Directional-Shift Dirichlet ARMA Models for Compositional Time Series with Structural Break Intervention

Compositional time series, vectors of proportions summing to unity observed over time, frequently exhibit structural breaks due to external shocks, policy changes, or market disruptions. Standard methods either ignore such breaks or handle them through ad-hoc dummy variables that cannot extrapolate beyond the estimation sample. We develop a Bayesian Dirichlet ARMA model augmented with a directional-shift intervention mechanism that captures structural breaks through three interpretable parameters: a unit direction vector specifying which components gain or lose share, an amplitude controlling the magnitude of redistribution, and a logistic gate governing the timing and speed of transition. The model preserves compositional constraints by construction, maintains innovation-form DARMA dynamics for short-run dependence, and produces coherent probabilistic forecasts during and after structural breaks. We establish that the directional shift corresponds to geodesic motion on the simplex and is invariant to the choice of ILR basis. A comprehensive simulation study with 400 fits across 8 scenarios demonstrates that when the shift direction is correctly identified (77.5% of cases), amplitude and timing parameters are recovered with near-zero bias, and credible intervals for the mean composition achieve nominal 80% coverage; we address the sign identification challenge through a hemisphere constraint. An empirical application to fee recognition lead-time distributions during COVID-19 compares baseline, fixed-effects, and intervention specifications in rolling forecast evaluation, demonstrating the intervention model's superior point accuracy (Aitchison distance 0.83 vs. 0.90) and calibration (87% vs. 71% coverage) during structural transitions.