["Oxford Bulletin of Economics and Statistics, EarlyView. ", "\nABSTRACT\nThis paper discusses semiparametric inference on hypotheses on the cointegration and the attractor spaces for I(1)$$ I(1) $$ linear processes with moderately large cross‐sectional dimension. The approach is based on sample canonical correlations and functional approximation of Brownian motions, and it can be applied both to the whole system and or to any set of linear combinations of it. The hypotheses of interest are cast in terms of the number of stochastic trends in specified subsystems, and inference is based either on selection criteria or on sequences of tests. This paper derives the limit distribution of these tests in the special one‐dimensional case, and discusses asymptotic properties of the derived inference criteria for hypotheses on the attractor space for sequentially diverging sample size and number of basis elements in the functional approximation. Finite sample properties are analysed via a Monte Carlo study and an empirical illustration on exchange rates is provided.\n"]