Fay–Herriot model

The Fay–Herriot model is a statistical model which includes some distinct variation for each of several subgroups of observations. It is an area-level model, meaning some input data are associated with sub-aggregates such as regions, jurisdictions, or industries. The model produces estimates about the subgroups. The model is applied in the context of small area estimation in which there is a lot of data overall, but not much for each subgroup.

The subgroups are determined in advance of estimation and are built into the model structure. The model combines, by averaging, estimates of fixed effects and of the random effects type. The model is typically used to adjust for group-related differences in some dependent variable.

In random effects models like the Fay–Herriot, estimation is built on the assumption that the effects associated with subgroups are drawn independently from a normal (Gaussian) distribution, whose variance is estimated from the data on each subgroup. It is more common to use a fixed-effects model instead for many systematically different groups. A mixed random effects model like the Fay–Herriot is preferred if there are not enough observations per group to reliably estimate the fixed effects, or if for some reason fixed effects would not be consistently estimated.

The Fay–Herriot is a two-stage hierarchical model. The parameters of the distributions within the groups are often assumed to be independent, or it is assumed that they are correlated to those measured for another variable.