A varying-coefficients regression model of the form \(Y\sum_{j=1}^p a_j(U)X_j+\varepsilon_j\) is considered, where \((U,X_1,\dots,X_p)\) are covariates, \(Y\) is a response variable, \(\text{Var}(\varepsilon \mid U,X_1,\dots,X_p)=\sigma^2(U)\).
Local least squares with a local Tailor series approximation of \(a_j\) are used to derive estimators for \(a_j\). Asymptotics of the estimators in the uniform norm are investigated. The results are applied to confidence bands construction and hypothesis testing. Results of simulations and analysis of an environmental data set are presented.