Abstract
ABSTRACT This paper explores the homogeneity of coefficient functions in nonlinear models with functional coefficients and identifies the underlying semiparametric modelling structure. With initial kernel estimates, we combine the classic hierarchical clustering method with a generalised version of the information criterion to estimate the number of clusters, each of which has a common functional coefficient, and determine the membership of each cluster. To identify a possible semi-varying coefficient modelling framework, we further introduce a penalised local least squares method to determine zero coefficients, non-zero constant coefficients and functional coefficients which vary with an index variable. Through the nonparametric kernel-based cluster analysis and the penalised approach, we can substantially reduce the number of unknown parametric and nonparametric components in the models, thereby achieving the aim of dimension reduction. Under some regularity conditions, we establish the asymptotic properties for the proposed methods including the consistency of the homogeneity pursuit. Numerical studies, including Monte-Carlo experiments and two empirical applications, are given to demonstrate the finite-sample performance of our methods.
Highlights
IntroductionWhen the number of functional coefficients is large or moderately large, it is wellknown that a direct nonparametric estimation of the potentially p different coefficient functions in model (1.1) would be unstable
We consider the functional-coefficient model defined byYt = X⊺tβ0(Ut) + εt, t = 1, · · ·, n, (1.1)where Yt is a response variable, Xt = (Xt1, · · ·, Xtp)⊺ is a p-dimensional vector of random covariates, ⊺β0(·) = β01(·), · · ·, β0p(·) is a p-dimensional vector of functional coefficients, Ut is a univariate index variable, and εt is an independent and identically distributed (i.i.d.) error term
There have been some extensive studies in the literature on selecting significant variables in functional-coefficient models (Fan, Ma and Dai, 2014; Liu, Li and Wu, 2014) or exploring certain rank-reduced structure in functional coefficients (Jiang et al, 2013; Chen, Li and Xia, 2019), both of which aim to reduce the dimension of unknown functional coefficients and improve estimation efficiency
Summary
When the number of functional coefficients is large or moderately large, it is wellknown that a direct nonparametric estimation of the potentially p different coefficient functions in model (1.1) would be unstable. To address this issue, there have been some extensive studies in the literature on selecting significant variables in functional-coefficient models (Fan, Ma and Dai, 2014; Liu, Li and Wu, 2014) or exploring certain rank-reduced structure in functional coefficients (Jiang et al, 2013; Chen, Li and Xia, 2019), both of which aim to reduce the dimension of unknown functional coefficients and improve estimation efficiency.
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