*3.2. In-Sample Estimation*

Table 2 provides posterior summaries for parameter estimates from the stochastic volatility with co-jumps (SVCJ) model for all cryptocurrency series. For the MCMC framework, there were 10,000 iterations with a burn-in of 2000 iterations to minimize the influence of the initial values. The initial values were as follows: *<sup>μ</sup>* ∼ *<sup>N</sup>*(0, 1), *<sup>κ</sup>* ∼ *<sup>N</sup>*(0, 1), *κθ* ∼ *<sup>N</sup>*(0, 1), *<sup>ρ</sup>* ∼ *<sup>u</sup>*(−1, 1), *<sup>σ</sup>*<sup>2</sup> *<sup>V</sup>* ∼ *IG*(2.5, 0.1), *<sup>μ</sup><sup>X</sup>* ∼ *<sup>N</sup>*(0, 100), *<sup>ρ</sup><sup>J</sup>* ∼ *<sup>N</sup>*(0, 4), *<sup>σ</sup>*<sup>2</sup> *<sup>X</sup>* ∼ *IG*(5, 20), *μ<sup>V</sup>* ∼ *G*(20, 10), and *λ* ∼ *B*(2, 40). The SVCJ model appears to be an ideal candidate for the cryptocurrencies, as indicated by the low MSE. The results show that the jump intensity *λ* is significant for all cryptocurrencies and is high for XRP and LTC, respectively 10.6% and 9.3%, and low for BCN and BTC, respectively 2.5% and 3.8%. The jump correlation *ρ<sup>J</sup>* is insignificant for all cryptocurrencies, similarly to the findings of Eraker et al. (2003) with stock prices.

The results also show a positive correlation, *ρ*, between the Brownian motions of returns and volatility for all cryptocurrencies except for XRP and DASH, where it is negative. This shows that a negative shock to returns increases volatility, and we can infer that the leverage effect contributes to the effectiveness in fitting the volatility of cryptocurrency returns. Figure 1 displays the jumps in returns and volatility for selected cryptocurrencies with high and low intensity of jumps. XRP and LTC have high intensity, and BTC and BCN have low intensity jumps.

<sup>8</sup> Our sample of cryptocurrencies captures market dynamics for various market capitalizations, ranging from high to low. Among the largest market caps (22 May 2019), we have Bitcoin (\$136.13 billion) and XRP (\$15.88 billion), in the middle market cap category, we have Litecoin (\$5.44 billion), and Bytecoin (\$0.169 billion) represents the small market cap category.

<sup>9</sup> It is important to acknowledge that there are significant differences in the quality of data that are available at multiple sites including CoinAPI, Cryptodatadownload, Cryptocompare, Coinmarketcap, and Coingecko. According to Alexander and Dakos (2019), some of these data are traded prices while others are non-traded prices issued by the exchanges, leading to questionable results in empirical studies.


**Table 2.** Parameter estimates of stochastic volatility with co-jumps (SVCJ).

Parameter estimates of SVCJ model are displayed along with the posterior means and the posterior standard deviations (in parentheses). The posterior sampling was carried out with 10,000 MCMC iterations and 2000 burn-in iterations.

**Figure 1.** Jumps in returns (**left columns**) and jumps in volatilities (**right columns**).

We have also estimated several AR(2) return models with various volatility specifications namely, asymmetric GARCH, IGARCH, TARCH, and GJR-GARCH, and by alternating between Student-*t* and Skewed Student-*t* errors. Table A1 (see Appendix A) displays the estimation results of these models for the cryptocurrencies. Each model was ranked on the basis of the log-likelihood function (higher the better) and the AIC (lower the better). Overall, the TGARCH with skewed *t*-distributed errors turns out to be the best volatility fitting model for the cryptocurrencies considered in this paper. These results contradict the findings of Chan et al. (2017) that IGARCH and GJR-GARCH models provide the best fits for the most popular and largest cryptocurrencies.

Table 3 summarizes these results by reporting the AR(2)-TGARCH(1,1)∼Skewed *t* estimated parameters. The parameters *α* and *β*, which represent short-run dynamics, are all significant for all cryptocurrencies. This suggests that the volatility is intensively reacting to market movements and that shocks to the conditional variance take time to die out. The leverage effect *γ* is statistically significant for all series except for XRP, DASH, and BCN. There were no remaining autocorrelations in both the standardized residuals and the squared standardized residuals.

**Table 3.** Parameter estimates of AR(2)-TGARCH(1,1)∼Skewed *t* volatility model.


Summary of the estimation results of the AR(2)-TGARCH(1,1)∼Skewed *t* for the cryptocurrencies. Standard errors are in parentheses and bold indicates insignificance at 5% and 1% levels.

The estimated volatility from these three distinctly different models are reported in Figure 2 for BTC, as an example. A visual examination shows that the volatility graphs are markedly different across models. The SVCJ model produces the smoothest plot because it includes all parameters of the volatility series. The plots generated from the remaining models are substantially jagged and show significant structural breaks, which can impede our estimation of tail risk.

**Figure 2.** Estimated Volatility from SVCJ, TGARCH, and RiskMetrics Models
