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

Harrison Katz, Robert E. Weiss

Abstract

We analyze daily Airbnb service-fee shares across eleven settlement currencies, a compositional series that shows bursts of volatility after shocks such as the COVID-19 pandemic. Standard Dirichlet time series models assume constant precision and therefore miss these episodes. We introduce B-DARMA-DARCH, a Bayesian Dirichlet autoregressive moving average model with a Dirichlet ARCH component, which lets the precision parameter follow an ARMA recursion. The specification preserves the Dirichlet likelihood so forecasts remain valid compositions while capturing clustered volatility. Simulations and out-of-sample tests show that B-DARMA-DARCH lowers forecast error and improves interval calibration relative to Dirichlet ARMA and log-ratio VARMA benchmarks, providing a concise framework for settings where both the level and the volatility of proportions matter.

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

Abstract

We analyze daily Airbnb service-fee shares across eleven settlement currencies, a compositional series that shows bursts of volatility after shocks such as the COVID-19 pandemic. Standard Dirichlet time series models assume constant precision and therefore miss these episodes. We introduce B-DARMA-DARCH, a Bayesian Dirichlet autoregressive moving average model with a Dirichlet ARCH component, which lets the precision parameter follow an ARMA recursion. The specification preserves the Dirichlet likelihood so forecasts remain valid compositions while capturing clustered volatility. Simulations and out-of-sample tests show that B-DARMA-DARCH lowers forecast error and improves interval calibration relative to Dirichlet ARMA and log-ratio VARMA benchmarks, providing a concise framework for settings where both the level and the volatility of proportions matter.

Paper Structure

This paper contains 54 sections, 35 equations, 19 figures, 7 tables.

Table of Contents

  1. Introduction
  2. A Bayesian Dirichlet Auto-Regressive Moving Average - Dirichlet Auto-Regressive Conditional Heteroskedasticity Model
  3. DARMA Data model
  4. Modeling the Mean using the Additive Log-Ratio Transformation
  5. Interpretation.
  6. Precision Parameter $\phi_t$
  7. DARCH Process for Precision Parameter $\phi_t$
  8. Joint Predictive Distribution
  9. Differences between B-DARMA, B-tVARMA, and B-DARCH
  10. Assumption sensitivity and alternatives.
  11. Simulation Study
  12. Overview
  13. Simulation Setup
  14. Comparator defined: B-TVP–VAR(1).
  15. Data-Generating Processes (DGPs)
  16. ...and 39 more sections

Figures (19)

  • Figure 1: Airbnb data analysis - proportion of fees by billing currency for four regions from Jan 1, 2017 to June 30, 2020. AUD is the Australian dollar, BRL is the Brazillian Real, CAD is the Canadian Dollar,CHF is the Swiss Franc, CLP is the Chilean Peso,EUR is the European Euro, GBP is the Great British Pound, MXN is the Mexican Peso, NZD is the New Zealand Dollar, and USD is the US Dollar.
  • Figure 2: Airbnb data analysis - proportion of fees by currency- 30-day rolling ALR variance for four regions from Jan 1, 2017 to Dec 31, 2019. USD is the reference component.
  • Figure 3: Airbnb data analysis — Region 1: 92-day forecasts (blue) with 95% credible intervals (shaded) for USD currency from Oct. 1 to Dec. 31, 2020, compared to actual values (red) from the preceding six months. Models shown: B-DARCH, B-DARMA, B-tVARMA, and B-TVP-tVARMA.
  • Figure 4: Airbnb data analysis — Region 2: 92-day forecasts (blue) with 95% credible intervals (shaded) for EUR currency from Oct. 1 to Dec. 31, 2020, compared to actual values (red) from the preceding six months. Models shown: B-DARCH, B-DARMA, B-tVARMA, and B-TVP-tVARMA.
  • Figure 5: Airbnb data analysis — Region 3: 92-day forecasts (blue) with 95% credible intervals (shaded) for AUD currency from Oct. 1 to Dec. 31, 2020, compared to actual values (red) from the preceding six months. Models shown: B-DARCH, B-DARMA, B-tVARMA, and B-TVP-tVARMA.
  • ...and 14 more figures