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

2025-10-20

AI Summary

In financial compositional time series modeling, the conventional B-DARMA moving average (MA) term—based on additive log-ratio (ALR) residuals—exhibits nonzero conditional mean under the Dirichlet likelihood, inducing mean-path bias and ambiguous coefficient interpretation. To address this, we propose a closed-form centered innovation scheme: analytically correcting ALR residuals via the digamma function to enforce strict zero conditional mean for MA regressors, while preserving the original Dirichlet likelihood and link function. This enhances parameter interpretability and recursive prediction stability. Empirical evaluation on the Federal Reserve’s H.8 bank asset composition data demonstrates that the proposed model substantially improves log predictive scores, achieves comparable point forecast accuracy, and yields more robust Hamiltonian Monte Carlo convergence and MCMC diagnostics.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Adaptive Behavior

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 $alr(mathbf{Y}_t)-oldsymbolη_t$ 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 $oldsymbolε_t^{circ}=alr(mathbf{Y}_t)-mathbb{E}{alr(mathbf{Y}_t)mid oldsymbolη_t,φ_t}$, 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 $oldsymbolμ_t$, 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.

Problem

• Addresses biased mean path in Dirichlet ARMA models
• Proposes centered innovation to restore zero-mean property
• Improves predictive accuracy for compositional financial time series

Innovation

• Centered MA innovation replaces raw ALR residuals
• Closed-form computation using digamma functions
• Retains likelihood and ALR link while improving forecasts