  In this paper, we study the model selection and structure specification for
the generalised semi-varying coefficient models (GSVCMs), where the number of
potential covariates is allowed to be larger than the sample size. We first
propose a penalised likelihood method with the LASSO penalty function to obtain
the preliminary estimates of the functional coefficients. Then, using the
quadratic approximation for the local log-likelihood function and the adaptive
group LASSO penalty (or the local linear approximation of the group SCAD
penalty) with the help of the preliminary estimation of the functional
coefficients, we introduce a novel penalised weighted least squares procedure
to select the significant covariates and identify the constant coefficients
among the coefficients of the selected covariates, which could thus specify the
semiparametric modelling structure. The developed model selection and structure
specification approach not only inherits many nice statistical properties from
the local maximum likelihood estimation and nonconcave penalised likelihood
method, but also computationally attractive thanks to the computational
algorithm that is proposed to implement our method. Under some mild conditions,
we establish the asymptotic properties for the proposed model selection and
estimation procedure such as the sparsity and oracle property. We also conduct
simulation studies to examine the finite sample performance of the proposed
method, and finally apply the method to analyse a real data set, which leads to
some interesting findings.
