We introduce a new methodology to conduct simultaneous inference of non-parametric trend in a partially linear time series regression model where the trend is a multivariate unknown function. In particular, we construct a simultaneous confidence region (SCR) for the trend 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 auto-regressive processes. We demonstrate the validity of our proposed inference approach by examining the finite-sample performance in the simulation study. The method is also applied to a real example in time series: the forward premium regression, where we construct the SCR for the foreign exchange risk premium in the exchange rate data.

翻译：我们采用新方法,在一个部分线性时间序列回归模型中同时对非参数趋势进行推断,该趋势是一个多变量的未知功能,特别是,我们通过将高斯高度近似扩展至具有连续指数集的依附进程,为趋势功能建立一个同时的互信区域;我们的结果允许与以往的工程相比,形成更普遍的依赖性结构,并广泛适用于各种线性和非线性自动回归过程;我们通过在模拟研究中审查有限抽样性能,表明我们提议的推论方法的有效性;这个方法还适用于一个实时序列中的实际例子:远期溢价回归,我们在汇率数据中为外汇风险溢价构建了SCR。