Ineconometrics, arandom effects model, also called avariance components model, is astatistical model where the model effects arerandom variables. It is a kind ofhierarchical 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 amixed model.

Contrast this to thebiostatistics definitions,^{[1][2][3][4][5]} 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).

Random effect models assist in controlling forunobserved 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.^[6]

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.^[6]

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

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

A simple way to model this variable is

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

where${\displaystyle \mu }$ is the average test score for the entire population.

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

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

${\displaystyle Y_{ij}=\mu +{\beta }_1{\mathrm{S}\mathrm{e}\mathrm{x}}_{ij}+{\beta }_2{\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}\mathrm{E}\mathrm{d}\mathrm{u}\mathrm{c}}_{ij}+U_i+W_{ij},}$

where${\displaystyle {\mathrm{S}\mathrm{e}\mathrm{x}}_{ij}}$ is a binarydummy variable and${\displaystyle {\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}\mathrm{E}\mathrm{d}\mathrm{u}\mathrm{c}}_{ij}}$records, say, the average education level of a child's parents. This is amixed model, not a purely random effects model, as it introducesfixed-effects terms for Sex and Parents' Education.

The variance of${\displaystyle Y_{ij}}$ is the sum of the variances${\displaystyle {\tau }^2}$ and${\displaystyle {\sigma }^2}$ of${\displaystyle U_i}$ and${\displaystyle W_{ij}}$ respectively.

Let

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

be the average, not of all scores at the${\displaystyle i}$-th school, but of those at the${\displaystyle i}$-th school that are included in therandom sample. Let

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

be thegrand average.

Let

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

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

be respectively the sum of squares due to differenceswithin groups and the sum of squares due to differencebetween groups. Then it can be shown^{[citation needed]} that

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

and

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

These "expected mean squares" can be used as the basis forestimation of the "variance components"${\displaystyle {\sigma }^2}$ and${\displaystyle {\tau }^2}$.

The${\displaystyle {\sigma }^2}$ parameter is also called theintraclass correlation coefficient.

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For random effects models themarginal likelihoods are important.^[7]

Random effects models used in practice include theBühlmann model of insurance contracts and theFay-Herriot model used forsmall area estimation.

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

• Baltagi, Badi H. (2008).Econometric Analysis of Panel Data (4th ed.). New York, NY: Wiley. pp. 17–22.ISBN 978-0-470-51886-1.
• Hsiao, Cheng (2003).Analysis of Panel Data (2nd ed.). New York, NY: Cambridge University Press. pp. 73–92.ISBN 0-521-52271-4.
• Wooldridge, Jeffrey M. (2002).Econometric Analysis of Cross Section and Panel Data. Cambridge, MA: MIT Press. pp. 257–265.ISBN 0-262-23219-7.
• Gomes, Dylan G.E. (20 January 2022)."Should I use fixed effects or random effects when I have fewer than five levels of a grouping factor in a mixed-effects model?".PeerJ.10 e12794.doi:10.7717/peerj.12794.PMC 8784019.PMID 35116198.

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

• Regression models
• Analysis of variance

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