This paper discusses semiparametric inference on hypotheses on the cointegration and the attractor spaces for I(1) linear processes, using canonical correlation analysis and functional approximation of Brownian Motions. It proposes inference criteria based on the estimation of the number of common trends in various subsets of variables, and compares them to sequences of tests of hypotheses. The exact limit distribution for one of the test statistics is derived in the univariate case. Properties of the inferential tools are discussed theoretically and illustrated via a Monte Carlo study. An empirical analysis of exchange rates is also included.

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