coefficientvarying

Coefficient varying, or varying-coefficient models, are regression models in which regression coefficients are allowed to vary smoothly with an index variable, such as time or another covariate. This approach captures nonstationary or heterogeneous effects, where the influence of a predictor changes across contexts or over time.

Mathematically, for data pairs (y_i, x_{i1}, ..., x_{ip}) with an index z_i, the model is y_i = sum_{k=1}^p x_{ik} beta_k(z_i) + epsilon_i, where the beta_k(. ) are unknown smooth functions. In practice, each beta_k is represented by a basis expansion, for example beta_k(z) = sum_m theta_{km} B_m(z), turning estimation into fitting a (generalized) linear model for the coefficients theta_{km} with smoothing penalties. Alternative estimation strategies include local regression or local likelihood, where beta_k(z) is estimated pointwise with weighted data around z. Extensions allow multivariate indices, interactions, or random effects. The concept and methods were developed in the statistical literature in the 1990s, notably by Hastie and Tibshirani in their work on varying-coefficient models.

Applications span econometrics, environmental and epidemiological studies, where time-varying or context-dependent effects are important. The framework offers greater flexibility than models with fixed coefficients and can be related to generalized additive models and functional data analysis, sharing ideas about smooth function estimation and basis representations.

Key considerations include choosing the index variable, selecting smoothing parameters, and guarding against overfitting. Computationally, estimation can be more intensive than standard regression, but modern software provides practical implementations.