  Precision matrices play important roles in many practical applications.
Motivated by temporally dependent multivariate data in modern social and
scientific studies, we consider the statistical inference of precision matrices
for high-dimensional time dependent observations. Specifically, we propose a
data-driven procedure to construct a class of simultaneous confidence regions
for the precision coefficients within an index set of interest. The confidence
regions can be applied to test for specific structures of a precision matrix
and to recover its nonzero components. We first construct an estimator of the
underlying precision matrix via penalized node-wise regressions, and then
develope the Gaussian approximation results on the maximal difference between
the estimated and true precision matrices. A computationally feasible
parametric bootstrap algorithm is developed to implement the proposed
procedure. Theoretical results indicate that the proposed procedure works well
without the second order cross-time stationary assumption on the data and
sparse structure conditions on the long-run covariance of the estimates.
Simulation studies and a real example on S&P 500 stock return data confirm the
performance of the proposed approach.
