Title: Centered MA Dirichlet ARMA for Financial Compositions: Theory & Empirical Evidence Show Chinese title

Title: 中心MA狄利克雷ARMA模型用于金融组成：理论与实证证据

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. Show Chinese abstract

Abstract: 观测驱动的狄利克雷模型用于组成时间序列，通常使用加性对数比（ALR）链接，并包含一个由ALR残差构建的移动平均（MA）项。 在标准的B--DARMA递归中，通常的MA回归量 $\alr(\mathbf{Y}_t)-\boldsymbol{\eta}_t$在狄利克雷似然下具有非零条件均值，这会偏差均值路径并模糊MA系数的解释。 我们提出一个最小改动：用一个 \emph{中心的}创新 $\boldsymbol{\epsilon}_t^{\circ}=\alr(\mathbf{Y}_t)-\mathbb{E}\{\alr(\mathbf{Y}_t)\mid \boldsymbol{\eta}_t,\phi_t\}$替换原始回归量，可通过digamma函数以闭合形式计算。 居中处理恢复了MA块的零均值创新，而不会改变似然或ALR链接。 我们提供了条件均值和预测递归的简单恒等式，展示了与digamma链接DARMA的一阶等价性，同时保留了 $\boldsymbol{\mu}_t$的闭合形式逆，还提供了可以直接使用的代码。 对联邦储备H.8银行资产构成进行每周应用，在固定保留和滚动一步起点下比较原始（原始MA）和居中规格。 居中公式在几乎相同的点误差下提高了对数预测得分，并显著改善了哈密顿蒙特卡罗诊断。