The Bubble Crash-GARCH model

This paper proposes the Bubble Crash–GARCH model, a volatility-forecasting framework that incorporates tail price events into GARCH-type specifications for returns. Bubbles and crashes are first detected and date-stamped through the Phillips, Shi, and Yu real-time monitoring procedure applied to price series, and the resulting indicators are included as regressors in the conditional mean equation of returns. Unlike standard ARMA-type mean specifications, the conditional mean is directly driven by bubble and crash indicators generated in real time, avoiding the use of lead variables typically required by noncausal approaches, thus allowing us to disentangle and quantify the effects of both periodic bubble collapses and crashes. The model is evaluated on major cryptocurrencies and some of the Magnificent Seven stocks, over the period from January 1, 2018 to April 30, 2026, with December 31, 2023 used as the forecasting cutoff date. The benchmark model is selected through a systematic assessment of alternative volatility specifications, including asymmetric GARCH models and two-regime Markov switching GARCH models under different assumptions on the innovation distribution. In addition to own-asset bubble and crash effects, the analysis investigates contagion channels by incorporating episodes driven by Bitcoin and Nvidia for cryptocurrencies and the Magnificent Seven, respectively. Forecast accuracy is assessed through one-step-ahead volatility predictions and formally tested using the Clark–West test. The empirical results confirm that accounting for bubble and crash episodes improves volatility forecasts relative to the selected benchmark, both through idiosyncratic tail-event effects and through cross-asset contagion channels. These findings suggest that explicitly disentangling extreme price episodes from regular volatility dynamics enhances the informational content of GARCH-type models and provides useful evidence for risk management and asset allocation.