Table 6 Parameter estimates for the...

Table 6

Parameter estimates for the bivariate models with explanatory variables 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), AICs, 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	–	4.4419 ***	–	4.5641 ***	ν	–	4.5534 ***	–	4.6996 ***
ν	–	(0.2271)	–	(0.2381)		–	(0.2407)	–	(0.2589)
α₁	–	–	−0.9220 ***	−0.0545		–	–	−0.5478 *	−0.0161
α₁	–	–	(0.2183)	(0.1103)		–	–	(0.2872)	(0.1059)
α₂	–	–	−1.0267 ***	−0.4129 ***		–	–	−1.5252 ***	−0.6353 ***
α₂	–	–	(0.3246)	(0.1206)		–	–	(0.3414)	(0.1224)
ω	0.0311	0.0452	0.2863 ***	0.0648		0.0292	0.0052	0.2443 *	0.0276
ω	(0.0713)	(0.0627)	(0.1018)	(0.0671)		(0.0907)	(0.1205)	(0.1364)	(0.1341)
a	0.0143 **	0.0261 **	0.0145 **	0.0287 **		0.0133 ***	0.0175 **	0.0170 ***	0.0206 **
a	(0.0059)	(0.0123)	(0.0061)	(0.0135)		(0.0044)	(0.0072)	(0.0062)	(0.0092)
b	0.9921 ***	0.9810 ***	0.9955 ***	0.9796 ***		0.9956 ***	0.9956 ***	0.9968 ***	0.9951 ***
b	(0.0079)	(0.0161)	(0.0050)	(0.0179)		(0.0042)	(0.0075)	(0.0035)	(0.0071)
ρ_GOLD	2.0794	3.0496 *	1.4236	3.2294 *		1.3638	1.4850	1.2627	1.6118
ρ_GOLD	(1.3081)	(1.6524)	(1.1271)	(1.6580)		(0.9965)	(1.2520)	(1.1795)	(1.2343)
ρ_DGS10	0.2931	0.5470	0.1870	0.5888		0.1731	0.2169	0.1488	0.2408
ρ_DGS10	(0.2276)	(0.3658)	(0.1803)	(0.4105)		(0.1608)	(0.2145)	(0.1716)	(0.2205)
ρ_T10Y3M	−0.0302	−0.1462	−0.0299	−0.1609		0.0062	0.0553	0.0061	0.0557
ρ_T10Y3M	(0.1157)	(0.2127)	(0.1113)	(0.2328)		(0.0902)	(0.1502)	(0.0963)	(0.1537)
ρ_T10YIE	−0.2540	−0.5656	−0.1107	−0.6168		−0.1938	−0.3701	−0.1689	−0.4077
ρ_T10YIE	(0.2085)	(0.3783)	(0.1566)	(0.5065)		(0.1447)	(0.2323)	(0.1804)	(0.2588)
LogLik	−5755.5913	−5458.6567	−5706.0112	−5451.3256		−5744.0306	−5468.1402	−5696.5973	−5452.6736
AIC	11525.1826	10933.3134	11430.0224	10922.6512		11502.0612	10952.2803	11411.1947	10925.3473
BIC	11564.5618	10978.3182	11480.6528	10978.9071		11541.4403	10997.2851	11461.8250	10981.6032
	(c) BTC-VIX					(d) BTC-WTI
ν	–	4.7458 ***	–	4.7815 ***		–	3.9342 ***	–	4.1129 ***
ν	–	(0.2643)	–	(0.2466)		–	(0.1576)	–	(0.1779)
α₁	–	–	−1.2741 ***	−0.1801 *		–	–	−0.1136	−0.0334
α₁	–	–	(0.0900)	(0.0969)		–	–	(0.1121)	(0.0794)
α₂	–	–	−0.1740	−0.7235 ***		–	–	2.5194 ***	1.1737 ***
α₂	–	–	(0.1756)	(0.1289)		–	–	(0.1743)	(0.1463)
ω	0.0411	0.0323	0.1376	0.0921 **		−0.0388	−0.0512	−0.0867	−0.0837
ω	(0.0307)	(0.0316)	(0.0884)	(0.0444)		(0.0853)	(0.0638)	(0.1412)	(0.0793)
a	0.0075	0.0069	0.0116	0.0104		0.0119 **	0.0254 **	0.0218 *	0.0361 ***
a	(0.0067)	(0.0117)	(0.0112)	(0.0145)		(0.0047)	(0.0099)	(0.0115)	(0.0135)
b	0.9643 ***	0.9548 ***	0.9653 ***	0.9546 ***		0.9935 ***	0.9834 ***	0.9945 ***	0.9826 ***
b	(0.0215)	(0.0445)	(0.0229)	(0.0369)		(0.0050)	(0.0112)	(0.0052)	(0.0119)
ρ_GOLD	1.4942	1.6100	1.6265	1.7492		−2.6100 **	−2.9660 **	−3.5025 **	−3.4686 **
ρ_GOLD	(1.1158)	(1.3137)	(1.2972)	(1.5691)		(1.0294)	(1.3306)	(1.4633)	(1.6348)
ρ_DGS10	0.3026	0.3993	0.3424	0.4383		−0.3670 **	−0.5621 *	−0.4570 *	−0.6716 *
ρ_DGS10	(0.1936)	(0.3274)	(0.2519)	(0.3370)		(0.1759)	(0.3038)	(0.2386)	(0.3765)
ρ_T10Y3M	−0.1114	−0.1377	−0.1566	−0.1611		0.0126	0.1523	0.0012	0.1846
ρ_T10Y3M	(0.1218)	(0.2022)	(0.1613)	(0.1991)		(0.0819)	(0.1732)	(0.1171)	(0.2157)
ρ_T10YIE	−0.1126	−0.2001	−0.0624	−0.2299		0.3486 **	0.4328	0.4639 **	0.5502
ρ_T10YIE	(0.2196)	(0.3058)	(0.2828)	(0.4041)		(0.1598)	(0.3273)	(0.2345)	(0.4185)
LogLik	−5815.7229	−5533.6659	−5774.8598	−5514.1492		−5755.9246	−5345.4034	−5633.5289	−5294.5868
AIC	11645.4459	11083.3319	11567.7196	11048.2985		11525.8492	10706.8068	11285.0578	10609.1737
BIC	11684.8250	11128.3366	11618.3500	11104.5544		11565.2283	10751.8116	11335.6882	10665.4296

	Normal	Student-t	Asymmetric normal	Asymmetric Student-t		Normal	Student-t	Asymmetric normal	Asymmetric Student-t
	(a) BTC-S&P500					(b) BTC-NASDAQ
ν	–	4.4419 ***	–	4.5641 ***	ν	–	4.5534 ***	–	4.6996 ***
ν	–	(0.2271)	–	(0.2381)		–	(0.2407)	–	(0.2589)
α₁	–	–	−0.9220 ***	−0.0545		–	–	−0.5478 *	−0.0161
α₁	–	–	(0.2183)	(0.1103)		–	–	(0.2872)	(0.1059)
α₂	–	–	−1.0267 ***	−0.4129 ***		–	–	−1.5252 ***	−0.6353 ***
α₂	–	–	(0.3246)	(0.1206)		–	–	(0.3414)	(0.1224)
ω	0.0311	0.0452	0.2863 ***	0.0648		0.0292	0.0052	0.2443 *	0.0276
ω	(0.0713)	(0.0627)	(0.1018)	(0.0671)		(0.0907)	(0.1205)	(0.1364)	(0.1341)
a	0.0143 **	0.0261 **	0.0145 **	0.0287 **		0.0133 ***	0.0175 **	0.0170 ***	0.0206 **
a	(0.0059)	(0.0123)	(0.0061)	(0.0135)		(0.0044)	(0.0072)	(0.0062)	(0.0092)
b	0.9921 ***	0.9810 ***	0.9955 ***	0.9796 ***		0.9956 ***	0.9956 ***	0.9968 ***	0.9951 ***
b	(0.0079)	(0.0161)	(0.0050)	(0.0179)		(0.0042)	(0.0075)	(0.0035)	(0.0071)
ρ_GOLD	2.0794	3.0496 *	1.4236	3.2294 *		1.3638	1.4850	1.2627	1.6118
ρ_GOLD	(1.3081)	(1.6524)	(1.1271)	(1.6580)		(0.9965)	(1.2520)	(1.1795)	(1.2343)
ρ_DGS10	0.2931	0.5470	0.1870	0.5888		0.1731	0.2169	0.1488	0.2408
ρ_DGS10	(0.2276)	(0.3658)	(0.1803)	(0.4105)		(0.1608)	(0.2145)	(0.1716)	(0.2205)
ρ_T10Y3M	−0.0302	−0.1462	−0.0299	−0.1609		0.0062	0.0553	0.0061	0.0557
ρ_T10Y3M	(0.1157)	(0.2127)	(0.1113)	(0.2328)		(0.0902)	(0.1502)	(0.0963)	(0.1537)
ρ_T10YIE	−0.2540	−0.5656	−0.1107	−0.6168		−0.1938	−0.3701	−0.1689	−0.4077
ρ_T10YIE	(0.2085)	(0.3783)	(0.1566)	(0.5065)		(0.1447)	(0.2323)	(0.1804)	(0.2588)
LogLik	−5755.5913	−5458.6567	−5706.0112	−5451.3256		−5744.0306	−5468.1402	−5696.5973	−5452.6736
AIC	11525.1826	10933.3134	11430.0224	10922.6512		11502.0612	10952.2803	11411.1947	10925.3473
BIC	11564.5618	10978.3182	11480.6528	10978.9071		11541.4403	10997.2851	11461.8250	10981.6032
	(c) BTC-VIX					(d) BTC-WTI
ν	–	4.7458 ***	–	4.7815 ***		–	3.9342 ***	–	4.1129 ***
ν	–	(0.2643)	–	(0.2466)		–	(0.1576)	–	(0.1779)
α₁	–	–	−1.2741 ***	−0.1801 *		–	–	−0.1136	−0.0334
α₁	–	–	(0.0900)	(0.0969)		–	–	(0.1121)	(0.0794)
α₂	–	–	−0.1740	−0.7235 ***		–	–	2.5194 ***	1.1737 ***
α₂	–	–	(0.1756)	(0.1289)		–	–	(0.1743)	(0.1463)
ω	0.0411	0.0323	0.1376	0.0921 **		−0.0388	−0.0512	−0.0867	−0.0837
ω	(0.0307)	(0.0316)	(0.0884)	(0.0444)		(0.0853)	(0.0638)	(0.1412)	(0.0793)
a	0.0075	0.0069	0.0116	0.0104		0.0119 **	0.0254 **	0.0218 *	0.0361 ***
a	(0.0067)	(0.0117)	(0.0112)	(0.0145)		(0.0047)	(0.0099)	(0.0115)	(0.0135)
b	0.9643 ***	0.9548 ***	0.9653 ***	0.9546 ***		0.9935 ***	0.9834 ***	0.9945 ***	0.9826 ***
b	(0.0215)	(0.0445)	(0.0229)	(0.0369)		(0.0050)	(0.0112)	(0.0052)	(0.0119)
ρ_GOLD	1.4942	1.6100	1.6265	1.7492		−2.6100 **	−2.9660 **	−3.5025 **	−3.4686 **
ρ_GOLD	(1.1158)	(1.3137)	(1.2972)	(1.5691)		(1.0294)	(1.3306)	(1.4633)	(1.6348)
ρ_DGS10	0.3026	0.3993	0.3424	0.4383		−0.3670 **	−0.5621 *	−0.4570 *	−0.6716 *
ρ_DGS10	(0.1936)	(0.3274)	(0.2519)	(0.3370)		(0.1759)	(0.3038)	(0.2386)	(0.3765)
ρ_T10Y3M	−0.1114	−0.1377	−0.1566	−0.1611		0.0126	0.1523	0.0012	0.1846
ρ_T10Y3M	(0.1218)	(0.2022)	(0.1613)	(0.1991)		(0.0819)	(0.1732)	(0.1171)	(0.2157)
ρ_T10YIE	−0.1126	−0.2001	−0.0624	−0.2299		0.3486 **	0.4328	0.4639 **	0.5502
ρ_T10YIE	(0.2196)	(0.3058)	(0.2828)	(0.4041)		(0.1598)	(0.3273)	(0.2345)	(0.4185)
LogLik	−5815.7229	−5533.6659	−5774.8598	−5514.1492		−5755.9246	−5345.4034	−5633.5289	−5294.5868
AIC	11645.4459	11083.3319	11567.7196	11048.2985		11525.8492	10706.8068	11285.0578	10609.1737
BIC	11684.8250	11128.3366	11618.3500	11104.5544		11565.2283	10751.8116	11335.6882	10665.4296

Note(s): The significance of each explanatory variable (GOLD, DGS10, T10Y3M and T10YIE) coefficient is indicated with stars. * 10%, ** 5%, and *** 1% levels

Source(s): Authors’ own work

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