Discontinuous movements and asymmetries in cryptocurrency markets

Figures & data

Figure 1. Cryptocurrencies’ total jumps.

Note: This Figure depicts the jumps in the five cryptocurrencies considered, namely Bitcoin, Ether, Litecoin, Ripple, and EOS, for the period from 1st July 2017–30th June 2019 (730 observations).

Table 1. Descriptive statistics of total jump series.

	Bitcoin	Ether	Ripple	Litecoin	EOS
Mean	0.0005	0.0006	0.0009	0.0008	0.0013
Median	0.0001	0.0002	0.0002	0.0002	0.0003
Minimum	0	0	0	0	0
Maximum	0.0153	0.0197	0.0371	0.0604	0.0231
Std. Dev.	0.0012	0.0015	0.0030	0.0027	0.0025
Skewness	7.4276	6.5790	7.4026	15.9131	4.0983
Kurtosis	80.9461	65.7113	68.5741	333.0877	24.5054
Jarque-Bera	191511.9***	124885.9***	137457.5***	3344946***	16110.69***
N	730	730	730	730	730

Notes: This Table reports descriptive statistics for the total jump series as defined in Appendix A over the period from 1^st July 2017–30^th June 2019. *** denotes significance at the 1% level.

Table 2. Effects of jumps in Bitcoin on jumps in altcoins.

	Ether	Ripple	Litecoin	EOS
Panel A
${\delta }^{-}$	−6.35e-04***	−9.52e-04***	−7.39e-04***	−2.13e-03***
s.e	(1.17e-04)	(2.53e-04)	(1.61e-04)	(4.02e-04)
t-stat	[−5.4194]	[−3.7689]	[−4.5932]	[−5.2898]
${\delta }^{+}$	−6.44e-04***	−1.07e-03***	−1.01e-03***	−2.38e-03***
s.e	(1.47e-04)	(2.34e-04)	(2.62e-04)	(3.99e-04)
t-stat	[−4.3790]	[−4.5965]	[−3.8435]	[−5.9621]
${\psi }^s$	−1.37e-03***	−7.03e-04**	−7.39e-04***	−1.11e-03***
s.e	(3.49e-04)	(3.45e-04)	(1.61e-04)	(3.00e-04)
t-stat	[−3.9248]	[−2.0358]	[−4.5932]	[−3.6807]
${\psi }^l$	7.81e-04***	1.35e-03***	1.08e-03***	1.23e-03***
s.e	(1.71e-04)	(3.66e-04)	(3.16e-04)	(2.38e-04)
t-stat	[4.5619]	[3.6862]	[3.4138]	[5.1746]
${\gamma }^{co}$	8.47e-04***	1.45e-03***	1.22e-03***	1.75e-03***
s.e	(1.15e-04)	(2.40e-04)	(2.17e-04)	(1.63e-04)
t-stat	[7.3428]	[6.0472]	[5.6327]	[10.708]
${\gamma }^{rc}$	0.0383**	0.0267	0.1059***	0.0703**
s.e	(0.0170)	(0.0203)	(0.0394)	(0.0285)
t-stat	[2.2570]	[1.3154]	[2.6869]	[2.4647]
Panel B
T1	1.3817	3.5585***	2.2706**	2.4026**
p-value	[0.1675]	[0.0004]	[0.0235]	[0.0165]
F1	29.394***	15.5133***	29.2455***	11.4538***
p-value	[0.0000]	[0.0001]	[0.0000]	[0.0000]
T2	5.4217***	3.9387***	5.4079***	131.19***
p-value	[0.0000]	[0.0001]	[0.0000]	[0.0000]
F2	0.0044	0.2279	0.9385	5.5012**
p-value	[0.9473]	[0.6332]	[0.3330]	[0.0193]
F3	1.9091	12.6632***	5.1554**	5.7726**
p-value	[0.1675]	[0.0004]	[0.0235]	[0.0165]

Notes: This Table reports the estimated effects (Panel A) and results of the tests for asymmetric effects (Panel B) of jumps in Bitcoin on jumps in Ether, Ripple, Litecoin, and EOS. ${\delta }^{-}$ and ${\delta }^{+}$ refer to the effects of downside and upside jumps occurring in Bitcoin, respectively, ${\psi }^s$ and ${\psi }^l$ refer to the effects of small and large jumps occurring in Bitcoin, accordingly, ${\gamma }^{co}$ refers to the effect of jumps occurring in Bitcoin when jumps in both Bitcoin and the corresponding altcoin take place at the same time, and ${\gamma }^{rc}$ refers to the effect of the realized correlation between Bitcoin and the corresponding altcoin. All estimators are Heteroskedasticity and Autocorrelation Consistent (HAC) robust. The asymmetric response of altcoins to downside and upside jumps in Bitcoin is tested using the T₁ and F₁ tests, where the null hypothesis is $H_0:{\delta }^{-}={\delta }^{+}$. The asymmetric response of altcoins to the small and large jumps in Bitcoin is tested using the T₂ and F₂ tests, where the null hypothesis is $H_0:{\psi }^s={\psi }^l$. The asymmetric effect of upside and downside jumps in Bitcoin and small and large jumps in Bitcoin is tested using the F₃ test, where the null hypothesis is $H_0:{\delta }^{-}+{\delta }^{+}={\psi }^s+{\psi }^l$. *** and ** indicate significance at the 1% and 5% level, respectively.

Table 3. Effects of jumps in altcoins on jumps in Bitcoin.

	Ether	Ripple	Litecoin	EOS
Panel A
${\delta }^{-}$	−4.53e-04***	−4.08e-04***	−3.84e-04***	7.26e-04***
s.e	(8.67e-05)	(7.94e-05)	(7.9oe-05)	(1.38e-04)
t-stat	[−5.2270]	[−5.1349]	[−4.8609]	[−5.2770]
${\delta }^{+}$	−5.68e-04***	−4.81e-04***	−6.32e-04***	−7.74e-04***
s.e	(1.10e-04)	(8.52e-05)	(1.16-04)	(1.43e-04)
t-stat	[−51524]	[−5.6513]	[−5.4586]	[−5.3611]
${\psi }^s$	−7.82e-04***	−6.59e-04***	−9.66-04***	−6.59e-04***
s.e	(1.66e-04)	(2.53e-04)	(2.82e-04)	(2.02e-04)
t-stat	[−4.7162]	[−2.6091]	[−3.4261]	[−3.2588]
${\psi }^l$	7.63e-04***	7.32e-04***	7.94e-04***	5.18e-04***
s.e	(1.44e-04)	(1.84e-04)	(1.47e-04)	(1.17e-04)
t-stat	[5.3110]	[3.9794]	[5.4003]	[4.4367]
${\gamma }^{co}$	6.12e-04***	6.33e-04***	5.59e-04***	6.17e-04***
s.e	(9.4e-05)	(7.08e-05)	(8.06e-05)	(8.99e-05)
t-stat	[6.5037]	[8.1080]	[6.9342]	[6.8562]
${\gamma }^{rc}$	0.0312**	0.0289**	0.0307***	0.0286***
s.e	(0.0137)	(0.0129)	(0.0105)	(9.64e-03)
t-stat	[2.2696]	[2.2403]	[2.9093]	[2.9723]
Panel B
T1	7.3158***	4.5044***	4.5584***	4.9705***
p-value	[0.0000]	[0.0000]	[0.0000]	[0.0000]
F1	53.521***	20.289***	20.779***	24.706***
p-value	[0.0000]	[0.0000]	[0.0000]	[0.0000]
T2	−1.0901	−0.8037	−2.2840**	−0.4593
p-value	[0.2760]	[0.4218]	[0.0227]	[0.6461]
F2	1.1885	0.6460	5.2169**	0.2109
p-value	[0.2760]	[0.4218]	[0.0227]	[0.6461]
F3	9.3408***	6.6785**	10.185***	13.185***
p-value	[0.0023]	[0.0100]	[0.0015]	[0.0003]

Notes: This Table reports the estimated effects (Panel A) and results of the tests for asymmetric effects (Panel B) of jumps in each altcoin on jumps in Bitcoin. ${\delta }^{-}$ and ${\delta }^{+}$ refer to the effects of downside and upside jumps occurring in altcoins, respectively, ${\psi }^s$ and ${\psi }^l$ refer to the effects of small and large jumps occurring in altcoins, accordingly, ${\gamma }^{co}$ refers to the effect of jumps occurring in each altcoin when jumps in both Bitcoin and the corresponding altcoin take place at the same time, and ${\gamma }^{rc}$ refers to the effect of the realized correlation between Bitcoin and the corresponding altcoin. All estimators are Heteroskedasticity and Autocorrelation Consistent (HAC) robust. The asymmetric response of Bitcoin to downside and upside jumps in altcoins is tested using the T₁ and F₁ tests, where the null hypothesis is $H_0:{\delta }^{-}={\delta }^{+}$. The asymmetric response of Bitcoin to the small and large jumps in altcoins is tested using the T₂ and F₂ tests, where the null hypothesis is $H_0:{\psi }^s={\psi }^l$. The asymmetric effect of upside and downside jumps in an altcoin and small and large jumps in the same altcoin is tested using the F₃ test, where the null hypothesis is $H_0:{\delta }^{-}+{\delta }^{+}={\psi }^s+{\psi }^l$. *** and ** indicate significance at the 1% and 5% level, respectively.

Table 4. Effects of jumps in altcoins on jumps in Ether.

	Ripple	Litecoin	EOS
Panel A
${\delta }^{-}$	−8.93e-04***	−5.84e-04***	−9.00e-04***
s.e	(1.46e-04)	(1.04e-04)	(1.63e-04)
t-stat	[−6.1370]	[−5.6190]	[−5.5097]
${\delta }^{+}$	−8.03e-04***	−7.32e-04***	−8.89e-04***
s.e	(1.29e-04)	(1.52e-04)	(1.78 e-04)
t-stat	[−6.2114]	[−4.8141]	[−5.0072]
${\psi }^s$	−1.42e-04**	−1.42e-03***	−8.74e-04***
s.e	(5.87e-04)	(4.31e-04)	(2.90e-04)
t-stat	[−2.4328]	[−3.3116]	[−3.0131]
${\psi }^l$	9.84e-04***	8.60e-04***	7.26e-04***
s.e	(2.36e-04)	(1.98e-04)	(1.77e-04)
t-stat	[4.1716]	[4.3394]	[4.0993]
${\gamma }^{co}$	1.34e-03***	7.80e-04***	1.01e-03***
s.e	(1.47e-04)	(9.82e-05)	(1.30e-04)
t-stat	[7.7201]	[7.9389]	[7.7615]
${\gamma }^{rc}$	0.0221*	0.0321**	0.0301**
s.e	(0.012660)	(0.0135)	(0.0121)
t-stat	[1.7522]	[2.3778]	[2.4880]
Panel B
T1	0.9162	−1.1192	0.0956
p-value	[0.3599]	[0.2634]	[0.9238]
F1	0.8394	1.2527	9.15e-03
p-value	[0.3599]	[0.2634]	[0.9238]
T2	3.9368***	4.4701***	5.6111***
p-value	[0.0001]	[0.0000]	[0.0000]
F2	15.4984***	19.9825***	31.4848***
p-value	[0.0001]	[0.0000]	[0.0000]
F3	2.8742*	2.3322	8.8349***
p-value	[0.0904]	[0.1272]	[0.0031]

Notes: This Table reports the estimated effects (Panel A) and results of the tests for asymmetric effects (Panel B) of jumps in each altcoin on jumps in Ether. ${\delta }^{-}$ and ${\delta }^{+}$ refer to the effects of downside and upside jumps occurring in altcoins, respectively, ${\psi }^s$ and ${\psi }^l$ refer to the effects of small and large jumps occurring in altcoins, accordingly, ${\gamma }^{co}$ refers to the effect of jumps occurring in each altcoin when jumps in both Ether and the corresponding altcoin take place at the same time, and ${\gamma }^{rc}$ refers to the effect of the realized correlation between Ether and the corresponding altcoin. All estimators are Heteroskedasticity and Autocorrelation Consistent (HAC) robust. The asymmetric response of Ether to downside and upside jumps in altcoins is tested using the T₁ and F₁ tests, where the null hypothesis is $H_0:{\delta }^{-}={\delta }^{+}$. The asymmetric response of Ether to the small and large jumps in altcoins is tested using the T₂ and F₂ tests, where the null hypothesis is $H_0:{\psi }^s={\psi }^l$. The asymmetric effect of upside and downside jumps in an altcoin and small and large jumps in the same altcoin is tested using the F₃ test, where the null hypothesis is $H_0:{\delta }^{-}+{\delta }^{+}={\psi }^s+{\psi }^l$. ***, ** and * indicate significance at the 1%, 5% and 10% level, respectively.

Table 5. Effects of jumps in altcoins on jumps in Ripple.

	Ether	Litecoin	EOS
Panel A
${\delta }^{-}$	−1.50e-03***	−1.50e-03***	−1.65e-03***
s.e	(3.51e-04)	(2.93e-04)	(3.82e-04)
t-stat	[−4.2825]	[−3.4774]	[−4.3254]
${\delta }^{+}$	−1.81e-03***	−1.73e-03***	−1.90e-03***
s.e	(4.12e-04)	(3.57e-04)	(4.06e-04)
t-stat	[−4.4177]	[−4.8766]	[−4.6938]
${\psi }^s$	−9.73e-04**	−8.04e-04*	−4.95e-04**
s.e	(4.45e-04)	(4.44e-04)	(2.37e-04)
t-stat	[−2.1859]	[−1.8207]	[−2.0929]
${\psi }^l$	1.16e-03***	1.53e-03***	9.71e-04***
s.e	(2.83e-04)	(3.43e-04)	(2.56e-04)
t-stat	[4.1147]	[4.4661]	[3.7957]
${\gamma }^{co}$	1.08e-03***	1.15e-03***	1.15e-03***
s.e	(1.89e-04)	(1.83e-04)	(1.66e-04)
t-stat	[5.7144]	[6.3170]	[6.9489]
${\gamma }^{rc}$	8.68e-03	−0.0146	0.0247
s.e	(0.0124)	(0.0306)	(0.0171)
t-stat	[0.6976]	[−0.4778]	[1.4483]
Panel B
T1	−1.4733	−2.5529**	−1.3938
p-value	[0.1411]	[0.0109]	[0.1638]
F1	2.1707	6.5175**	1.9429
p-value	[0.1411]	[0.0109]	[0.1638]
T2	3.6122***	3.8206***	4.7700***
p-value	[0.0003]	[0.0001]	[0.0000]
F2	13.0480***	14.5969***	22.753***
p-value	[0.0003]	[0.0001]	[0.0000]
F3	13.7425***	16.0192***	19.5624***
p-value	[0.0002]	[0.0001]	[0.0000]

Notes: This Table reports the estimated effects (Panel A) and results of the tests for asymmetric effects (Panel B) of jumps in each altcoin on jumps in Ripple. ${\delta }^{-}$ and ${\delta }^{+}$ refer to the effects of downside and upside jumps occurring in altcoins, respectively, ${\psi }^s$ and ${\psi }^l$ refer to the effects of small and large jumps occurring in altcoins, accordingly, ${\gamma }^{co}$ refers to the effect of jumps occurring in each altcoin when jumps in both Ripple and the corresponding altcoin take place at the same time, and ${\gamma }^{rc}$ refers to the effect of the realized correlation between Ripple and the corresponding altcoin. All estimators are Heteroskedasticity and Autocorrelation Consistent (HAC) robust. The asymmetric response of Ripple to downside and upside jumps in altcoins is tested using the T₁ and F₁ tests, where the null hypothesis is $H_0:{\delta }^{-}={\delta }^{+}$. The asymmetric response of Ripple to the small and large jumps in altcoins is tested using the T₂ and F₂ tests, where the null hypothesis is $H_0:{\psi }^s={\psi }^l$. The asymmetric effect of upside and downside jumps in an altcoin and small and large jumps in the same altcoin is tested using the F₃ test, where the null hypothesis is $H_0:{\delta }^{-}+{\delta }^{+}={\psi }^s+{\psi }^l$. ***, ** and * indicate significance at the 1%, 5% and 10% level, respectively.

Table 6. Effects of jumps in altcoins on jumps in Litecoin.

	Ether	Ripple	EOS
Panel A
${\delta }^{-}$	−7.52e-04***	−9.17e-04***	−1.24e-03***
s.e	(1.38e-04)	(2.39e-04)	(3.57e-04)
t-stat	[−5.4673]	[−3.8354]	[−3.4842]
${\delta }^{+}$	−1.02e-03***	−8.88e-04***	−1.08e03***
s.e	(3.05e-04)	(1.56e-04)	(3.60e-04)
t-stat	[−3.3645]	[−5.6867]	[−3.0056]
${\psi }^s$	−1.15e-03***	−1.48e-03***	−6.98e-04***
s.e	(3.41e-04)	(4.93e-04)	(2.45e-04)
t-stat	[−3.3840]	[−3.0190]	[−2.8483]
${\psi }^l$	1.23e-03***	1.25e-03***	6.54e-04***
s.e	(4.09e-04)	(4.40e-04)	(1.85e-04)
t-stat	[3.0087]	[2.8490]	[3.5451]
${\gamma }^{co}$	1.25e-03***	1.36e-03***	1.09e-03***
s.e	(2.23e-04)	(2.45e-04)	(1.38e-04)
t-stat	[5.6224]	[5.5872]	[7.9629]
${\gamma }^{rc}$	0.0572***	0.0489***	0.0494***
s.e	(0.0168)	(0.012645)	(0.0138)
t-stat	[3.3890]	[3.871115]	[3.5583]
Panel B
T1	−0.8135	0.1750	1.3339
p-value	[0.4162]	[0.8611]	[0.1827]
F1	0.6617	0.0306	1.7793
p-value	[0.4162]	[0.8611]	[0.1827]
T2	6.3282***	4.3966***	4.5178***
p-value	[0.0000]	[0.0000]	[0.0000]
F2	40.0466***	19.330***	20.4111***
p-value	[0.0000]	[0.0000]	[0.0000]
F3	4.2762**	2.8127*	8.6414***
p-value	[0.0390]	[0.0940]	[0.0034]

Notes: This Table reports the estimated effects (Panel A) and results of the tests for asymmetric effects (Panel B) of jumps in each altcoin on jumps in Litecoin. ${\delta }^{-}$ and ${\delta }^{+}$ refer to the effects of downside and upside jumps occurring in altcoins, respectively, ${\psi }^s$ and ${\psi }^l$ refer to the effects of small and large jumps occurring in altcoins, accordingly, ${\gamma }^{co}$ refers to the effect of jumps occurring in each altcoin when jumps in both Litecoin and the corresponding altcoin take place at the same time, and ${\gamma }^{rc}$ refers to the effect of the realized correlation between Litecoin and the corresponding altcoin. All estimators are Heteroskedasticity and Autocorrelation Consistent (HAC) robust. The asymmetric response of Litecoin to downside and upside jumps in altcoins is tested using the T₁ and F₁ tests, where the null hypothesis is $H_0:{\delta }^{-}={\delta }^{+}$. The asymmetric response of Litecoin to the small and large jumps in altcoins is tested using the T₂ and F₂ tests, where the null hypothesis is $H_0:{\psi }^s={\psi }^l$. The asymmetric effect of upside and downside jumps in an altcoin and small and large jumps in the same altcoin is tested using the F₃ test, where the null hypothesis is $H_0:{\delta }^{-}+{\delta }^{+}={\psi }^s+{\psi }^l$. *** , ** and * indicate significance at the 1%, 5% and 10% level, respectively.

Table 7. Effects of jumps in altcoins on jumps in EOS.

	Ether	Ripple	Litecoin
Panel A
${\delta }^{-}$	−1.62e-03***	−1.42e-03***	−1.62e-03***
s.e	(2.76e-04)	(2.42e-04)	(2.85e-04)
t-stat	[−5.9057]	[−5.8692]	[−5.7028]
${\delta }^{+}$	−1.82e-03***	−1.78e-03***	−2.01e-03***
s.e	(2.87e-04)	(2.63e-04)	(3.34e-04)
t-stat	[−6.3651]	[−6.7821]	[−6.0378]
${\psi }^s$	−1.51e-03***	−1.39e-03**	−1.10e-03***
s.e	(3.34e-04)	(5.62e-04)	(3.76e-04)
t-stat	[−4.5326]	[−2.4884]	[−2.9293]
${\psi }^l$	1.43e-03***	1.40e-03***	1.41e-03***
s.e	(2.53e-04)	(2.84e-04)	(2.63e-04)
t-stat	[5.655937]	[4.9325]	[5.3654]
${\gamma }^{co}$	1.53e-03***	2.06e-03***	1.66e-03***
s.e	(1.69e-04)	(1.91e-04)	(1.66e-04)
t-stat	[9.1028]	[10.8214]	[10.0548]
${\gamma }^{rc}$	0.0684***	0.0581***	0.0591**
s.e	(0.0249)	(0.0191)	(0.0252)
t-stat	[2.7402]	[3.0304]	[2.3416]
Panel B
T1	−1.1140	−2.2287**	−2.2806**
p-value	[0.2656]	[0.0261]	[0.0229]
F1	1.2411	4.9673**	5.2015**
p-value	[0.2656]	[0.0261]	[0.0229]
T2	7.7456***	4.4392***	5.4848***
p-value	[0.0000]	[0.0000]	[0.0000]
F2	59.9951***	19.7071***	30.0831***
p-value	[0.0000]	[0.0000]	[0.0000]
F3	20.8655***	17.029***	26.2906***
p-value	[0.0000]	[0.0000]	[0.0000]

Notes: This Table reports the estimated effects (Panel A) and results of the tests for asymmetric effects (Panel B) of jumps in each altcoin on jumps in EOS. ${\delta }^{-}$ and ${\delta }^{+}$ refer to the effects of downside and upside jumps occurring in altcoins, respectively, ${\psi }^s$ and ${\psi }^l$ refer to the effects of small and large jumps occurring in altcoins, accordingly, ${\gamma }^{co}$ refers to the effect of jumps occurring in each altcoin when jumps in both EOS and the corresponding altcoin take place at the same time, and ${\gamma }^{rc}$ refers to the effect of the realized correlation between EOS and the corresponding altcoin. All estimators are Heteroskedasticity and Autocorrelation Consistent (HAC) robust. The asymmetric response of EOS to downside and upside jumps in altcoins is tested using the T₁ and F₁ tests, where the null hypothesis is $H_0:{\delta }^{-}={\delta }^{+}$. The asymmetric response of EOS to the small and large jumps in altcoins is tested using the T₂ and F₂ tests, where the null hypothesis is $H_0:{\psi }^s={\psi }^l$. The asymmetric effect of upside and downside jumps in an altcoin and small and large jumps in the same altcoin is tested using the F₃ test, where the null hypothesis is $H_0:{\delta }^{-}+{\delta }^{+}={\psi }^s+{\psi }^l$. *** and ** indicate significance at the 1% and 5% level, respectively.

Table 8. Asset allocation based on the Sortino ratio.

	Optimal			Weights
	Sortino Ratio	Mean	Bitcoin	Ether	Litecoin	Ripple	EOS
Portfolio	0.0611	0.0019	0.9120	0.0100	0.0220	0.0450	0.0110

Notes: This Table presents the optimal Sortino ratio and mean portfolio return as well as the weights of the optimal portfolio consisting of the five cryptocurrencies considered in our study, i.e. Bitcoin, Ether, Litecoin, Ripple, and EOS, over the period from 1^st July 2017–30^th June 2019.

Figure 2. Optimal mean and Sortino ratio in a rolling window.

Note: These Figures depict the optimal mean portfolio return and optimal Sortino ratio of the portfolio consisting of the five cryptocurrencies considered, namely Bitcoin, Ether, Litecoin, Ripple, and EOS, in a rolling window application for the period from 1st July 2018–30th June 2019 (365 observations).

Figure 3. Weights of the optimal portfolio in a rolling window.

Note: This Figure depicts the weights of the optimal portfolio consisting of the five cryptocurrencies considered in our study, namely Bitcoin, Ether, Litecoin, Ripple, and EOS, in a rolling window application for the period from 1st July 2018–30th June 2019 (365 observations).

Table 9. Effects of jumps in the S&P 500 equity index on jumps in cryptocurrencies.

	Bitcoin	Ether	Ripple	Litecoin	EOS
Panel A
${\delta }^{-}$	−4.57e-04**	−6.14e-04***	−5.93e-04	−5.60e-04	−1.18e-03***
s.e	(1.78e-04)	(2.04e-04)	(3.69e-04)	(3.73e-04)	(3.28e-04)
t-stat	[−2.5706]	[−3.0155]	[−1.6051]	[−1.4998]	[−3.6247]
${\delta }^{+}$	−5.44e-04***	−5.82e-04***	−1.35e-03***	−9.32e-04**	−8.16e-04**
s.e	(1.74e-04)	(2.04e-04)	(3.81e-04)	(3.89e-04)	(3.49e-04)
t-stat	[−3.1318]	[−2.8528]	[−3.5598]	[−2.3969]	[−2.340142]
${\psi }^s$	1.24 e-04	7.77e-05	7.66e-04***	2.15e-04	3.21e-05
s.e	(1.01e-04)	(1.3e-04)	(2.60e-04)	(2.37e-04)	(2.14e-04)
t-stat	[1.2384]	[0.5980]	[2.9503]	[0.9089]	[0.1497]
${\psi }^l$	−5.79e-05	−1.48e-04	−1.65e-04	−2.52e-04	−6.15e-04**
s.e	1.41e-04	(1.82e-04)	(3.62e-04)	(3.30e-04)	(2.99e-04)
t-stat	[−0.4101]	[−0.8089]	[−0.4568]	[−0.7631]	[−2.0539]
${\gamma }^{co}$	6.41 e-04***	9.66e-04***	1.74e-03***	1.26e-03***	1.96e-03***
s.e	(1.56 e-04)	(1.78e-04)	(3.24e-04)	(3.28e-04)	(2.93e-04)
t-stat	[4.1161]	[5.4367]	[−1.6051]	[3.8507]	[6.6807]
Panel B
T1	−0.6951	0.1998	−2.3970**	−1.2866	1.4198
p-value	[0.4872]	[0.8416]	[0.0168]	[0.1986]	[0.1561]
F1	0.4832	0.0399	5.7458**	1.6555	2.0159
p-value	[0.4872]	[0.8416]	[0.0168]	[0.1986]	[0.1561]
T2	−1.0603	−1.0103	−2.1057**	−1.1574	−1.7681*
p-value	[0.2893]	[0.3127]	[0.0356]	[0.2475]	[0.0775]
F2	1.1244	1.0207	4.4340**	1.3396	3.1264*
p-value	[0.2893]	[0.3127]	[0.0356]	[0.2475]	[0.0775]
F3	6.5119**	5.0645**	7.0662***	2.3765	2.8915*
p-value	[0.0109]	[0.0247]	[0.0080]	[0.1236]	[0.0895]

Notes: This Table reports the estimated effects (Panel A) and results of the tests for asymmetric effects (Panel B) of jumps in the S&P 500 index on jumps in cryptocurrencies. ${\delta }^{-}$ and ${\delta }^{+}$ refer to the effects of downside and upside jumps occurring in the S&P 500 index, respectively, ${\psi }^s$ and ${\psi }^l$ refer to the effects of small and large jumps occurring in the S&P 500 index, accordingly, and ${\gamma }^{co}$ refers to the effect of jumps occurring in the S&P 500 index when jumps in both the S&P 500 index and the corresponding cryptocurrency take place at the same time. All estimators are Heteroskedasticity and Autocorrelation Consistent (HAC) robust. The asymmetric response of cryptocurrencies to downside and upside jumps in the S&P 500 index is tested using the T₁ and F₁ tests, where the null hypothesis is $H_0:{\delta }^{-}={\delta }^{+}$. The asymmetric response of cryptocurrencies to the small and large jumps in the S&P 500 index is tested using the T₂ and F₂ tests, where the null hypothesis is $H_0:{\psi }^s={\psi }^l$. The asymmetric effect of upside and downside jumps in the S&P 500 index and small and large jumps in the index is tested using the F₃ test, where the null hypothesis is $H_0:{\delta }^{-}+{\delta }^{+}={\psi }^s+{\psi }^l$. ***, ** and * indicate significance at the 1%, 5% and 10% level, respectively.