Simultaneous inference of a partially linear model in time series

We introduce a new methodology to conduct simultaneous inference of the non‐parametric component in partially linear time series regression models where the non‐parametric part is a multi‐variate unknown function. In particular, we construct a simultaneous confidence region (SCR) for the multi‐variate 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 non‐linear 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.