Estimating the number of common trends in large T and N factor models via canonical correlations analysis

Asymptotics for squared canonical correlations is considered when the analysis is performed between levels and cumulated levels of N variables generated by a factor model with s common trends. It is first shown that, for fixed N (and s) and as T → ∞, the largest s squared canonical correlations converge to a non-degenerate distribution while the remaining N − s squared canonical correlations converge in probability to 0. Further it is shown that, when T → ∞ is followed by N − s → ∞ sequentially, all the largest s squared canonical correlations converge in probability to 1. This feature allows to estimate the number of common trends as the number for which consecutive squared canonical correlations have the largest decrease, equal to 1 in the lmit; the criterion is shown to be consistent. A Monte Carlo simulation study illustrates the findings.