Forecasting realized volatility of bitcoin returns: tail events and asymmetric loss

Abstract

We use intraday data to construct measures of the realized volatility of bitcoin returns. We then construct measures that focus exclusively on relatively large realizations of returns to assess the tail shape of the return distribution, and use the heterogeneous autoregressive realized volatility (HAR-RV) model to study whether these measures help to forecast subsequent realized volatility. We find that mainly forecasters suffering a higher loss in case of an underprediction of realized volatility (than in case of an overprediction of the same absolute size) benefit from using the tail measures as predictors of realized volatility, especially at a short and intermediate forecast horizon. This result is robust controlling for jumps and realized skewness and kurtosis, and it also applies to downside (bad) and upside (good) realized volatility.

Keywords:

• Bitcoin
• realized volatility
• forecasting
• tail events

JEL classifications:

• C22
• F31
• F37

Acknowledgments

We thank an associate editor and two anonymous reviewers for helpful comments. The usual disclaimer applies.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 In this research, we use the term realized volatility to refer to the realized variance rather than the realized standard deviation of bitcoin returns. In Section 4.5, we shall present results for the the realized standard deviation of bitcoin returns.

2 With the term heavy-tailed, we refer to those distributions that do not have all their moments finite.

3 In order to find the appropriate sample fractions $k_{t,N}$, we apply a failure-to-reject method based on the idea of Choulakian and Stephens (2012). As such, we conduct a sensitivity analysis by selecting a number of possible sample fraction values. We then we select the lowest sample fraction for which the average value of the daily realized estimates (considering all the available trading days in the sample) stabilizes. Very low sample fraction values are not selected to avoid small sample errors which can give a false sense of accuracy. In any case, results suggest that the exact choice of the optimal sample fraction $k_{t,N}$ does not matter much.

4 We arrange the data matrix such that we have exactly the same number of observations for all three forecasting horizons. We compute all estimation results that we report in this research using the R programing environment (R Core Team 2017). We compute the p-values of the Diebold–Mariano test using the R package ‘forecast’ (Hyndman 2017; Hyndman and Khandakar 2008), with the code changed to account for the asymmetry of the loss function.

5 While we report results for the medium and long forecast horizon based on multiple-step forecasts of $y_{t+h}$, results for forecasts of $\text{mean}(y_{t+1}+\cdots +y_{t+h})$ are similar. Moreover, as suggested by an anonymous reviewer, we also consider as an extension HAR-RV models that includes returns as an additional control variable. Results for these extended HAR-RV models are similar to those reported in Figure 2. Complete results are available upon request.

6 In few cases, the Diebold–Mariano test produces a negative estimate of the variance of the loss differential. These cases are not used for the computation of the p-values.

7 One way to identify the optimal length of the rolling-estimation window is to compute, given the asymmetry parameter and the forecast horizon, the cumulated loss for the models that feature the realized tail indices and divide it by the cumulated loss obtained for the baseline model. The minimum of this ratio gives the optimal length of the rolling-estimation window for every asymmetry parameter and every forecast horizon (results are available upon request).

8 As one would have expected, the variance of forecasts increases as the rolling-estimation window gets shorter. For this reason, we do not to consider a mean forecast (on which the forecasts from the short rolling-estimation window would have a relatively large effect) but rather a median forecast.