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

TL;DR

This paper introduces a Bayesian Dirichlet ARMA model with a directional-shift intervention mechanism to effectively model compositional time series with structural breaks, capturing the direction, magnitude, and timing of shifts.

Contribution

The paper develops a novel intervention mechanism within a Dirichlet ARMA framework that models structural breaks as geodesic shifts on the simplex, maintaining compositional constraints.

Findings

Simulation studies show near-zero bias and good coverage when shift direction is correctly identified.

Empirical Airbnb data applications demonstrate improved calibration in monotone break scenarios.

The model performs comparably or better than fixed effects in capturing structural transitions.

Abstract

Compositional time series frequently exhibit structural breaks due to external shocks, policy changes, or market disruptions. Standard methods either ignore such breaks or handle them through fixed effects that cannot extrapolate beyond the sample, or step-function dummies that impose instantaneous adjustment. We develop a Bayesian Dirichlet ARMA model augmented with a directional-shift intervention mechanism that captures structural breaks through three interpretable parameters: a direction vector specifying which components gain or lose share, an amplitude controlling redistribution magnitude, and a logistic gate governing transition timing and speed. The model preserves compositional constraints by construction, maintains DARMA dynamics for short-run dependence, and produces coherent probabilistic forecasts through and after structural breaks. The intervention trajectory corresponds to geodesic motion on the simplex and is invariant to the choice of ILR basis. A simulation study with 400 fits across 8 scenarios shows near-zero amplitude bias and nominal 80\% credible interval coverage when the shift direction is correctly identified (77.5\% of cases); supplementary studies confirm robustness across extreme transition speeds and non-monotone DGPs. Two empirical applications to COVID-era Airbnb data characterize performance relative to simpler alternatives. Where the break is monotone and ongoing, the intervention model achieves near-nominal calibration (79.6\%) while the fixed effect substantially under-covers (66.1\%). Where post-break dynamics are non-monotone, both models are acceptably calibrated and the fixed effect outperforms on point accuracy. The intervention model's advantages are thus specific to settings with roughly monotone structural transitions.