Adaptive varying–coefficient linear models

Varying–coefficient linear models arise from multivariate nonparametric regression, non–linear time series modelling and forecasting, functional data analysis, longitudinal data analysis and others. It has been a common practice to assume that the varying coefficients are functions of a given variable, which is often called an index. To enlarge the modelling capacity substantially, this paper explores a class of varying–coefficient linear models in which the index is unknown and is estimated as a linear combination of regressors and/or other variables. We search for the index such that the derived varying–coefficient model provides the least squares approximation to the underlying unknown multidimensional regression function. The search is implemented through a newly proposed hybrid backfitting algorithm. The core of the algorithm is the alternating iteration between estimating the index through a one–step scheme and estimating coefficient functions through one–dimensional local linear smoothing. The locally significant variables are selected in terms of a combined use of the t–statistic and the Akaike information criterion. We further extend the algorithm for models with two indices. Simulation shows that the methodology proposed has appreciable flexibility to model complex multivariate non–linear structure and is practically feasible with average modern computers. The methods are further illustrated through the Canadian mink–muskrat data in 1925–1994 and the pound—dollar exchange rates in 1974–1983.