When Standardization Spreads Factor Variation Across the Spectrum

Abstract

Standardizing variables by their sample standard deviations is routine in multivariate analysis and removes marginal scale differences before dependence is assessed. The resulting correlation matrix captures both dominant common components and dependence beyond them. A central inferential difficulty arises because the same random sample variances normalize the entire spectrum. In spiked covariance models, the sampling errors of leading and residual eigenvalues can become asymptotically independent. Sample standardization fundamentally changes this behavior and can preserve their dependence, allowing strong factors to remain a major source of uncertainty even after the leading eigenvalues are excluded. To obtain valid inference for both residual and full spectrum statistics, we establish their joint sampling law for sample correlation matrices with diverging spikes. The theory reveals exact trace compensation and statistic dependent transitions between asymptotic independence and dependence as the spikes grow. It leads to tests of an identity correlation matrix under strong factors and a feasible two sample comparison of residual spectral concentration. A sex balanced HapMap analysis illustrates the substantial effect of factor variation on residual spectrum inference.