Varying-coefficient linear models arise from multivariate nonparametric regression, nonlinear time series modelling and forecasting, functional data analysis, longitudinal data analysis, and others. It has been a common practice to assume that the vary-coefficients are functions of a given variable which is often called an index. A frequently asked question is which variable should be used as the index. In this paper, we explore the class of the varying-coefficient linear models in which the index is unknown and is estimated as a linear combination of regression and/or other variables. This will enlarge the modelling capacity substantially. We search for the index such that the derived varying-coefficient model provides the best approximation to the underlying unknown multi-dimensional regression function in the least square sense. The search is implemented through the newly proposed hybrid backfitting algorithm. The core of the algorithm is the alternative iteration between estimating the index through a one-step scheme and estimating coefficient functions through a one-dimensional local linear smoothing. The generalised cross-validation method for choosing bandwidth is efficiently incorporated into the algorithm. The locally significant variables are selected in terms of the combined use of t-statistic and Akaike information criterion. We further extend the algorithm for the models with two indices. Simulation shows that the proposed methodology has appreciable flexibility to model complex multivariate nonlinear 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.
