We study statistical inference on unit roots and cointegration for time series in a Hilbert space. We develop statistical inference on the number of common stochastic trends embedded in the time series, i.e., the dimension of the nonstationary subspace. We also consider tests of hypotheses on the nonstationary and stationary subspaces themselves. The Hilbert space can be of an arbitrarily large dimension, and our methods remain asymptotically valid even when the time series of interest takes values in a subspace of possibly unknown dimension. This has wide applicability in practice; for example, to cointegrated vector time series that are either high-dimensional or of finite dimension, to high-dimensional factor models that include a finite number of nonstationary factors, to cointegrated curve-valued (or function-valued) time series, and to nonstationary dynamic functional factor models. We include two empirical illustrations to the term structure of interest rates and labor market indices, respectively.

翻译：我们研究了希尔伯特空间中时间序列的单位根与协整的统计推断。我们发展了关于时间序列中嵌入的共同随机趋势数量（即非平稳子空间的维数）的统计推断方法。同时，我们还考虑了针对非平稳子空间和平稳子空间本身的假设检验。希尔伯特空间的维数可以是任意大的，且即使所关注的时间序列取值于一个可能未知维数的子空间中，我们的方法仍保持渐近有效性。这在实际应用中具有广泛的适用性；例如，适用于高维或有限维的协整向量时间序列、包含有限数量非平稳因子的高维因子模型、协整的曲线值（或函数值）时间序列，以及非平稳动态函数因子模型。我们分别以利率期限结构和劳动力市场指数为例进行了两项实证说明。