Signal-plus-noise theory for cointegration with nonzero rank Π

Develop a spiked (signal-plus-noise) asymptotic theory for CCA-based cointegration analysis when the VAR(1) coefficient matrix Π has positive rank, including characterization of outlier canonical correlations beyond the Wachter bulk and the limiting behavior of the associated sample canonical vectors in the proportional-growth regime.

Background

The survey establishes proportional-growth asymptotics for the null (Π=0) via Wachter limits and Airy_1 edge statistics for the modified procedure. Simulations with Π≠0 show apparent spectral spikes, suggesting a parallel to spiked models in CCA and PCA.

However, a full signal-plus-noise asymptotic theory for cointegration (Π≠0) has not yet been completed as of 2024.

References

As of 2024, such a theory has yet to be fully established.

— Canonical Correlation Analysis: review (2411.15625 - Bykhovskaya et al., 23 Nov 2024) in Subsection “Outlook: signal plus noise for cointegration” (Chapter 5)