  The correlation matrix is the key element in optimal portfolio allocation and
risk management. In particular, the eigenvectors of the correlation matrix
corresponding to large eigenvalues can be used to identify the market mode,
sectors and style factors. We investigate how these eigenvalues depend on the
time scale of securities returns in the U.S. market. For this purpose,
one-minute returns of the largest 533 U.S. stocks are aggregated at different
time scales and used to estimate the correlation matrix and its spectral
properties. We propose a simple lead-lag factor model to capture and reproduce
the observed time-scale dependence of eigenvalues. We reveal the emergence of
several dominant eigenvalues as the time scale increases. This important
finding evidences that the underlying economic and financial mechanisms
determining the correlation structure of securities depend as well on time
scales.
