  We introduce a new methodology to conduct simultaneous inference of the
nonparametric component in partially linear time series regression models where
the nonparametric part is a multivariate unknown function. In particular, we
construct a simultaneous confidence region (SCR) for the multivariate function
by extending the high-dimensional Gaussian approximation to dependent processes
with continuous index sets. Our results allow for a more general dependence
structure compared to previous works and are widely applicable to a variety of
linear and nonlinear autoregressive processes. We demonstrate the validity of
our proposed methodology by examining the finite-sample performance in the
simulation study. Finally, an application in time series, the forward premium
regression, is presented, where we construct the SCR for the foreign exchange
risk premium from the exchange rate and macroeconomic data.
