and Program Participation">Survey of Income and Program Participation</a> (SIPP)</li>
<li><a href="LLMDB" title="LLMDB">Lifelong Labour Market Database</a> (LLMDB)</li>
<li><a href="Panel_Study_of_Income_Dynamics" title="Panel Study of Income Dynamics">Panel Study of Income Dynamics</a> (PSID)</li>
<li><a href="China_Family_Panel_Studies" title="China Family Panel Studies">China Family Panel Studies</a> (CFPS)</li>
<li><a href="National_Longitudinal_Surveys" title="National Longitudinal Surveys">National Longitudinal Surveys</a> (NLSY)</li>
<li><a href="Labour_Force_Survey" title="Labour Force Survey">Labour Force Survey</a> (LFS)</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Data_sets_which_have_a_multi-dimensional_panel_design">Data sets which have a multi-dimensional panel design</h2></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Multidimensional_panel_data" title="Multidimensional panel data">Multidimensional panel data</a></div>
<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>
<style data-mw-deduplicate="TemplateStyles:r1239543626">
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</style><div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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.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("./mw/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("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/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}}
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</style><cite id="CITEREFDiggleHeagertyLiangZeger2002" class="citation book cs1">Diggle, Peter J.; Heagerty, Patrick; Liang, Kung-Yee; Zeger, Scott L. (2002). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/analysislongitud00digg_730"><i>Analysis of Longitudinal Data</i></a></span> (2nd ed.). Oxford University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/analysislongitud00digg_730/page/n19">2</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-852484-6</bdi>.</cite></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFFitzmauriceLairdWare2004" class="citation book cs1">Fitzmaurice, Garrett M.; Laird, Nan M.; Ware, James H. (2004). <i>Applied Longitudinal Analysis</i>. Hoboken: John Wiley & Sons. p. 2. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-21487-6</bdi>.</cite></span>
</li>
</ol></div></div>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li><cite id="CITEREFBaltagi2008" class="citation book cs1">Baltagi, Badi H. (2008). <i>Econometric Analysis of Panel Data</i> (Fourth ed.). Chichester: John Wiley & Sons. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-470-51886-1</bdi>.</cite></li>
<li><cite id="CITEREFDaviesLahiri1995" class="citation journal cs1">Davies, A.; Lahiri, K. (1995). "A New Framework for Testing Rationality and Measuring Aggregate Shocks Using Panel Data". <i><a href="Journal_of_Econometrics" title="Journal of Econometrics">Journal of Econometrics</a></i>. <b>68</b> (1): <span class="nowrap">205–</span>227. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0304-4076%2894%2901649-K">10.1016/0304-4076(94)01649-K</a>.</cite></li>
<li><cite id="CITEREFDaviesLahiri2000" class="citation book cs1">Davies, A.; Lahiri, K. (2000). "Re-examining the Rational Expectations Hypothesis Using Panel Data on Multi-Period Forecasts". <i>Analysis of Panels and Limited Dependent Variable Models</i>. Cambridge: Cambridge University Press. pp. <span class="nowrap">226–</span>254. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-63169-6</bdi>.</cite></li>
<li><cite id="CITEREFFrees2004" class="citation book cs1">Frees, E. (2004). <i>Longitudinal and Panel Data: Analysis and Applications in the Social Sciences</i>. New York: Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-82828-7</bdi>.</cite></li>
<li><cite id="CITEREFHsiao2003" class="citation book cs1"><a href="Cheng_Hsiao" title="Cheng Hsiao">Hsiao</a>, Cheng (2003). <i>Analysis of Panel Data</i> (Second ed.). New York: Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-52271-4</bdi>.</cite></li></ul>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><a rel="nofollow" class="external text" href="http://psidonline.isr.umich.edu/">PSID</a></li>
<li><a rel="nofollow" class="external text" href="http://www.kli.re.kr/klips/">KLIPS</a></li>
<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110719101922/http://www.pairfam.uni-bremen.de/en/study.html">pairfam</a></li>
<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140416182301/http://survey.keis.or.kr/ENCOMAM0000N.do">Korea Employment Survey</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
This article is issued from <a class="external text" title="Last edited on 2025-05-24" href="https://en.wikipedia.org/wiki/?title=Panel_data&oldid=1291885903">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.
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