Random effects model

 ( Regression Models ) 

In econometrics, a random effects model, also called a variance components model, is a statistical model where the model effects are . It is a kind of hierarchical linear model, which assumes that the data being analysed are drawn from a hierarchy of different populations whose differences relate to that hierarchy. A random effects model is a special case of a mixed model.

Contrast this to the biostatistics definitions,

(2026). 9780471214878, John Wiley & Sons. ISBN 9780471214878

as biostatisticians use "fixed" and "random" effects to respectively refer to the population-average and subject-specific effects (and where the latter are generally assumed to be unknown, latent variables).

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Qualitative description Random effect models assist in controlling for unobserved heterogeneity when the heterogeneity is constant over time and not correlated with independent variables. This constant can be removed from longitudinal data through differencing, since taking a first difference will remove any time invariant components of the model.

Jeffrey Wooldridge (2026). 9780262232586, MIT Press. ISBN 9780262232586

Two common assumptions can be made about the individual specific effect: the random effects assumption and the fixed effects assumption. The random effects assumption is that the individual unobserved heterogeneity is uncorrelated with the independent variables. The fixed effect assumption is that the individual specific effect is correlated with the independent variables.

If the random effects assumption holds, the random effects estimator is more efficient than the fixed effects model.

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Simple example Suppose $m$ large elementary schools are chosen randomly from among thousands in a large country. Suppose also that $n$ pupils of the same age are chosen randomly at each selected school. Their scores on a standard aptitude test are ascertained. Let $Y\_\{ij\}$ be the score of the $j$-th pupil at the $i$-th school.

A simple way to model this variable is

$$

   Y_{ij} = \mu + U_i + W_{ij},\,
 
     

where $\backslash mu$ is the average test score for the entire population.

In this model $U\_i$ is the school-specific random effect: it measures the difference between the average score at school $i$ and the average score in the entire country. The term $W\_\{ij\}$ is the individual-specific random effect, i.e., it's the deviation of the $j$-th pupil's score from the average for the $i$-th school.

The model can be augmented by including additional explanatory variables, which would capture differences in scores among different groups. For example:

$$

   Y_{ij} = \mu + \beta_1 \mathrm{Sex}_{ij} + \beta_2 \mathrm{ParentsEduc}_{ij} + U_i + W_{ij},\,
 
     

where $\backslash mathrm\{Sex\}\_\{ij\}$ is a binary dummy variable and $\backslash mathrm\{ParentsEduc\}\_\{ij\}$records, say, the average education level of a child's parents. This is a mixed model, not a purely random effects model, as it introduces fixed-effects terms for Sex and Parents' Education.

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Variance components The variance of $Y\_\{ij\}$ is the sum of the variances $\backslash tau^2$ and $\backslash sigma^2$ of $U\_i$ and $W\_\{ij\}$ respectively.

Let

$\backslash overline\{Y\}\_\{i\backslash bullet\}\; =\; \backslash frac\{1\}\{n\}\backslash sum\_\{j=1\}^n\; Y\_\{ij\}$

be the average, not of all scores at the $i$-th school, but of those at the $i$-th school that are included in the random sample. Let

$\backslash overline\{Y\}\_\{\backslash bullet\backslash bullet\}\; =\; \backslash frac\{1\}\{mn\}\backslash sum\_\{i=1\}^m\backslash sum\_\{j=1\}^n\; Y\_\{ij\}$

be the grand average.

Let

$SSW\; =\; \backslash sum\_\{i=1\}^m\backslash sum\_\{j=1\}^n\; (Y\_\{ij\}\; -\; \backslash overline\{Y\}\_\{i\backslash bullet\})^2\; \backslash ,$

$SSB\; =\; n\backslash sum\_\{i=1\}^m\; (\backslash overline\{Y\}\_\{i\backslash bullet\}\; -\; \backslash overline\{Y\}\_\{\backslash bullet\backslash bullet\})^2\; \backslash ,$

be respectively the sum of squares due to differences within groups and the sum of squares due to difference between groups. Then it can be shown that

$\backslash frac\{1\}\{m(n\; -\; 1)\}E(SSW)\; =\; \backslash sigma^2$

and

$\backslash frac\{1\}\{(m\; -\; 1)n\}E(SSB)\; =\; \backslash frac\{\backslash sigma^2\}\{n\}\; +\; \backslash tau^2.$

These "expected mean squares" can be used as the basis for estimation of the "variance components" $\backslash sigma^2$ and $\backslash tau^2$.

The $\backslash sigma^2$ parameter is also called the .

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Marginal likelihood /ref>

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Applications Random effects models used in practice include the Bühlmann model of insurance contracts and the Fay-Herriot model used for small area estimation.

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See also

• Bühlmann model
• Hierarchical linear modeling
• Fixed effects
• MINQUE
• Covariance estimation
• Conditional variance
• Panel analysis

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Further reading

• Badi H. Baltagi (2026). 9780470518861, Wiley. ISBN 9780470518861
• Cheng Hsiao (2026). 9780521522717, Cambridge University Press. . ISBN 9780521522717
• Jeffrey M. Wooldridge (2026). 9780262232197, MIT Press. . ISBN 9780262232197

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External links

• Fixed and random effects models
• How to Conduct a Meta-Analysis: Fixed and Random Effect Models

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