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

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 $1/\phi$) between the centered specification and a digamma-link DARMA at fixed parameters, under explicit interior and lag-stability conditions. The result clarifies why the two specifications should be predictively indistinguishable in the high-precision regime 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, $T=522$ weeks), predictive performance is statistically indistinguishable across $104$ rolling weekly origins on every accuracy metric examined, while 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 confirms that predictive equivalence is reference-invariant and that the geometric advantage of centering is preserved across references but varies with the prevalence of pathological raw fits, from a substantial reduction at the loans reference to parity at the cash reference. The practical implication is operational rather than predictive: 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-in, and requires only a local change to the MA innovation calculation.