Centered‐Innovation MA for Bayesian Dirichlet ARMA: Theoretical Equivalence and an Application to Bank‐Asset Shares

ABSTRACT

We study a minimal change to an observation‐driven Bayesian Dirichlet ARMA (B–DARMA) for compositional time series: replace the raw additive log‐ratio (ALR) residual in the moving‐average block with a centered innovation that subtracts the Dirichlet conditional ALR mean, available in closed form via digamma identities. We prove a recursion‐level first‐order equivalence (in ) between the centered specification and a digamma‐link DARMA at fixed parameters, under explicit interior and lag‐stability conditions. The result motivates expecting small recursion‐induced predictive differences in high‐precision settings but does not by itself govern the geometry of the Bayesian posteriors that re‐estimation produces. On weekly Federal Reserve H.8 bank‐asset shares (October 2015 through October 2025, weeks), paired comparisons across 104 rolling weekly origins detect no statistically significant mean differences between the Centered–MA and Raw–MA specifications on any predictive performance metric examined. Hamiltonian Monte Carlo divergent transitions are approximately an order of magnitude more frequent under the raw specification, driven by isolated rolling fits at which the raw posterior exhibits localized pathologies. A four‐reference sensitivity analysis shows that no statistically significant mean difference is detected on any predictive metric at any of the four references, while the geometric advantage of centering varies with the prevalence of pathological raw fits, from a substantial reduction at the loans reference to parity at the cash reference. The main practical implication in this application is operational: Centering avoids the catastrophic raw‐MA divergence spikes that occur at isolated rolling origins, which matters for production workflows in which posterior simulation feeds downstream stress tests. The adjustment is analytic and plug‐ins and requires only a local change to the MA innovation calculation.