Over the last decade, Bitcoin has attracted a great deal of public interest and along with this, the Bitcoin market has grown rapidly. Its speculative price movements have also drawn the interest of many researchers as well as financial investors. Accordingly, numerous studies have been devoted to the analysis of Bitcoin, more exactly the volatility modelling of Bitcoin returns.
The Bitcoin returns are highly volatile and exhibit volatility clustering. To capture the stylized characteristics, many studies commonly employed various GARCH type models. See, for e.g., Glaser
Many studies have compared and evaluated various models under the assumption that parameters do not change. However, it is widely recognized that time series often suffer from structural or parameter changes in underlying models. Due to its importance in statistical inferences and actual practice, change point problem has attracted much attention from many researchers, particularly in time series analysis. For a general review, see Aue and Horváth (2013), Horváth and Rice (2014), and the references therein. In financial time series, parameter change often occurs due to, for e.g., changes of monetary policy or critical events. Considering the various events in the Bitcoin markets, it needs to take into account parameter change in analysis of Bitcoin. Indeed, several recent studies such as Thies and Molnár (2018), Canh
Our interest also lies in examining whether change points exist in the Bitcoin return series. For this, we use a test for parameter change to detect change points, but we emphasize in this study that it needs to use a test that is robust against deviating observations. This is because another characteristic of Bitcoin returns that should be noted is the existence of outlying observations, which are mainly caused by some events or incidents such as Mt. Gox incident. As is well known in the statistical literature, outlying observations can lead to serious bias in estimation and erroneous conclusion in testing. See, for e.g., Tsay (1988), Chen and Liu (1993), and Franses and Ghijsels (1999). This problem is still present in change point analysis. When outlying observations are included in a data set being suspected of having parameter changes, it is actually not so easy to determine whether the results are due to genuine changes or not. For more details on the parameter change test in the presence of outliers, see Fearnhead and Rigaill (2019) and Song and Kang (2021).
In this study, we fit GARCH models to Bitcoin returns and then detect change points. In order to reduce the undesirable influence of deviating observations, we conduct a robust parameter change test proposed by Song and Kang (2021) and report some change points that are not detected by the existing tests. We evaluate the models with and without parameter changes via AIC and compare value at risk (VaR) forecasting performance.
The remainder of this paper is organized as follows: Section 2 introduces data and the methodology, Section 3 presents the empirical results, and Section 4 concludes.
The data analyzed in this study consists of daily closing prices from January 2013 to December 2020, total 2,922 observations, which is available at coinmetrics, and the data after October 2020 is used in out-of-sample analysis. The prices {
We consider the standard GARCH (1, 1) model and use the robust test introduced by Song and Kang (2021) to detect change points. In order to model the increase in price over the observation period, non zero mean term
where {
The test statistics is constructed based on the so-called minimum density power divergence estimator (MDPDE), which is given by, for
where
Here,
MDPDE with
The test statistics employed for testing the null hypothesis of no change in the parameter
where
Under the null hypothesis,
One of the main characteristics of the test
The score test for parameter change is performed based on the ML-estimate. Since the MLE is sensitive to outliers, it is not easy to distinguish whether the testing result of the score test is due to outliers or genuine change in parameter. On the other hand, the test
In order to find multiple changes, we need to use the binary segmentation procedure as follows: we first perform the test
In the empirical analysis below, we use the AIC modified by Ninomiya (2015) to evaluate the adequacies of candidate models with and without change point. In the above GARCH (1, 1) model with
We conduct the test
where
In the present analysis, we consider
For each
MDPD estimates and descriptive statistics for each subperiod divided by
We now calculate one-step ahead 95% VaR for the period from October 2020 to December 2020, total 92 observations, and evaluate out-of-sample forecasting performance. The following models are considered: the above GARCH (1, 1) model with parameter changes obtained by
where
The one-step ahead conditional variance of the models
where
The one-step-ahead 95% VaR at time
In this study, we located some change points for the fitted GARCH models. In particular, considering the existence of some outlying observations in Bitcoin return series, we used the robust test for parameter change to detect change points. Based on the AIC that gives more penalty to the model with change points, we selected the model with optimal change points. The empirical results, including estimation results for each subperiod, showed that the whole period is meaningfully divided by the obtained change points, and the model with parameter changes provided a more accurate one-step-ahead 95% VaR compared to the VaRs from the naive models without any break. Our findings emphasize that the model allowing for parameter change can be better fitted to the Bitcoin data and thus can improve the accuracy of VaR forecasting.
The binary segmentation results of
Iteration no. | Start pt. | End pt. | P-value | Chg. pt. ( |
---|---|---|---|---|
1 | 1 | 2829 | 0.017 | 1801 |
2 | 1 | 1801 | 0.044 | 1389 |
3 | 1 | 1389 | 0.021 | 352 |
4 | 1 | 352 | 0.160 | · |
5 | 353 | 1389 | 0.040 | 1045 |
6 | 353 | 1045 | 0.187 | · |
7 | 1046 | 1389 | 0.134 | · |
8 | 1390 | 1801 | 0.014 | 1449 |
9 | 1390 | 1449 | 0.983 | · |
10 | 1450 | 1801 | 0.308 | · |
11 | 1802 | 2829 | 0.099 | 2187 |
12 | 1802 | 2187 | 0.389 | · |
13 | 2188 | 2829 | 0.039 | 2389 |
14 | 2188 | 2389 | 0.001 | 2309 |
15 | 2188 | 2309 | 0.845 | · |
16 | 2310 | 2389 | 0.772 | · |
17 | 2390 | 2829 | 0.112 | · |
MDPD estimates and descriptive statistics for each subperiod
No. | Period | N | Mean | Std | Skew. | Kurt. | ||||
---|---|---|---|---|---|---|---|---|---|---|
1 | 2013/01/01 – 2013/12/19 | 0.612 | 0.226 | 0.235 | 0.766 | 352 | 1.12 | 7.87 | −1.92 | 18.03 |
2 | 2013/12/20 – 2015/11/12 | −0.038 | 0.414 | 0.133 | 0.774 | 693 | −0.10 | 3.83 | −0.57 | 7.14 |
3 | 2015/11/13 – 2016/10/21 | 0.089 | 0.096 | 0.088 | 0.815 | 344 | 0.18 | 2.78 | −0.46 | 7.73 |
4 | 2016/10/22 – 2016/12/20 | 0.352 | 0.000 | 0.000 | 0.944 | 60 | 0.40 | 1.73 | −0.89 | 4.47 |
5 | 2016/12/21 – 2017/12/07 | 1.003 | 1.315 | 0.192 | 0.718 | 352 | 0.87 | 4.65 | 0.08 | 3.20 |
6 | 2017/12/08 – 2018/12/28 | −0.078 | 0.000 | 0.066 | 0.917 | 386 | −0.38 | 4.55 | −0.34 | 1.54 |
7 | 2018/12/29 – 2019/04/29 | 0.151 | 0.016 | 0.010 | 0.958 | 122 | 0.23 | 2.76 | 1.29 | 12.03 |
8 | 2019/04/30 – 2019/07/18 | 1.424 | 1.105 | 0.180 | 0.802 | 80 | 0.91 | 5.47 | −0.29 | 0.63 |
9 | 2019/07/19 – 2020/09/30 | −0.021 | 0.632 | 0.031 | 0.861 | 440 | 0.00 | 3.94 | −3.52 | 46.18 |
2013/01/01 – 2020/09/30 | 0.180 | 1.0727 | 0.171 | 0.789 | 2829 | 0.24 | 4.65 | −1.45 | 23.73 |