Advancing Bayesian Forecasting: A Bayesian Dirichlet Auto-Regressive Conditional Heteroskedasticity Model, a Bayesian Dirichlet Auto-Regressive Moving Average Model, and Other Innovations

This dissertation introduces new Bayesian time series models for compositional and high-dimensional data with dynamic structure and potential heteroskedasticity. It comprises four papers that offer methodological advances, simulation results, and applications in hospitality and finance.Paper 1 introduces the Bayesian Dirichlet Auto-Regressive Moving Average (B-DARMA) model, motivated by the need to forecast the proportion of future fees recognized in future monthly intervals using daily Airbnb data. By embedding Auto-Regressive Moving Average (ARMA) components on an additive log-ratio scale within a Dirichlet likelihood, the model enforces compositional constraints and yields reasonable forecasts. Simulation studies highlight B-DARMA's predictive performance, and empirical analysis shows more accurate lead-time predictions compared with standard VARMA-based methods—vector auto-regressive moving average models that jointly capture relationships among multiple time series—thereby guiding resource allocation and strategic planning.Paper 2 extends B-DARMA to a Bayesian Dirichlet Auto-Regressive Conditional Heteroskedasticity (B-DARCH) model by incorporating a Generalized Auto-Regressive Conditional Heteroskedasticity (GARCH)-like process for the Dirichlet precision parameter. Empirical analysis of Airbnb’s currency-fee data demonstrates that B-DARCH achieves higher forecast accuracy than both B-DARMA and standard VARMA-based methods. Simulation studies further confirm that modeling time-varying volatility significantly improves predictive coverage relative to simpler B-DARMA or transformed VARMA models.Paper 3 conducts a sensitivity analysis of B-DARMA under several priors—normal, Laplace, horseshoe, spike-and-slab, and hierarchical. Six simulation studies highlight shrinkage as crucial for pruning unneeded parameters while showing that prior choice alone cannot fix model misspecification. An application to S\&P 500 sector allocations illustrates how prior-based shrinkage manages complexity in high-dimensional or limited-sample scenarios.Paper 4 examines high-dimensional vector auto-regressive processes, comparing horseshoe, lasso, and hierarchical priors with ridge and nonparametric shrinkage methods in three distinct simulations. In Canadian macroeconomic data, horseshoe priors outperform other approaches by shrinking smaller coefficients while retaining major signals, enhancing forecast accuracy.Collectively, these four papers form a cohesive suite of Bayesian methods for compositional and high-dimensional time series, addressing interpretability, over-parameterization, and volatility. They offer theoretically grounded, empirically tested frameworks for accurate inference, improved risk management, and deeper strategic insights in hospitality, macroeconomics, and finance.

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