Canonical correlation analysis of stochastic trends via functional approximation

This paper proposes a novel approach for semiparametric inference on the number s of common trends and their loading matrix ψ in I(1)/I(0) systems. It combines functional approximation of limits of random walks and canonical correlations analysis, performed between the p observed time series of length T and the first K discretized elements of an L2 basis. Tests and selection criteria on s, and estimators and tests on ψ are proposed; their properties are discussed as T and K diverge sequentially for fixed p and s. It is found that tests on s are asymptotically pivotal, selection criteria of s are consistent, estimators of ψ are T-consistent, mixed-Gaussian and efficient, so that Wald tests on ψ are asymptotically Normal or χ2. The paper also discusses asymptotically pivotal misspecification tests for checking model assumptions. The approach can be coherently applied to subsets or aggregations of variables in a given panel. Monte Carlo simulations show that these tools have reasonable performance for T≥10p and p≤300. An empirical analysis of 20 exchange rates illustrates the methods.