We use the GARCH-MIDAS model to extract the long- and short-term volatility components of cryptocurrencies. As potential drivers of Bitcoin volatility, we consider measures of volatility and risk in the US stock market as well as a measure of global economic activity. We find that S&P 500 realized volatility has a negative and highly significant effect on long-term Bitcoin volatility. The finding is atypical for volatility co-movements across financial markets. Moreover, we find that the S&P 500 volatility risk premium has a significantly positive effect on long-term Bitcoin volatility. Finally, we find a strong positive association between the Baltic dry index and long-term Bitcoin volatility. This result shows that Bitcoin volatility is closely linked to global economic activity. Overall, our findings can be used to construct improved forecasts of long-term Bitcoin volatility.

In this article we derive necessary and sufficient conditions for the nonnegativity of the conditional variance in the fractionally integrated generalized autoregressive conditional heteroskedastic (p, d, q) (FIGARCH) model of the order p ≤ 2 and sufficient conditions for the general model. These conditions can be seen as being analogous to those derived by Nelson and Cao (1992, Journal of Business & Economic Statistics 10, 229–235) for the GARCH(p, q) model. However, the inequality constraints which we derive for the FIGARCH model illustrate two remarkable properties of the FIGARCH model which are in contrast to the GARCH model: (i) even if all parameters are nonnegative, the conditional variance can become negative and (ii) even if all parameters are negative (apart from d), the conditional variance can be nonnegative almost surely. In particular, the conditions for the (1, d, 1) model substantially enlarge the sufficient parameter set provided by Bollerslev and Mikkelsen (1996, Journal of Econometrics 73, 151–184). The importance of the result is illustrated in an empirical application of the FIGARCH(1, d, 1) model to Japanese yen versus U.S. dollar exchange rate data.