  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.
