Centered MA Dirichlet ARMA for Financial Compositions: Theory & Empirical Evidence

Abstract: Observation-driven Dirichlet models for compositional time series often use the additive log-ratio (ALR) link and include a moving-average (MA) term built from ALR residuals. In the standard B--DARMA recursion, the usual MA regressor $\alr(\mathbf{Y}_t)-\boldsymbol{\eta}_t$ has nonzero conditional mean under the Dirichlet likelihood, which biases the mean path and blurs the interpretation of MA coefficients. We propose a minimal change: replace the raw regressor with a \emph{centered} innovation $\boldsymbol{\epsilon}_t^{{\circ}=\alr(\mathbf{Y}_t)-\mathbb{E}{\alr(\mathbf{Y}_t)\mid} \boldsymbol{\eta}_t,\phi_t}$, computable in closed form via digamma functions. Centering restores mean-zero innovations for the MA block without altering either the likelihood or the ALR link. We provide simple identities for the conditional mean and the forecast recursion, show first-order equivalence to a digamma-link DARMA while retaining a closed-form inverse to $\boldsymbol{\mu}_t$, and give ready-to-use code. A weekly application to the Federal Reserve H.8 bank-asset composition compares the original (raw-MA) and centered specifications under a fixed holdout and rolling one-step origins. The centered formulation improves log predictive scores with essentially identical point error and markedly cleaner Hamiltonian Monte Carlo diagnostics.