In this section, we report on an in sample and out of sample forecast analysis of the Bitcoin’s realized volatility using several Heterogeneous Autoregressive (HAR) specifications. HAR has been originally introduced by
Corsi (
2008) in order to approximate the slow decay of the autocorrelation function of realized volatility. The model builds on the assumption of three different types of investors creating three different types of volatility. The investors are: (i) short-term traders with daily activity; (ii) medium investors who typically regulate their portfolio once a week; and (iii) long-term investors with horizon around a month or longer.
Corsi (
2008) and
Corsi et al. (
2012) argue that while the level of short-term volatility does not affect the long-term traders, the level of long-term volatility does affect the short-term traders, as it determines the expectation to the future size of trends and risks. Hence, the short-term volatility is dependent on the longer horizon volatility, while the long-term volatility only consist of an
structure, then the model can be written in a hierarchical system defined by
where
are the daily, weekly and monthly realized volatility and
, and
are the volatility innovations for the daily, weekly and monthly horizons, respectively. The economic interpretation of this hierarchical system is that each horizon volatility component consists of two parameters: (i) the expectation to the next period volatility; and (ii) an expectation for the longer horizon volatility, which is shown to have an impact on the future volatility. The HAR model can be written in a cascade of previous values for one day, one week and one month. By straightforward recursive substitutions we obtain a forecasting model for the realized volatility as:
where
is the forecast horizon and
is a zero mean serially uncorrelated shocks and
, and
are the weekly and monthly volatility, respectively.
5 This model is labelled as “HAR-RV”.
Andersen et al. (
2007) extended the HAR-RV model to include a jump component in the cascade of lagged volatility measures. Jumps are defined as:
where:
with
is the bipower variation introduced by
Barndorff-Nielsen and Shephard (
2003) and
Barndorff-Nielsen (
2004). By including the jump component into the HAR model we obtain the “HAR-RV-J” defined as:
A related specification has been further introduced by
Barndorff-Nielsen et al. (
2006) by including the so called “significant jumps” component. Specifically, let:
be the realized tripower quarticity where
. The significant jump component at level
is defined as:
where
is the indicator function equal to 1 if
A is true and 0 otherwise, and:
is the feasible test statistics arising from the asymptotic distribution of the difference between the realized volatility and the bipower variation, see
Barndorff-Nielsen et al. (
2006) for more details. Finally, the new HAR model with continuous jumps, HAR-RV-CJ, is defined as:
where:
selects
if
and
if
. We perform a sensitivity analysis similar to that reported in
Andersen et al. (
2007) and set
.
4.1. Including a Leverage Component
A well known stylized fact of equity financial returns is the so called leverage effect, see
Black (
1976),
Nelson (
1991) and
Zakoian (
1994), among others. The leverage effect relates to the different reaction of the volatility of a firm to past positive and negative news. Its original formulation relates to the reaction of the volatility to changes in the debt to equity ratio of a traded company. Specifically, when a bad news arrives, the value of the firm decreases while its debt remains unchanged. This leads to an increase of the debt to equity ratio corresponding to an increase of the riskiness of the firm which translates in more volatility. Of course, the original interpretation of the leverage effect cannot be applied to Bitcoin since it does not have any capital structure. However, previous empirical works have found evidence of leverage effect for Bitcoin, see
Catania and Grassi (
2017),
Katsiampa (
2017),
Bariviera (
2017), and
Ardia et al. (
2018). We follow
Corsi et al. (
2012) and introduce a leverage component in the HAR specification by defining:
which indicates the minimum return over the trading day. The variable
along with its weekly
and monthly
averages are included linearly in the HAR-RV, HAR-RV-J, and HAR-RV-CJ specifications. For example, the HAR-RV specification with leverage, HAR-RV-L, is defined as:
4.2. In Sample Results
We consider the realized variance of Bitcoin from 17 March 2013 for Bitstamp and from 2 February 2015 for Coinbase up to 18 March 2018. Similar to previous results, we also consider the Hype period from 1 January 2017 to 18 March 2018. Results are also reported for the realized standard deviation,
and the logarithmic realized variance,
.
Figure 6 displays: (i) the time series of the log realized variance; (ii) the feasible test statistics; and (iii) the significant logarithmic jump series,
over the full sample for Bitstamp and Coinbase. We find that the logarithmic realized variance for the Coinbase exchange displays an increasing pattern, with the highest values in the end, and especially around December 2017 where the underline value increased significantly. Interestingly, we find that realized volatility is lower during the bubble period of 2017 compared to the bear market period of 2018.
Panel
reports the test statistics from Equation (
9) for
. The red horizontal line indicates
, i.e., the threshold after which jumps are classified as significant. Interestingly, we find a very large proportion of jumps for Bitcoin compared to the proportion usually found in other asset classes, see e.g.,
Andersen et al. (
2007). Indeed, the proportion of jumps ranges from 27% to 92% depending on different choices of
. When
, the proportion of jumps over the full period is around 79% for Bitstamp and 85% for Coinbase. If we focus on the Hype period the proportion of jumps is halved for both exchanges. This results further indicates the growing trade intensity and the increased stability of the market over time.
Table 5 reports the summary statistics for the realized variance and its transformations. We find that both the median and the standard deviation of the realized variance and jump component are higher during the Hype period. We also find that similar to
Andersen et al. (
2001a) and
Andersen et al. (
2001b), we are not able to reject the null hypothesis of normality for the logarithmic realized variance according to the Jarque-Bera test statistics.
Model Estimation
We now estimate by OLS the HAR-RV, HAR-RV-J, and HAR-RV-CJ models to the realized variance, realize standard deviation and logarithmic realized variance over the full sample for the two exchanges. Specifications that include the leverage component are also estimated and indicated with the additional label “-L”. Estimation results are reported in
Table 6. Estimated coefficients are in line with those usually found in the literature for other asset classes. Interestingly, we find that specifications that include the leverage component outperform their counterpart without leverage. Regarding the estimated leverage coefficients, we see that these are negative and statistically significant at standard confidence levels. This finding is somehow in contrast with previous results by results by
Catania and Grassi (
2017) and
Ardia et al. (
2018) who document an “inverted” leverage effect for Bitcoin. To further investigate this aspect, in
Figure 7 we report the empirical autocorrelation at different lags between realized variance and the leverage component, i.e.,
for
. The plot is reported for the two exchanges for the full sample as well as for the Hype period. Results indicate that correlations are negative and statistically different from zero up to
when computed over the full sample. However, when we focus on the Hype period, evidence of correlation between
and
is less strong. This result suggests that the leverage effect has changed over time for Bitcoin and somehow confirms the findings of
Ardia et al. (
2018).
4.3. Out of Sample Results
We now conduct an out of sample analysis studying the predictability of Bitcoin realized variance at different horizons. Predictions are made by the models previously introduced at horizons
(one day),
(one week), and
(one month) using the direct method of forecast, see
Marcellino et al. (
2006). We start making prediction from 21 April 2014 for Bitstamp and 17 March 2016 for Coinbase and than update model parameters each time a new observation becomes available during the whole forecast periods using a fixed rolling window. The length of the out of sample is
and
for Bitstamp and Coinbase, respectively. Results are compared with the Random Walk (RW) specification defined by:
Let
be the prediction made at time
t for time
. Comparison among different specification is performed according to the mean absolute forecast error (MAFE) and root mean square forecast error (RMSFE). MAFE at horizon
h is defined as:
while RMSFE as:
where
T is the length of the in sample period. Models with lower MAFE and RMSFE are preferred.
Table 7 reports the results computed over the full sample. Results for the Hype period are similar and are available upon request to the second author.
Along with the MAFE and RMSFE measures, the table also reports the
of the Mincer–Zarnowitz regression defined by:
as well as the
Diebold and Mariano (
1994) test statistics of each model with respect to the benchmark RW (DM1) and with respect to the plain HAR-RV model (DM2). Results indicate that predictability is higher for lower forecast horizons. Indeed, looking at the
we find that when
, up to
of the log realized variance variability can be predicted with the HAR-RV-CJ model. However, when
the
decreases to only
. Overall, the inclusion of jumps does not always translate in better predictions. In this respect, results are a bit mixed. Differently, models that include the leverage component seem to generally perform better than the standard HAR-RV model. Looking at the Diebold Mariano test statistic with respect to the benchmark model (DM1), we find strong evidence of predictability of all specifications. Differently, when we focus on predictability with respect to the plain HAR-RV model, results are mixed and do not show a clear pattern. Comparing results between the two exchanges indicates that realized variance is easier to predict in the Coinbase exchange.
To conclude our analysis we study the stability of prediction gains with respect to the RW benchmark over time. To do so, we compute the cumulative absolute error of a forecast model over the cumulative absolute error of the benchmark model. Specifically, the ratio of cumulative absolute errors RCAE at time
f is defined as:
where
is the forecast error of generic model
j at time
s and
is the forecast error of the benchmark specification. Results are reported for
and
. Values of
below one indicate outperformance with respect to the benchmark and viceversa.
Figure 8 displays the
for the log realized variance for different forecast horizons and the two exchanges. In the top graph of each sub-figure the comparison is performed with respect to RW, while in the bottom graphs we use HAR-RV as the benchmark. Results are very clear and show that predictability of the realized variance is increased over time. Indeed, at the start of the sample we observe large losses of all models with respect to RW and HAR-RV probably due to uncertainty in estimated parameters. However, at the end of the forecasting period those losses seem to vanish suggesting that volatility becomes more easy to predict. Across the different specifications we observe that HAR-L and HAR-CJ-L are the top performer. This result confirms the in sample findings and indicates that the leverage component is important for volatility prediction of Bitcoin. A comparison across the two exchanges also suggests that volatility in the Coinbase exchange is easier to predict.