Simultaneous Confidence Bands and Hypothesis Testing in Varying‐coefficient Models

Regression analysis is one of the most commonly used techniques in statistics. When the dimension of independent variables is high, it is difficult to conduct efficient non‐parametric analysis straightforwardly from the data. As an important alternative to the additive and other non‐parametric models, varying‐coefficient models can reduce the modelling bias and avoid the “curse of dimensionality” significantly. In addition, the coefficient functions can easily be estimated via a simple local regression. Based on local polynomial techniques, we provide the asymptotic distribution for the maximum of the normalized deviations of the estimated coefficient functions away from the true coefficient functions. Using this result and the pre‐asymptotic substitution idea for estimating biases and variances, simultaneous confidence bands for the underlying coefficient functions are constructed. An important question in the varying coefficient models is whether an estimated coefficient function is statistically significantly different from zero or a constant. Based on newly derived asymptotic theory, a formal procedure is proposed for testing whether a particular parametric form fits a given data set. Simulated and real‐data examples are used to illustrate our techniques.