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Abstract:
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We study the asymptotic behaviour of the squared canonical correlation coefficients between high-dimensional Gaussian random walk and its own innovations when the dimensionality and the number of observation increase proportionally. These coefficients can be interpreted as the eigenvalues of a product of two interconnected high-dimensional projection matrices. We find that their empirical distribution converges to the Wachter distribution, that there are no eigenvalues outside the support of the Wachter distribution, asymptotically, and that corresponding regular linear spectral statistics are asymptotically normal. Our results have important applications in econometric analysis of co-integration, where rejecting the hypothesis of no co-integration can be interpreted as the existence of a long-run equilibrium relationship between analysed non-stationary time series.
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