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A Bayesian Dirichlet Auto-Regressive Conditional Heteroskedasticity Model for Forecasting Currency Shares

This paper introduces the B-DARMA-DARCH model, a Bayesian Dirichlet time series framework that incorporates an ARCH component to capture volatility clustering in compositional data like Airbnb service-fee shares, demonstrating superior forecasting accuracy and interval calibration compared to standard benchmarks.

Original authors: Harrison Katz, Robert E. Weiss

Published 2026-03-13

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CC BY 4.0

Original authors: Harrison Katz, Robert E. Weiss

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are running a massive global marketplace, like Airbnb, where people from all over the world book stays. They pay in their own local currencies: dollars, euros, yen, Brazilian reals, and so on.

To run the business, you need to know exactly how much money you'll have in US Dollars at the end of the day. But here's the tricky part: you don't just need to know how many bookings you got; you need to know the mix of currencies. If 90% of your bookings are in Euros today, but tomorrow 90% are in Yen, your total US Dollar revenue changes drastically, even if the number of bookings stays the same.

This "mix" is what the paper calls compositional data. It's like a pie chart where the slices must always add up to 100%. If one slice gets bigger, the others must get smaller.

The Problem: The Pie Chart is Wobbly

The authors noticed that this currency mix isn't stable. Sometimes it's calm, but other times—like during a pandemic or a sudden economic crash—it goes wild. The "slices" of the pie swing wildly from day to day.

Standard forecasting tools are like trying to predict the weather using a thermometer that only works on sunny days. They assume the "wobble" (volatility) of the currency mix is constant. But in reality, the wobble gets huge during crises and tiny during calm times. When the model assumes the wobble is constant, it gets confused and makes bad predictions.

The Solution: A Smart, Self-Adjusting Model

The authors built a new model called B-DARCH. Think of it as a smart, self-adjusting weather forecast for your currency pie.

Here is how it works, using a simple analogy:

1. The "Pie" Constraint (The Simplex)

Most math models try to predict numbers that can go up or down forever. But currency shares are a pie. You can't have 110% of the pie.

Old Models: They often try to stretch the pie into a straight line to make the math easier, then try to bend it back. Sometimes, this stretching breaks the pie, leading to impossible predictions (like negative percentages).
The New Model: It respects the pie shape from the start. It knows the slices must always add up to 100%.

2. The "Wobble" Detector (Volatility)

This is the secret sauce. Imagine you are driving a car.

The Old Way (B-DARMA): The car has a suspension system that is set to "medium" stiffness. If you hit a small bump, it's fine. If you hit a massive pothole (a market crash), the car bounces wildly, and the driver (the model) gets thrown off course.
The New Way (B-DARCH): This car has adaptive suspension.
- When the road is smooth, the suspension is stiff and precise, giving a very accurate ride.
- When the road gets bumpy (volatility spikes), the suspension instantly softens and absorbs the shock. It doesn't just guess the next bump; it feels the road changing and adjusts its own sensitivity in real-time.

In math terms, the model has a "precision dial." When the currency mix is chaotic, the dial turns down the confidence (admitting "I'm not sure, things are crazy right now") and widens the safety net. When things are calm, it turns the dial up for a sharper, more precise prediction.

Why Does This Matter?

The authors tested this model against three other types of models using Airbnb's real data and some fake "disaster" scenarios.

Accuracy: The new model predicted the currency mix much better than the others, especially when things got chaotic.
Safety: It gave better "confidence intervals." Imagine a weather forecast saying, "It will rain between 2 PM and 4 PM." The old models might say "2 PM to 2:05 PM" (too narrow, likely to be wrong) or "All day" (too wide, useless). The new model adjusted its range based on how stormy the day was, hitting the mark more often.
Less "Hangover": After a big shock (like a sudden news event), the old models kept making mistakes for days afterward because they were slow to realize the rules had changed. The new model adapted almost immediately.

The Bottom Line

The paper introduces a tool that treats financial data like a living, breathing thing rather than a static machine.

If the market is calm: Simple models work fine.
If the market is chaotic: You need a model that can change its own rules on the fly.

The B-DARCH model is that flexible tool. It's like giving your financial forecast a nervous system that feels the tremors of the market and adjusts its balance instantly, ensuring that when the world gets messy, your predictions stay reliable.

1. Problem Statement

The paper addresses the challenge of forecasting compositional time series in the context of marketplace finance, specifically Airbnb's daily currency fee shares.

The Core Issue: Revenue forecasting requires converting bookings made in various local currencies into a consolidated U.S. Dollar (USD) amount. This depends on two factors: the total volume of bookings and the proportion (share) of those bookings paid in each currency.
Constraints: Currency shares are compositional data; they are non-negative and must sum to one (lying on a simplex). Standard time-series methods (like Gaussian VARMA) often fail to respect these constraints or require transformations (e.g., Additive Log-Ratio, ALR) that can distort uncertainty or violate simplex properties upon back-transformation.
Volatility Dynamics: The data exhibits conditional heteroskedasticity (volatility clustering). Currency shares experience recurrent bursts of dispersion, particularly during macroeconomic disruptions (e.g., the onset of COVID-19). Existing models often assume fixed precision (constant variance) or deterministic variance, failing to capture the dynamic nature of these shocks.

2. Methodology: The B-DARCH Model

The authors propose a Bayesian Dirichlet Auto-Regressive Moving Average with Dirichlet Auto-Regressive Conditional Heteroskedasticity (B-DARCH) model. This framework combines a mean model on the simplex with a dynamic precision component.

A. The Data Model (Mean Dynamics)

Likelihood: The observations $y_t$ (currency shares) follow a Dirichlet distribution: $y_t | \mu_t, \phi_t \sim \text{Dirichlet}(\phi_t, \mu_t)$ . This ensures predictions remain valid proportions on the simplex.
Mean Structure: The mean vector $\mu_t$ $μ_{t}$ is modeled using the Additive Log-Ratio (ALR) transformation to map the simplex to $\mathbb{R}^{J-1}$ $R^{J - 1}$ .
- $\eta_t = \text{alr}(\mu_t)$ is modeled via an observation-driven VARMA recursion:
  $\eta_t = \sum_{p=1}^P A_p(\text{alr}(y_{t-p}) - X_{t-p}\beta) + \sum_{q=1}^Q B_q(\text{alr}(y_{t-q}) - \eta_{t-q}) + X_t\beta$
- This captures autoregressive (AR) and moving average (MA) dependencies in the mean.

B. The Precision Model (Volatility Dynamics)

The key innovation is the DARCH component, which models the precision parameter $\phi_t$ (inverse of variance) dynamically, analogous to GARCH models in finance.

Log-Precision Recursion: Instead of a fixed or covariate-driven precision, $\log(\phi_t)$ follows an ARMA process driven by past innovations and past precision:
$\log(\phi_t) = \sum_{l=1}^L \alpha_l(\log(\phi_{t-l}) - z'_{t-l}\gamma) + \sum_{k=1}^K \tau_k (\text{alr}(y_{t-k}) - \eta_{t-k})'(\text{alr}(y_{t-k}) - \eta_{t-k}) + z'_t\gamma$
Mechanism: The term $(\text{alr}(y_{t-k}) - \eta_{t-k})'$ $(alr (y_{t - k}) - η_{t - k})^{'}$ represents the squared innovation (shock). This allows the model to:
1. Capture volatility clustering (periods of high variance followed by high variance).
2. Adapt rapidly to regime shifts (e.g., pandemic onset).
3. Maintain compositional constraints while modeling time-varying uncertainty.

C. Inference

The model is estimated using a full Bayesian framework via Markov Chain Monte Carlo (MCMC) in Stan.
It produces joint predictive distributions, allowing for the calculation of credible intervals that automatically widen during volatile periods.

3. Key Contributions

Novel Model Class: Introduction of B-DARCH, the first Bayesian Dirichlet time-series model to incorporate a GARCH-like recursion for precision on the simplex.
Handling Compositional Constraints: Unlike transformed-Gaussian approaches, B-DARCH keeps the likelihood on the simplex, ensuring forecasts are always valid proportions.
Dynamic Uncertainty Quantification: The model explicitly separates mean dynamics from volatility dynamics, allowing for robust risk assessment when "the noise itself moves."
Comprehensive Benchmarking: The study compares B-DARCH against:
- B-DARMA: Standard Bayesian Dirichlet ARMA (fixed/deterministic precision).
- B-tVARMA: Transformed-Gaussian VARMA (constant variance in ALR space).
- B-TVP-tVARMA: Time-varying parameter VARMA (drifting mean, constant variance).

4. Results

A. Simulation Studies

The authors simulated data under four scenarios: DARMA and tVARMA data-generating processes (DGPs), each perturbed by misreported observations and temporary regime shifts.

Forecast Accuracy: B-DARCH consistently achieved the lowest Forecast Root Mean Squared Error (FRMSE) and Forecast Mean Absolute Error (FMAE) across all scenarios.
Residual Diagnostics: B-DARCH showed the weakest residual persistence in the Partial Autocorrelation Function (PACF) of squared standardized residuals. While other models exhibited lingering autocorrelation after shocks, B-DARCH adapted quickly, dissipating residual dependence faster.
Robustness: Even when the true DGP had constant variance, B-DARCH performed competitively, demonstrating its ability to absorb shocks without overfitting.

B. Airbnb Data Application

The model was applied to daily currency fee shares across four geographic regions (2017–2020).

Performance: B-DARCH achieved the lowest aggregate FMAE and Forecast Residual Sum of Squares (FRSS) in all four regions.
- In Region 1, B-DARCH reduced total FMAE by >60% compared to B-DARMA and >80% compared to B-tVARMA.
Uncertainty Calibration: B-DARCH provided the most accurate 95% credible interval coverage (closest to the nominal 0.95), whereas other models frequently under-covered during volatile periods (e.g., 0.65 coverage in Region 1 for B-TVP-tVARMA).
Residual Analysis: The PACF of squared residuals for B-DARCH showed minimal structure beyond lag 1, confirming that the DARCH component successfully captured the volatility clustering present in the real-world data.

5. Significance and Implications

Practical Utility: For financial reporting and treasury risk management, B-DARCH offers a superior tool for forecasting revenue in a multi-currency environment. It allows teams to better hedge against FX fluctuations by accurately quantifying the uncertainty of currency mix shifts.
Theoretical Advancement: The paper bridges the gap between compositional data analysis and financial volatility modeling (GARCH). It demonstrates that modeling precision as a dynamic process is crucial for compositional time series, not just for financial asset prices.
Guidance for Practitioners: The authors propose a practical rule:
- Use Transformed-Gaussian (B-tVARMA) models when volatility is stable and computational speed is paramount.
- Use Time-Varying Parameter (B-TVP) models when the mean drifts significantly.
- Use B-DARCH when volatility clusters or when the noise itself is non-stationary, as this yields the best accuracy and calibration.

In conclusion, the B-DARCH model provides a flexible, unified approach to dynamic heteroskedasticity on the simplex, significantly outperforming standard benchmarks in both simulated and real-world scenarios involving structural breaks and volatility shifts.