Nonlinear Random Effects Model
Also known as: nonlinear RE model, NLRE model, random effects nonlinear panel model, mixed nonlinear panel model
The nonlinear random effects model extends classical random effects estimation to settings where the outcome variable is binary, count-based, censored, or otherwise non-continuously distributed across panel units. It accounts for unobserved individual heterogeneity by treating unit-specific effects as random draws from a distribution, then integrating them out to form a likelihood that can be maximised over the structural parameters.
Key highlights
- Handles non-continuous and limited dependent variables (binary, count, censored) in panel settings that linear random effects cannot model correctly.
- Consistent and efficient in short panels (small T) where fixed effects incur incidental parameters bias.
- Directly identifies the variance of the individual-specific effect, quantifying the importance of unobserved heterogeneity.
- Average partial effects can be computed and compared across specifications.
- Accommodates unbalanced panels naturally within the likelihood framework.
Intuition
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How it works
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When to use it
Use the nonlinear random effects model when you have repeated observations on the same units (panel data) and a non-continuous outcome such as a binary indicator, an event count, or a censored variable, and when you are willing to assume that the unobserved individual heterogeneity is uncorrelated with the regressors. It is especially valuable in short panels (small T, large N) where fixed effects estimation would consume too many degrees of freedom or suffer from incidental parameters bias. Do not use it when the regressors are likely correlated with the unobserved individual effect — use a correlated random effects (Mundlak-Chamberlain) specification or a conditional fixed effects estimator instead.
Strengths & limitations
- Handles non-continuous and limited dependent variables (binary, count, censored) in panel settings that linear random effects cannot model correctly.
- Consistent and efficient in short panels (small T) where fixed effects incur incidental parameters bias.
- Directly identifies the variance of the individual-specific effect, quantifying the importance of unobserved heterogeneity.
- Average partial effects can be computed and compared across specifications.
- Accommodates unbalanced panels naturally within the likelihood framework.
- Consistency depends on the strict assumption that c_i is orthogonal to the regressors; violation leads to biased estimates.
- Numerical integration of the marginal likelihood can be computationally demanding and sensitive to the number of quadrature points.
- Unlike linear random effects, there is no simple Hausman test; testing against fixed effects alternatives requires more complex procedures.
- Individual partial effects are not separately identified — only average partial effects can be reported.
- Incidental parameters bias is avoided, but only because the random effects assumption is imposed, which may itself be misspecified.
Common pitfalls
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Applications
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Frequently asked
What is the incidental parameters problem and why does it motivate random effects?
In nonlinear fixed effects models, each individual requires its own parameter. When N is large but T is small, these incidental parameters are estimated from very few observations, causing bias that does not vanish as the sample grows. The random effects approach avoids this by integrating out individual effects rather than estimating them directly.
How do I test whether random effects or fixed effects is more appropriate?
A Mundlak-Chamberlain test adds the within-individual means of time-varying regressors to the random effects model. A joint test of their significance evaluates whether the random effects assumption is violated. Conditional likelihood estimators (e.g., conditional logit) provide a fixed effects alternative for binary outcomes but do not identify time-invariant regressors.
How many quadrature points should I use?
Start with 12 points and refit with 20 and 24. If coefficients change substantially, the default is insufficient — increase until estimates stabilise. Adaptive quadrature (available in Stata's xtprobit and R's lme4) typically requires fewer points than standard Gauss-Hermite quadrature.
Can I use a random effects logit instead of probit?
Yes. Both are consistent under their respective distributional assumptions for the random effect. The logit assumption is slightly more convenient computationally, but the probit assumption (normal random effect) is more natural and aligns with the integration that most software implements. In practice the two give similar average partial effects.
What software packages support nonlinear random effects models?
Stata offers xtprobit, xtlogit, xtpoisson, and xttobit with quadrature-based integration. R provides glmer (lme4), gnm, and the gamlss package. SAS provides PROC NLMIXED, which is highly flexible for custom nonlinear likelihoods.
Sources
- 1.Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data (2nd ed.). MIT Press.ISBN 978-0262232586
- 2.Hsiao, C. (2014). Analysis of Panel Data (3rd ed.). Cambridge University Press.ISBN 978-1107038691
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Cite this page
ScholarGate. (2026, June 3). Nonlinear Random Effects Model. ScholarGate. https://scholargate.app/econometrics/nonlinear-random-effects-model