  To better understand the spatial structure of large panels of economic and
financial time series and provide a guideline for constructing semiparametric
models, this paper first considers estimating a large spatial covariance matrix
of the generalized $m$-dependent and $\beta$-mixing time series (with $J$
variables and $T$ observations) by hard thresholding regularization as long as
${{\log J \, \cx^*(\ct)}}/{T} = \Co(1)$ (the former scheme with some time
dependence measure $\cx^*(\ct)$) or $\log J /{T} = \Co(1)$ (the latter scheme
with some upper bounded mixing coefficient). We quantify the interplay between
the estimators' consistency rate and the time dependence level, discuss an
intuitive resampling scheme for threshold selection, and also prove a general
cross-validation result justifying this. Given a consistently estimated
covariance (correlation) matrix, by utilizing its natural links with graphical
models and semiparametrics, after "screening" the (explanatory) variables, we
implement a novel forward (and backward) label permutation procedure to cluster
the "relevant" variables and construct the corresponding semiparametric model,
which is further estimated by the groupwise dimension reduction method with
sign constraints. We call this the SCE (screen - cluster - estimate) approach
for modeling high dimensional data with complex spatial structure. Finally we
apply this method to study the spatial structure of large panels of economic
and financial time series and find the proper semiparametric structure for
estimating the consumer price index (CPI) to illustrate its superiority over
the linear models.
