Adaptive varying-coefficient linear models |
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Authors: | Jianqing Fan Qiwei Yao Zongwu Cai |
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Affiliation: | University of North Carolina, Chapel Hill, USA; London School of Economics and Political Science, UK; University of North Carolina, Charlotte, USA |
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Abstract: | Summary. 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. |
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Keywords: | Akaike information criterion Backfitting algorithm Generalized cross-validation Local linear regression Local significant variable selection One-step estimation Smoothing index |
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