  We consider the estimation of the transition matrix in the high-dimensional
time-varying vector autoregression (TV-VAR) models. Our model builds on a
general class of locally stationary VAR processes that evolve smoothly in time.
We propose a hybridized kernel smoothing and $\ell^1$-regularized method to
directly estimate the sequence of time-varying transition matrices. Under the
sparsity assumption on the transition matrix, we establish the rate of
convergence of the proposed estimator and show that the convergence rate
depends on the smoothness of the locally stationary VAR processes only through
the smoothness of the transition matrix function. In addition, for our
estimator followed by thresholding, we prove that the false positive rate (type
I error) and false negative rate (type II error) in the pattern recovery can
asymptotically vanish in the presence of weak signals without assuming the
minimum nonzero signal strength condition. Favorable finite sample performances
over the $\ell^2$-penalized least-squares estimator and the unstructured
maximum likelihood estimator are shown on simulated data. We also provide two
real examples on estimating the dependence structures on financial stock prices
and economic exchange rates datasets.
