Paper: arXiv 2510.18903

Authors: Harrison Katz

Abstract

Observation-driven Dirichlet models for compositional time series commonly use the additive log-ratio (ALR) link and include a moving-average (MA) term based on ALR residuals. In the standard Bayesian Dirichlet Auto-Regressive Moving-Average (B-DARMA) recursion, this MA regressor has a nonzero conditional mean under the Dirichlet likelihood, which biases the mean path and complicates interpretation of the MA coefficients. We propose a minimal change: replace the raw regressor with a centered innovation equal to the ALR residual minus its conditional expectation, computable in closed form using digamma functions. Centering restores mean-zero innovations for the MA block without altering either the likelihood or the ALR link. We provide closed-form identities for the conditional mean and forecast recursion, show first-order equivalence to a digamma-link DARMA while retaining a simple inverse back to the mean composition, and supply ready-to-use code. In a weekly application to the Federal Reserve H.8 bank-asset composition, the centered specification improves log predictive scores with virtually identical point accuracy and markedly cleaner Hamiltonian Monte Carlo diagnostics.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 9.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents advanced mathematical derivations using Dirichlet distributions, digamma functions, and closed-form identities, while also demonstrating strong empirical rigor with a real-world dataset (Federal Reserve H.8), predictive evaluation metrics (ELPD, coverage), code availability, and robust out-of-sample testing.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Resolve MA Bias in<br>Dirichlet Compositional Models"] --> B{"Methodology"}
  B --> C["Standard B-DARMA<br>Use Raw ALR Residuals"]
  B --> D["Centered MA Dirichlet ARMA<br>Use Centered Innovations"]
  
  C --> E["Computational Process<br>Bayesian MCMC / HMC<br>Convergence Issues"]
  D --> E
  
  subgraph Inputs ["Data & Theory"]
      F["Weekly Fed H.8<br>Bank Asset Composition"]
      G["Dirichlet Likelihood<br>ALR Link Function"]
  end
  
  F --> E
  G --> E
  
  E --> H["Key Outcomes"]
  
  H --> I["Point Accuracy<br>Virtually Identical"]
  H --> J["Predictive Score<br>Improved Log Score"]
  H --> K["Diagnostics<br>Cleaner HMC Convergence"]
  
  style A fill:#e1f5fe
  style H fill:#e8f5e8
  style D fill:#fff3e0