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

Harrison Katz

TL;DR

This paper addresses bias in MA terms within observation-driven Dirichlet ARMA models for compositional time series by introducing centered innovations $\boldsymbol{\epsilon}_t^{\circ}=\operatorname{alr}(\mathbf{y}_t)-\mathbb{E}\{\operatorname{alr}(\mathbf{Y}_t)\mid \boldsymbol{\mu}_t, \phi_t\}$ computed via digamma functions. The centered MA preserves the Dirichlet likelihood and ALR link, yields mean-zero shocks, and leads to a forecast recursion where only already-realized shocks influence the mean path; first-order equivalence to a digamma-link DARMA is established through a digamma–ALR expansion. The approach improves density forecasts and coverage in a weekly H.8 bank-asset shares case study, with cleaner Hamiltonian Monte Carlo diagnostics and essentially unchanged point accuracy, suggesting practical gains in probabilistic forecasting for financial compositions. The paper also provides ready-to-use code and discusses limitations and possible extensions, including multi-step horizons and richer precision dynamics.

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.

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

TL;DR

This paper addresses bias in MA terms within observation-driven Dirichlet ARMA models for compositional time series by introducing centered innovations computed via digamma functions. The centered MA preserves the Dirichlet likelihood and ALR link, yields mean-zero shocks, and leads to a forecast recursion where only already-realized shocks influence the mean path; first-order equivalence to a digamma-link DARMA is established through a digamma–ALR expansion. The approach improves density forecasts and coverage in a weekly H.8 bank-asset shares case study, with cleaner Hamiltonian Monte Carlo diagnostics and essentially unchanged point accuracy, suggesting practical gains in probabilistic forecasting for financial compositions. The paper also provides ready-to-use code and discusses limitations and possible extensions, including multi-step horizons and richer precision dynamics.

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.
Paper Structure (24 sections, 7 theorems, 29 equations, 4 figures, 2 tables)

This paper contains 24 sections, 7 theorems, 29 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Contribution to forecasting.
  3. Model recap and centered innovations
  4. Main properties
  5. Dirichlet log–moment identity and conditional ALR mean
  6. Mean–zero innovations and forecast recursion
  7. Digamma–ALR expansion and first–order equivalence
  8. Case study: forecasting weekly bank‑asset shares
  9. Data and preprocessing
  10. Exploratory composition dynamics
  11. ALR reference and unit‑root check
  12. Precision regressor construction
  13. Implementation constants.
  14. Competing specifications, estimation, and scoring
  15. Raw–MA B--DARMA.
  16. ...and 9 more sections

Key Result

Lemma 1

If $\mathbf{Y}\sim\mathrm{Dir}(\boldsymbol{\alpha})$ with $\alpha_0=\sum_{k=1}^J \alpha_k$, then

Figures (4)

  • Figure 1: H.8 bank assets as weekly shares of total assets. Shaded bands show cash, securities, loans, and the residual other category over the last decade.
  • Figure 2: Rolling one–step cumulative ELPD difference (Centered $-$ Raw). Positive values favor the centered specification.
  • Figure 3: Rolling one–step total–share RMSE by origin. The two series are nearly indistinguishable; both spike briefly in early 2025.
  • Figure 4: Fixed–holdout comparison of total–share RMSE (bars) with MAE/ELPD/coverage annotations. Point accuracy is essentially tied; log score and coverage slightly favor the centered model.

Theorems & Definitions (14)

  • Lemma 1: Dirichlet log–moment identity
  • proof
  • Proposition 1: Conditional ALR mean
  • proof
  • Proposition 2: Mean–zero MA innovations
  • proof
  • Proposition 3: Forecast recursion
  • proof
  • Remark : One‑step case
  • Remark : Forecast interpretation for practitioners
  • ...and 4 more