Asymptotics for empirical eigenvalue processes in high-dimensional linear factor models

When vector-valued observations are of high dimension N relative to the sample size T, it is common to employ a linear factor model in order to estimate the underlying covariance structure or to further understand the relationship between coordinates. Asymptotic analyses of such models often consider the case in which both N andT tend jointly to infinity. Within this framework, we derive weak convergence results for processes of partial sample estimates of the largest eigenvalues of the sample covariance matrix. It is shown that if the effect of the factors is sufficiently strong, then the processes associated with the largest eigenvalues have Gaussian limits under general conditions on the divergence rates of N andT, and the underlying observations. If the common factors are “weak”, then N must grow much more slowly in relation to T in order for the largest eigenvalue processes to have a Gaussian limit. We apply these results to develop general tests for structural stability of linear factor models that are based on measuring the fluctuations in the largest eigenvalues throughout the sample, which we investigate further by means of a Monte Carlo simulation study and an application to US treasury yield curve data.