🤖 AI Summary
This paper addresses unit root and cointegration inference for functional time series in Hilbert spaces, focusing on identifying the number of common stochastic trends—i.e., the dimension of the nonstationary subspace—and conducting hypothesis tests for the nonstationary and stationary subspaces. We systematically extend classical unit root and cointegration theory to arbitrary-dimensional Hilbert spaces for the first time, proposing a projection-operator spectral analysis method that is asymptotically efficient, fully data-driven (requiring no prior dimension specification), and uniformly applicable to high-dimensional vector time series, dynamic functional factor models, and curve-valued time series. We further develop theoretically grounded dimension-selection criteria and valid test statistics with rigorous asymptotic properties. Empirical applications to the U.S. yield curve and labor market indices robustly identify key common nonstationary trends, demonstrating the method’s interpretability and practical utility for real-world high-dimensional functional time series.
📝 Abstract
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 the case of cointegrated vector time series that are either high-dimensional or of finite dimension, to high-dimensional factor model that includes 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.