Correlated Random-Effects Logit and Probit

Why Allow Correlation Between Effects and Covariates?

In a panel, the same individual is observed repeatedly, and a binary outcome may depend on persistent characteristics such as preferences, underlying health, or ability. A random-intercept logit or probit model accounts for this persistent heterogeneity by adding an individual-specific effect. Its usual independence assumption, however, can be implausible: people with persistently high covariate values may also differ in unobserved ways that affect the outcome.

A correlated random-effects model relaxes that independence assumption in a structured way. It models the association between the individual effect and observed covariates using each individual’s covariate means. This allows the model to distinguish associations arising from changes within an individual over time from associations between individuals with different average covariate levels. The means do not automatically remove all confounding or make an association causal; the adequacy of the model’s assumptions remains important.

Model and Assumptions

Write \(F(z)=1/(1+\exp(-z))\) for the logistic cumulative distribution function and \(F(z)=\Phi(z)\) for the standard normal cumulative distribution function. A common correlated random-effects specification is

Here \(X_i\) denotes the individual’s observed covariate history, \(\beta\) is the coefficient vector on current covariates, \(\theta\) is the coefficient vector on their individual means, and \(a_i\) is the remaining random intercept. The logit model uses the logistic \(F\); the probit model uses the normal \(F\). In a standard random-effects likelihood, outcomes are conditionally independent over time given the covariate history and \(a_i\), and the distribution of \(a_i\) is specified, often as normal with mean zero and variance \(\sigma_a^2\).

The key working assumption is that, after accounting for the included mean vector, the remaining effect \(a_i\) is independent of the covariate history. Equivalently, the model represents the part of the individual effect associated with observed covariates through their means and treats the remainder as independent. This is a restriction, not a mathematical guarantee: if dependence also involves other features of the covariate history, such as trends or particular lags, means alone may not capture it.

Because the same \(a_i\) applies across an individual’s observations, the likelihood contribution for that individual integrates the joint conditional likelihood over the distribution of \(a_i\). In general, this integration does not reduce to fitting an ordinary logit or probit to independent person-period observations. For logit with a normally distributed random effect, the integral generally has no closed-form expression and is evaluated numerically.

How the Means Separate Within and Between Associations

For clarity, first consider one time-varying covariate \(x_{it}\), with coefficient \(\beta\) and mean coefficient \(\theta\). The covariate terms in the index can be rearranged:

The deviation \(x_{it}-\bar{x}_i\) describes how the individual’s current value differs from their own average. The mean \(\bar{x}_i\) describes how that individual’s average compares across individuals. Thus \(\beta\) is the within-individual association on the model’s index scale, and \(\beta+\theta\) is the between-individual association on that scale. The coefficient \(\theta\) by itself is the adjustment to the current-value coefficient that makes the between association differ from the within association.

This decomposition is a statement about the index, not a claim that either coefficient is a constant change in outcome probability. In logit and probit models, probabilities are nonlinear functions of the index. Probability changes depend on the starting index and, when averaging over random effects, on the random-effect distribution as well.

Interpreting Probabilities and Random Effects

There are two distinct prediction targets. A probability conditional on a specified random effect is \(F(X_{it}'\beta+\bar{X}_i'\theta+a_i)\). A population-averaged probability for an individual with given covariate history integrates over the random-effect distribution:

where \(G\) is the assumed distribution of \(a_i\). These quantities need not agree. In particular, setting \(a_i=0\) does not generally produce the probability averaged over \(a_i\), because applying a nonlinear function and then averaging is not the same as applying the function at the average.

For a continuous covariate, the derivative of the conditional probability with respect to its current value, holding its individual mean and \(a_i\) fixed, is \(f(\eta_{it})\beta\), where \(f\) is the density corresponding to \(F\). The derivative for a population-averaged probability integrates this derivative over \(G\). For discrete covariates, a probability change is computed by comparing predictions at the two values. In either case, the probability-scale result depends on the covariate profile and should be interpreted with units and context.

What the Mean Terms Tell You

A nonzero mean coefficient indicates that the covariate’s individual average contributes to the response index, conditional on the current covariate value and the other included terms. In that sense, it is evidence that a model using current covariates alone would not represent the association captured by the specified mean terms. It is not, by itself, proof that all dependence between the individual effect and covariates has been handled, nor is it evidence of a causal effect.

For several time-varying covariates, the decomposition applies component by component. A joint test that their mean coefficients are all zero can assess whether the included means add explanatory information under the model. For one mean coefficient, a Wald test uses its estimate divided by its standard error; for several coefficients, a joint Wald test uses their estimated covariance matrix. Such tests rely on the fitted model and its inference procedure being appropriate.

Model Checks and Common Pitfalls

The mean terms are most informative when the covariate varies within individuals. For a time-invariant covariate, its individual mean equals its value, so including both as separate regressors creates exact collinearity; its separate current and mean coefficients cannot be identified in that specification. In an unbalanced panel, define means over the observed periods used for the analysis, and consider whether differing observation patterns affect their interpretation. Time indicators or other controls may also be needed when common time changes affect outcomes and covariates.

A central pitfall is to call the model “fixed effects” merely because it includes individual means. For example, one might reason, “The means absorb every stable unobserved difference, so the remaining coefficient is automatically causal.” That conclusion does not follow. The model still specifies a distribution for a random intercept and assumes its remaining component is independent of the observed covariate history conditional on the included means. Misspecified dependence, omitted time-varying confounders, or other violations can still bias interpretation. The within and between labels describe how the covariate terms enter the index; they do not, on their own, supply a causal design.

A second mistake is to interpret \(\beta\) as a probability change. In a logit model, a one-unit increase changes the conditional log odds by \(\beta\), holding the other index terms fixed; in a probit model, it changes the probit index by \(\beta\). Neither statement is a constant probability change. To communicate probability-scale implications, calculate contrasts or marginal effects for specified covariate profiles and state whether predictions condition on or integrate over the random effect.