Temporal Coarse-Graining of Multi-Sector Default Count Data Generates Posterior-Implied Copulas

Sectoral default dependence is usually described by a static correlation matrix, a static copula, or a small number of common factors. Such representations, when specified separately at each observation horizon, do not by themselves explain why the effective dependence observed in monthly credit data differs from that observed after annual aggregation. This paper proposes a dynamic low-rank state-space model for monthly multi-sector default-count data and studies the dependence structure induced by temporal coarse-graining. The leading eigenvectors of the monthly sectoral default-rate correlation matrix are used as fixed loading directions for persistent AR(1) latent credit-state factors, and defaults are modeled through a binomial observation layer. Survival aggregation of monthly posterior probability paths induces horizon-dependent distributions of sectoral default-probability vectors, from which effective correlation matrices, eigenvalue spectra, and posterior-implied rank copulas are obtained. Applied to S\&P monthly sector-level default-count data from 1981--01 to 2021--09, a two-factor specification captures the dominant market-wide and sector-rotation modes, reproduces the annual amplification of the leading eigenvalues, and generates heterogeneous copula structures across sector pairs. In an annual forecast evaluation, the dynamic factor specifications reduce the under-dispersion of static binomial and beta-binomial baselines, improving interval coverage and CRPS for aggregate portfolio counts. In log-score-based forecast comparisons, the one-factor specification is highly competitive, whereas the two-factor specification improves sector-level calibration as measured by per-sector CRPS.