randomparameters

Random parameters are model parameters treated as random variables rather than fixed constants. In statistical modeling, this approach allows the effect of a predictor to vary across entities such as individuals, groups, or time. This stands in contrast to fixed-parameter models, where all parameters are assumed identical across observations.

In practice, random parameters appear in several forms. In linear mixed models and generalized linear mixed models, some parameters—called random effects—vary by group and are assumed to follow a specified distribution, typically multivariate normal. In random-parameter or mixed logit models, coefficients themselves are random and distributed across individuals, capturing heterogeneity in preferences or behaviors. Estimation generally involves integrating over the distribution of the random parameters, using methods such as maximum simulated likelihood, hierarchical Bayes, or expectation-maximization algorithms.

Interpretation centers on variance components. The estimated variance of a random parameter measures how much the effect diverges across units; a larger variance indicates greater heterogeneity. Correlations among random parameters can reveal related patterns of variation.

Applications are widespread in economics, marketing, psychology, and biostatistics. Examples include modeling varying price sensitivities in demand estimation, allowing for individual differences in treatment effects, or capturing school- or region-level variation in outcomes.

Challenges include identifiability, the need for sufficient data to estimate variance components, and computational demands due to integration over parameter distributions. Nonetheless, random parameters provide a flexible framework to model real-world heterogeneity.