Varyingcoefficient

Varying-coefficient models are regression models in which the regression coefficients are allowed to vary smoothly with an index variable, most commonly time. This framework captures how relationships between predictors and a response can change over the index, providing a flexible alternative to constant-coefficient models.

In the linear form, the model can be written as y_i = sum_j beta_j(z_i) x_{ij} + error_i, or equivalently E[y_i | x_i, z_i] = x_i^T beta(z_i). When a link function is used for non-Gaussian responses, g(E[y_i | x_i, z_i]) = x_i^T beta(z_i), where beta_j(z) are smooth functions of the index z. The functions beta_j(z) are estimated from data and are typically assumed to be sufficiently smooth.

Estimation methods commonly involve representing beta_j(z) with basis expansions, such as beta_j(z) = sum_k c_{jk} B_k(z), where B_k are spline bases. The coefficients c_{jk} are then estimated via penalized likelihood or penalized least squares, with smoothing parameters chosen by cross-validation or information criteria. Alternative approaches use local regression or local likelihood with kernel weights around z. Conceptually, the model can be viewed as a generalized additive model with coefficient functions instead of additive terms, and can be implemented within a mixed-effects framework by treating the coefficients as random functions.

Extensions of the idea include generalized varying-coefficient models for GLMs, models for longitudinal data with time-varying effects, and survival models with time-varying coefficients. Applications span economics, epidemiology, environmental science, and finance, where dynamic or context-dependent relationships are important. The concept was popularized by Hastie and Tibshirani in the early 1990s and has since become a standard tool for modeling nonstationary relationships.