Bayesian Analyses of the Two-Stage CARR-Return Models with Applications to COVID-19 Impact on the Cryptocurrency Market

Abstract Volatility is one of the main risk measures as it describes the variability of returns; thereby it drives many investment decisions in the financial markets. Yet, the COVID-19 pandemic introduces a new source of market risk, and its impact remains in focus. This paper studies the volatility and return series of Bitcoin and Ethereum to gain a better understanding of the impact of this widespread global event on the financial markets particularly when central banks’ stimulus measures led to concerns about inflation, boosting interest in Bitcoin as the “digital gold”. We extend the conditional autoregressive range (CARR) volatility model by incorporating efficient realised range-based volatility measures taken at higher frequencies, which capture stronger short-term memory. These conditional volatility estimates are then used as inputs in return models, forming a two-stage model that models volatility and returns separately. This approach serves as a compelling alternative to the commonly used GARCH models. The key improvement comes from the choice of error distribution – the Generalized Beta Type two, which can be specified in the CARR model but not in GARCH models. For the return model, the Student-t distribution is typically the best choice. The two-stage model demonstrates strong out-of-sample forecasting performance, particularly in terms of Value-at-Risk (VaR) and Volatility-at-Risk (VoaR), underscoring its practical effectiveness. Model parameters are estimated using a Bayesian MCMC approach via the RStan package.