Parameter estimates for the bivariate models fitted on the pairs (BTC-S&P500, BTC-NASDAQ, BTC-VIX, BTC-WTI), comparing normal, Student-t, asymmetric normal, and asymmetric Student-t distributions. The table includes estimates for the degrees of freedom (ν), asymmetry parameters (α1, α2), and other model parameters (ω, a, b), along with log-likelihood (LogLik), AIC, and BIC values
| Normal | Student-t | Asymmetric normal | Asymmetric Student-t | Normal | Student-t | Asymmetric normal | Asymmetric Student-t | |
|---|---|---|---|---|---|---|---|---|
| (a) BTC-S&P500 | (b) BTC-NASDAQ | |||||||
| ν | – | 4.4332 *** | – | 4.5571 *** | – | 4.5494 *** | – | 4.6963 *** |
| – | (0.2159) | – | (0.2357) | – | (0.2361) | – | (0.2716) | |
| α1 | – | – | −0.9206 *** | −0.0598 | – | – | −0.7642 ** | −0.0151 |
| – | – | (0.1950) | (0.1145) | – | – | (0.3392) | (0.1129) | |
| α2 | – | – | −1.0431 *** | −0.4110 *** | – | – | −1.2605 ** | −0.6354 *** |
| – | – | (0.3009) | (0.1208) | – | – | (0.5042) | (0.1194) | |
| ω | 0.0925 | 0.0318 | 0.3207 ** | 0.0565 | 0.1024 | 0.0413 | 0.3359 *** | 0.0671 |
| (0.0864) | (0.1439) | (0.1308) | (0.1479) | (0.0987) | (0.1502) | (0.1129) | (0.1514) | |
| a | 0.0121 *** | 0.0165 *** | 0.0130 *** | 0.0178 *** | 0.0128 *** | 0.0162 *** | 0.0149 *** | 0.0191 *** |
| (0.0037) | (0.0059) | (0.0038) | (0.0064) | (0.0035) | (0.0052) | (0.0045) | (0.0062) | |
| b | 0.9972 *** | 0.9976 *** | 0.9985 *** | 0.9974 *** | 0.9973 *** | 0.9979 *** | 0.9983 *** | 0.9978 *** |
| (0.0021) | (0.0031) | (0.0017) | (0.0032) | (0.0022) | (0.0025) | (0.0019) | (0.0030) | |
| LogLik | −5758.5180 | −5461.9997 | −5707.4198 | −5454.6774 | −5746.2173 | −5470.4683 | −5697.7899 | −5454.9885 |
| AIC | 11523.0359 | 10931.9993 | 11424.8396 | 10921.3548 | 11498.4346 | 10948.9366 | 11405.5798 | 10921.9771 |
| BIC | 11539.9127 | 10954.5017 | 11452.9675 | 10955.1084 | 11515.3114 | 10971.4390 | 11433.7078 | 10955.7307 |
| (c) BTC-VIX | (d) BTC-WTI | |||||||
| ν | – | 4.7397 *** | – | 4.7762 *** | – | 3.9191 *** | – | 4.0983 *** |
| – | (0.2487) | – | (0.2491) | – | (0.1454) | – | (0.1656) | |
| α1 | – | – | −1.2765 *** | −0.1801 | – | – | −0.0705 | −0.0379 |
| – | – | (0.0972) | (0.1117) | – | – | (0.1236) | (0.0785) | |
| α2 | – | – | −0.1514 | −0.7243 *** | – | – | 2.5309 *** | 1.1691 *** |
| – | – | (0.1565) | (0.1317) | – | – | (0.1649) | (0.1407) | |
| ω | 0.0524 | 0.0544 * | 0.1380 | 0.1159 ** | −0.1457 | −0.0750 | −0.2379 | −0.1380 |
| (0.0337) | (0.0300) | (0.0847) | (0.0468) | (0.1054) | (0.1457) | (0.1451) | (0.1456) | |
| a | 0.0031 | 0.0070 | 0.0032 | 0.0101 | 0.0112 *** | 0.0190 ** | 0.0219 *** | 0.0283 *** |
| (0.0048) | (0.0081) | (0.0045) | (0.0110) | (0.0037) | (0.0075) | (0.0077) | (0.0105) | |
| b | 0.9930 *** | 0.9755 *** | 0.9958 *** | 0.9735 *** | 0.9976 *** | 0.9968 *** | 0.9976 *** | 0.9960 *** |
| (0.0202) | (0.0387) | (0.0066) | (0.0336) | (0.0023) | (0.0041) | (0.0022) | (0.0043) | |
| LogLik | −5818.1159 | −5535.7538 | −5777.3546 | −5516.1544 | −5762.9108 | −5349.8262 | −5640.6314 | −5299.0091 |
| AIC | 11642.2317 | 11079.5076 | 11564.7092 | 11044.3088 | 11531.8216 | 10707.6524 | 11291.2629 | 10610.0183 |
| BIC | 11659.1085 | 11102.0100 | 11592.8372 | 11078.0624 | 11548.6983 | 10730.1548 | 11319.3908 | 10643.7718 |
| Normal | Student- | Asymmetric normal | Asymmetric Student- | Normal | Student- | Asymmetric normal | Asymmetric Student- | |
|---|---|---|---|---|---|---|---|---|
| – | 4.4332 *** | – | 4.5571 *** | – | 4.5494 *** | – | 4.6963 *** | |
| – | (0.2159) | – | (0.2357) | – | (0.2361) | – | (0.2716) | |
| – | – | −0.9206 *** | −0.0598 | – | – | −0.7642 ** | −0.0151 | |
| – | – | (0.1950) | (0.1145) | – | – | (0.3392) | (0.1129) | |
| – | – | −1.0431 *** | −0.4110 *** | – | – | −1.2605 ** | −0.6354 *** | |
| – | – | (0.3009) | (0.1208) | – | – | (0.5042) | (0.1194) | |
| 0.0925 | 0.0318 | 0.3207 ** | 0.0565 | 0.1024 | 0.0413 | 0.3359 *** | 0.0671 | |
| (0.0864) | (0.1439) | (0.1308) | (0.1479) | (0.0987) | (0.1502) | (0.1129) | (0.1514) | |
| 0.0121 *** | 0.0165 *** | 0.0130 *** | 0.0178 *** | 0.0128 *** | 0.0162 *** | 0.0149 *** | 0.0191 *** | |
| (0.0037) | (0.0059) | (0.0038) | (0.0064) | (0.0035) | (0.0052) | (0.0045) | (0.0062) | |
| 0.9972 *** | 0.9976 *** | 0.9985 *** | 0.9974 *** | 0.9973 *** | 0.9979 *** | 0.9983 *** | 0.9978 *** | |
| (0.0021) | (0.0031) | (0.0017) | (0.0032) | (0.0022) | (0.0025) | (0.0019) | (0.0030) | |
| LogLik | −5758.5180 | −5461.9997 | −5707.4198 | −5454.6774 | −5746.2173 | −5470.4683 | −5697.7899 | −5454.9885 |
| AIC | 11523.0359 | 10931.9993 | 11424.8396 | 10921.3548 | 11498.4346 | 10948.9366 | 11405.5798 | 10921.9771 |
| BIC | 11539.9127 | 10954.5017 | 11452.9675 | 10955.1084 | 11515.3114 | 10971.4390 | 11433.7078 | 10955.7307 |
| – | 4.7397 *** | – | 4.7762 *** | – | 3.9191 *** | – | 4.0983 *** | |
| – | (0.2487) | – | (0.2491) | – | (0.1454) | – | (0.1656) | |
| – | – | −1.2765 *** | −0.1801 | – | – | −0.0705 | −0.0379 | |
| – | – | (0.0972) | (0.1117) | – | – | (0.1236) | (0.0785) | |
| – | – | −0.1514 | −0.7243 *** | – | – | 2.5309 *** | 1.1691 *** | |
| – | – | (0.1565) | (0.1317) | – | – | (0.1649) | (0.1407) | |
| 0.0524 | 0.0544 * | 0.1380 | 0.1159 ** | −0.1457 | −0.0750 | −0.2379 | −0.1380 | |
| (0.0337) | (0.0300) | (0.0847) | (0.0468) | (0.1054) | (0.1457) | (0.1451) | (0.1456) | |
| 0.0031 | 0.0070 | 0.0032 | 0.0101 | 0.0112 *** | 0.0190 ** | 0.0219 *** | 0.0283 *** | |
| (0.0048) | (0.0081) | (0.0045) | (0.0110) | (0.0037) | (0.0075) | (0.0077) | (0.0105) | |
| 0.9930 *** | 0.9755 *** | 0.9958 *** | 0.9735 *** | 0.9976 *** | 0.9968 *** | 0.9976 *** | 0.9960 *** | |
| (0.0202) | (0.0387) | (0.0066) | (0.0336) | (0.0023) | (0.0041) | (0.0022) | (0.0043) | |
| LogLik | −5818.1159 | −5535.7538 | −5777.3546 | −5516.1544 | −5762.9108 | −5349.8262 | −5640.6314 | −5299.0091 |
| AIC | 11642.2317 | 11079.5076 | 11564.7092 | 11044.3088 | 11531.8216 | 10707.6524 | 11291.2629 | 10610.0183 |
| BIC | 11659.1085 | 11102.0100 | 11592.8372 | 11078.0624 | 11548.6983 | 10730.1548 | 11319.3908 | 10643.7718 |
Note(s): * 10%, ** 5%, and *** 1% significance levels
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