This paper develops an observed-data likelihood for applying moving-aggregate modified autoregressive (MAGMAR) copula time-series models to discrete sovereign rating-migration counts with time-varying exposure. An annual count identifies a probability-integral-transform interval rather than a unique latent point, so the likelihood integrates the latent process over the complete sequence of count intervals. A guided sequential Monte Carlo implementation incorporates the parameter-dependent stationary marginal adjustment of MAGMAR directly inside this integration. In sovereign ratings from Fitch, Moody's, and S\&P over 1963--2025, the migration counts are strongly overdispersed relative to a binomial margin and, under an exposure-conditioned static beta-binomial margin, MAGMAR improves substantially on MAG. The interpretation changes once the marginal mean is allowed to depend on the lagged migration rate: residual serial correlation in the count margin disappears, a reference dynamic-margin fit gives only a small MAGMAR improvement, and information criteria favor MAG. Numerical profiles, higher-precision refits, and parameter-recovery experiments show that marginal persistence and copula persistence are only weakly separated in this short discrete series. The results therefore support interval integration as the correct observation layer while showing that conclusions about latent dynamic dependence must be conditioned on the specification of the count margin.

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