*3.2. Data and Descriptive Statistics*

We selected daily data of bitcoin and 14 asset prices from 6 June 2013, to 2 August 2021 as the sample, with the number of observations being 1703. Data were obtained from the Wind and Yahoo Finance databases. These assets cover four categories, namely, stock, bond, commodity and currency. The stock sample included the MSCI world index, S&P 500, FTSE 100, DAX 30, Nikkei 225 and SSEC. The bond sample includes the US bond index, non-U.S. bond index and emerging markets bond index, which are measured by the prices of ETFs that track the three indices. The commodity sample includes the S&P GSCI, CRB commodity index, Brent oil and gold. The currency sample is the U.S. dollar index.

Figure 1 plots the time series of the prices of bitcoin versus each asset. The correlation between bitcoin and most asset prices is not stable, with some periods moving in the same direction and some moving in the opposite direction. For example, bitcoin was positively correlated with the S&P 500 during 2017–2018 and then became negatively correlated in 2019. This suggests that the linkages between bitcoin and these traditional asset prices are time-varying, thus, necessitating the use of an (asymmetric) DCC model to more accurately capture the dynamic correlation between bitcoin and each asset. The daily percentage returns of all asset prices are calculated based on the following equation:

$$r\_t = 100 \ast \left(\ln P\_t - \ln P\_{t-1}\right) \tag{10}$$

where *rt* represents the return of each asset and *Pt* represents the original asset price. Table 1 presents descriptive statistics for the return of each asset. Bitcoin had a much higher mean return than other assets, which reflected the long-term upward trend in bitcoin prices over the sample period. Bitcoin also had a much higher standard deviation than other assets, indicating that bitcoin prices were extremely volatile. The level of skewness in all asset returns was not high, except for the emerging markets bond index, which exhibited a clear left skew. Most asset returns had high kurtosis. The Jarque–Bera (J-B) test significantly rejected the assumption of normality of the distribution for all series.

GARCH modeling requires data stationarity to ensure the validity of the estimation, and having ARCH effects is also a prerequisite for GARCH modeling. We used the augmented Dickey–Fuller (ADF) and Phillips–Perron (PP) tests to perform unit root tests on the return series (i.e., the first-order difference of asset price) and the Portmanteau test to perform an ARCH effect test on the return series, with the results reported in Tables 2 and 3, respectively. All return series were stationary at the 1% significance level and had significant ARCH effects at lags of order 5, 10 and 15, thus, satisfying the GARCH modeling conditions.

**Figure 1.** Time series of the prices of bitcoin versus various traditional financial assets. In each graph, the bitcoin price series is marked with a black line, corresponding to the right axis; the traditional asset price series is marked with a gray line, corresponding to the left axis.


### **Table 1.** Descriptive statistics.

Note: The J-B statistic is used to test the normality of the distribution of variables, and its null hypothesis is that the variable follows a normal distribution. The *p* value corresponds to the J-B test.

**Table 2.** Unit root test.


### **Table 3.** ARCH effect test.


Note: This table reports the results of the ARCH effect test at lags of order 5, 10 and 15. The null hypothesis for this test is "no ARCH effect".

Given the extremely high short-term volatility of bitcoin prices, bitcoin's short-term correlation with other assets is likely to be disturbed by its sharp short-term volatility. Intuitively, the long-term correlation between bitcoin and other asset prices is likely to be more

stable than the short-term correlation. To this end, we not only performed ADCC-GARCH analysis on daily frequency data of bitcoin and other asset prices, but also further performed ADCC-GARCH analysis on weekly and semi-monthly frequency data and then compared the results at different time frequencies to examine the differences in the linkage between bitcoin and various assets at different time frequencies. The weekly and semi-monthly frequency samples were obtained by taking the weekly and semi-monthly end-of-period values of the daily frequency samples, respectively, and the number of observations for both samples was 422 and 196, respectively. In addition to the weekly and semi-monthly frequencies, we also established an ADCC-GARCH model for the monthly frequency sample. However, since the number of observations for the monthly sample was only 97, the algorithm could not converge when performing ADCC-GARCH estimation. Therefore, it was abandoned.
