Structural change in the correlation, return and volatility spillovers: evidence from the oil, stock and exchange rate markets in the United States

Figures & data

Figure 1. The trend of return and likelihood ration (a) Wti (b) GasNyh (c) Heat (d) Dj (e) Nasdaq (f) Sp500.

Source: Author.

Table 1. Descriptive statistics of daily return for the overall, pre-SB and post-SB periods.

	Mean	SD	Max.	Min.	SK	KUR	J-B	$Q^2(24)$	Obs.
Panel A. The overall period
Wti	0.0136	2.3569	16.413	−12.826	−0.0533	4.8501^c	2880.09^c	3156.1^c	2937
GasNyh	0.0218	2.6364	23.505	−17.889	0.1445^c	6.5263^c	5222.61^c	1374.7^c	2937
Heat	0.0183	2.1441	12.212	−11.421	−0.0224	2.3382^c	669.350^c	871.5^c	2937
Dj	0.0200	1.1262	10.325	−8.200	−0.1947^c	11.194^c	15355.0^c	6232.1^c	2937
Nasdaq	0.0331	1.3247	9.106	−9.587	−0.4168^c	6.5337^c	5309.27^c	5497.8^c	2937
Sp500	0.0237	1.2257	10.423	−9.469	−0.4269^c	11.719^c	16896.7^c	6204.3^c	2937
UDI	0.0015	0.4875	2.155	−4.107	−0.2865^c	3.7235^c	1736.84^c	809.6^c	2937
Panel B. The pre-SB period
Wti	0.1302	2.0667	8.408	−12.390	−0.2366^c	1.487^c	115.5^c	56.0^c	1138
GasNyh	0.1167	2.9131	23.505	−17.889	0.3348^c	7.306^c	2583.7^c	587.5^c	1152
Heat	0.1373	2.3060	11.022	−9.042	0.1959^c	0.601^c	25.0^c	136.2^c	1167
Dj	0.0086	1.2657	10.325	−8.200	−0.1094^a	10.738^c	8445.3^c	3857.8^c	1757
Nasdaq	0.0101	1.4869	9.106	−9.587	−0.3693^c	6.075^c	2636.4^c	3388.2^c	1689
Sp500	0.0000	1.4176	10.423	−9.469	−0.3670^c	11.407^c	8182.5^c	3378.0^c	1503
UDI	–	–	–	–	–	–	–	–	–
Panel C. The post-SB period
Wti	-0.060	2.5211	16.413	−12.826	0.040	5.5312^c	2293.7^c	2337.4^c	1799
GasNyh	-0.039	2.4402	17.367	−15.690	−0.099^a	4.8234^c	1733.3^c	1189.7^c	1785
Heat	-0.060	2.0271	12.212	−11.421	−0.268^c	4.0149^c	1210.0^c	729.6^c	1770
Dj	0.0368	0.8784	4.153	−5.706	−0.472^c	4.4146^c	1002.1^c	980.9^c	1180
Nasdaq	0.0643	1.0663	5.159	−7.148	−0.444^c	4.0049^c	875.1^c	893.8^c	1248
Sp500	0.0485	0.9850	4.631	−6.895	−0.458^c	4.5617^c	1293.6^c	1025.2^c	1434
UDI	–	–	–	–	–	–	–	–	–

Notes: 1. The superscripts a, b and c at a specific descriptive statistics denote the descriptive statistics is significant at the 10%, 5% and 1% levels, respectively. 2. SK and KUR denote the skewness and excess kurtosis, respectively. 3. J-B statistics are based on Jarque and Bera (1987) and are asymptotically chi-squared-distributed with 2 degrees of freedom. 4. $Q^2(24)$ statistics are asymptotically chi-squared-distributed with 24 degrees of freedom. 5. Obs. denotes the number of observation. 6. The dates of structural breaks for the Wti, GasNyh, Heat, Sp500, Nasdaq, and Dj are 2008/5/21, 2008/6/11, 2008/7/2, 2009/11/4, 2010/8/4, and 2010/11/10, respectively. 7. The overall period is from Oct. 22, 2003 to Aug. 5, 2015. 8. The bold (resp. italic) font in column Mean or SD denotes the larger (resp. smaller) value of mean or standard deviation, when the values of Mean or SD for the overall period are compared with those for the pre-SB or post-SB period.

Source: Author.

Table 2. Empirical results of B-GJR-GARCH-SB model for the Oil-stock paired market data.

	Wti- Dj	Wti- Nasdaq	Wti- Sp500	GasNyh- Dj	GasNyh- Nasdaq	GasNyh- Sp500	Heat- Dj	Heat- Nasdaq	Heat- Sp500
Panel A. Return spillover
${\varphi }_{12}^B$	0.1134 (0.061)^a	0.1182 (0.048)^b	0.1486 (0.058)^b	0.1716 (0.079)^b	0.1777 (0.062)^c	0.1872 (0.074)^b	0.1031 (0.063)	0.1357 (0.050)^c	0.1193 (0.060)^b
${\varphi }_{12}^A$	−0.0314 (0.052)	−0.0137 (0.040)	0.0021 (0.041)	−0.0046 (0.057)	0.0150 (0.044)	−0.0101 (0.045)	−0.0012 (0.041)	0.0028 (0.032)	0.0112 (0.034)
${\varphi }_{21}^B$	−0.0064 (0.009)	−0.0116 (0.012)	−0.0044 (0.009)	−0.0050 (0.006)	−0.0037 (0.008)	−0.0010 (0.006)	−0.0092 (0.008)	−0.0186 (0.010)^a	−0.0062 (0.008)
${\varphi }_{21}^A$	0.0128 (0.010)	0.0173 (0.013)	0.0126 (0.010)	0.0127 (0.010)	0.0209 (0.012)^a	0.0163 (0.009)^a	0.0009 (0.012)	0.0026 (0.015)	0.0019 (0.011)
Panel B. Risk premium
${\tau }_1^B$	0.1343 (0.048)^c	0.1382 (0.040)^c	0.1465 (0.053)^c	−0.0099 (0.053)	0.0082 (0.055)	−0.0254 (0.072)	0.0871 (0.048)^a	0.0928 (0.043)^b	0.0913 (0.046)^b
${\tau }_1^A$	0.0826 (0.064)	0.0764 (0.048)	0.0979 (0.065)	−0.1300 (0.078)^a	−0.1122 (0.081)	−0.1347 (0.109)	0.0619 (0.082)	0.0620 (0.068)	0.0881 (0.067)
${\tau }_2^B$	0.0410 (0.022)^a	0.0852 (0.057)	0.0927 (0.026)^c	0.0300 (0.025)	0.0695 (0.055)	0.0832 (0.028)^c	0.0274 (0.019)	0.0633 (0.054)	0.0723 (0.022)^c
${\tau }_2^A$	0.0127 (0.018)	0.0693 (0.039)^a	0.0558 (0.023)^b	0.0304 (0.025)	0.0698 (0.046)	0.0569 (0.029)^a	0.0308 (0.020)	0.0674 (0.045)	0.0629 (0.025)^b
Panel C. Volatility spillover
${\theta }_{12}^B$	0.0649 (0.022)^c	0.0385 (0.015)^b	0.0488 (0.019)^b	0.1225 (0.056)^b	0.0861 (0.039)^b	0.1039 (0.057)^a	0.0200 (0.015)	0.0161 (0.008)^a	0.0155 (0.012)
${\theta }_{12}^A$	−0.0047 (0.012)	−0.0060 (0.008)	−0.0011 (0.009)	−0.0095 (0.015)	−0.0064 (0.011)	0.0011 (0.012)	−0.0046 (0.004)	−0.0035 (0.003)	−0.0017 (0.003)
${\theta }_{21}^B$	0.0016 (0.000)^b	0.0013 (0.001)	0.0002 (0.000)	0.0002 (0.000)	−0.0001 (0.000)	−0.0000 (0.000)	0.0001 (0.000)	0.0000 (0.000)	−0.0003 (0.000)
${\theta }_{21}^A$	0.0014 (0.000)	0.0003 (0.001)	0.0003 (0.000)	0.0004 (0.000)	−0.0012 (0.001)	0.0000 (0.000)	0.0006 (0.001)	0.0000 (0.001)	0.0003 (0.001)
Panel D. Correlation
${\omega }_{12}^B$	−0.0052 (0.003)^a	−0.0046 (0.003)	−0.0013 (0.002)	−0.0058 (0.003)	−0.0053 (0.004)	−0.0013 (0.003)	−0.0056 (0.003)^a	−0.0048 (0.003)	−0.0015 (0.003)
${\omega }_{12}^A$	0.0084 (0.002)^c	0.0113 (0.003)^c	0.0110 (0.003)^c	0.0093 (0.003)^b	0.0089 (0.003)^c	0.0098 (0.003)^c	0.0057 (0.002)^c	0.0068 (0.002)^b	0.0075 (0.002)^c
Panel E. Leverage effect
${\mathrm{\zeta }}_1^B$	0.0474 (0.009)^c	0.0411 (0.010)^c	0.0511 (0.010)^c	0.0097 (0.014)	−0.0003 (0.014)	0.0160 (0.015)	0.0210 (0.010)^b	0.0183 (0.009)^a	0.0213 (0.009)^b
${\mathrm{\zeta }}_1^A$	0.0436 (0.008)^c	0.0430 (0.009)^c	0.0421 (0.008)^c	−0.0016 (0.013)	−0.0079 (0.012)	−0.0158 (0.011)	0.0036 (0.008)	0.0072 (0.007)	0.0010 (0.007)
${\mathrm{\zeta }}_2^B$	0.1758 (0.017)^c	0.1386 (0.018)^c	0.1535 (0.014)^c	0.1841 (0.018)^c	0.1437 (0.020)^c	0.1714 (0.018)^c	0.1648 (0.017)^c	0.1314 (0.019)^c	0.1488 (0.014)^c
${\mathrm{\zeta }}_2^A$	0.1756 (0.019)^c	0.1346 (0.018)^c	0.1554 (0.015)^c	0.1862 (0.020)^c	0.1366 (0.019)^c	0.1766 (0.019)^c	0.1606 (0.019)^c	0.1186 (0.018)^c	0.1443 (0.015)^c
Panel F. The others information
$\mathrm{\nu }$	8.8100 (0.687)^c	9.4889 (0.778)^c	8.3085 (0.648)^c	8.9611 (0.781)^c	9.2629 (0.847)^c	8.4186 (0.674)^c	10.1589 (0.954)^c	10.4301 (1.016)^c	9.4691 (0.841)^c
$Q_1^2(24)$	25.555	25.988	25.512	74.986^c	72.908^c	65.101^c	78.295^c	82.578^c	79.687^c
$Q_2^2(24)$	31.170	32.881	32.638	30.759	33.022	31.068	31.535	31.715	30.497
Panel G. The settings for the pre- and post-SB periods
Pre-SB	2003/10/22- 2008/5/21	2003/10/22- 2008/5/21	2003/10/22- 2008/5/21	2003/10/22- 2008/6/11	2003/10/22- 2008/6/11	2003/10/22- 2008/6/11	2003/10/22- 2008/7/02	2003/10/22- 2008/7/02	2003/10/22- 2008/7/02
Post-SB	2010/11/10- 2015/08/05	2010/08/04- 2015/08/05	2009/11/04- 2015/08/05	2010/11/10- 2015/08/05	2010/08/04- 2015/08/05	2009/11/04- 2015/08/05	2010/11/10- 2015/08/05	2010/08/04- 2015/08/05	2009/11/04- 2015/08/05

Note: 1. The superscripts a, b and c at a specific parameter denote the parameter is significant at the 10%, 5% and 1% levels, respectively. 2. Numbers in parentheses at the row of parameters are the standard errors. 3. $Q_i^2\left(24\right)$ for $\mathrm{i}=1,2\mathrm{ }$ statistics are asymptotically chi-squared-distributed with 24 degrees of freedom. 4. The date period at the row ‘Pre-SB’ (respectively, ‘Post-SB’) denotes the starting date and ending date for the pre-SB (respectively, post-SB) period.

Source: Author.

Table 3. Empirical results of B-GJR-GARCH-SB model for the Oil-FX and Stock-FX types of paired market data.

	Wti- UDI	GasNyh- UDI	Heat- UDI	Dj- UDI	Nasdaq- UDI	Sp500- UDI
Panel A. Return spillover
${\varphi }_{12}^B$	−0.0508 (0.122)	0.1036 (0.155)	0.0486 (0.135)	−0.0555 (0.038)	−0.0570 (0.052)	−0.0518 (0.045)
${\varphi }_{12}^A$	−0.0086 (0.088)	0.0576 (0.100)	−0.1426 (0.079)^a	0.0024 (0.044)	−0.0173 (0.057)	−0.0047 (0.041)
${\varphi }_{21}^B$	−0.0207 (0.005)^c	−0.0125 (0.004)^c	−0.0165 (0.005)^c	−0.0731 (0.009)^c	−0.0627 (0.008)^c	−0.0823 (0.009)^c
${\varphi }_{21}^A$	−0.0096 (0.004)^b	−0.0056 (0.004)	−0.0073 (0.005)	−0.0249 (0.011)^b	−0.0188 (0.010)^a	−0.0233 (0.010)^b
Panel B. Risk premium
${\tau }_1^B$	0.1156 (0.054)^b	0.0532 (0.055)	0.0896 (0.047)^a	0.0291 (0.022)	0.0473 (0.039)	0.0601 (0.020)^c
${\tau }_1^A$	0.0152 (0.055)	−0.0383 (0.066)	0.0025 (0.054)	0.0222 (0.022)	0.0287 (0.041)	0.0252 (0.022)
${\tau }_2^B$	0.0157 (0.013)	0.0142 (0.011)	0.0144 (0.011)	0.0154 (0.010)	0.0119 (0.013)	0.0086 (0.011)
${\tau }_2^A$	−0.0069 (0.011)	−0.0091 (0.009)	−0.0087 (0.009)	−0.0046 (0.008)	−0.0061 (0.011)	−0.0111 (0.010)
Panel C. Volatility spillover
${\theta }_{12}^B$	0.6403 (0.223)^c	1.8544 (0.531)^c	0.5122 (0.174)^c	0.0436 (0.014)^c	0.1050 (0.050)^b	0.0391 (0.018)^b
${\theta }_{12}^A$	0.2667 (0.151)^a	0.5996 (0.263)^b	0.0959 (0.079)	0.0467 (0.017)^c	0.1285 (0.048)^c	0.0690 (0.016)^c
${\theta }_{21}^B$	0.0002 (0.000)^a	0.0000 (0.000)	0.0002 (0.000)	0.0009 (0.000)^b	0.0009 (0.000)^c	0.0008 (0.000)^c
${\theta }_{21}^A$	0.0006 (0.000)^c	0.0005 (0.000)^c	0.0010 (0.000)^c	0.0001 (0.000)	0.0000 (0.000)	0.0002 (0.000)
Panel D. Correlation
${\omega }_{12}^B$	−0.0006 (0.000)	−0.0003 (0.000)	−0.0006 (0.000)	−0.0002 (0.000)	0.0000 (0.000)	−0.0001 (0.000)
${\omega }_{12}^A$	−0.0002 (0.000)	−0.0003 (0.000)	−0.0001 (0.000)	−0.0000 (0.000)	0.0001 (0.000)	−0.0001 (0.000)
Panel E. Leverage effect
${\mathrm{\zeta }}_1^B$	0.0397 (0.013)^c	0.0014 (0.020)	0.0144 (0.012)	0.1753 (0.012)^c	0.1724 (0.023)^c	0.1717 (0.017)^c
${\mathrm{\zeta }}_1^A$	0.0573 (0.012)^c	0.0175 (0.014)	0.0248 (0.009)^c	0.1568 (0.014)^c	0.1416 (0.022)^c	0.1466 (0.016)^c
${\mathrm{\zeta }}_2^B$	0.0070 (0.009)	0.0036 (0.010)	0.0032 (0.010)	−0.0157 (0.007)^b	−0.0207 (0.007)^c	−0.0119 (0.008)
${\mathrm{\zeta }}_2^A$	−0.0079 (0.007)	−0.0180 (0.007)^b	−0.0204 (0.007)^c	−0.0162 (0.007)^b	−0.0123 (0.007)^a	−0.0092 (0.006)
Panel F. The others information
$\mathrm{\nu }$	10.1214 (0.893)^c	11.0052 (1.084)^c	11.1321 (1.117)^c	9.9458 (1.001)^c	11.3468 (1.332)^c	9.7158 (0.938)^c
$Q_1^2(24)$	27.223	50.022^c	63.008^c	26.707	29.900	24.782
$Q_2^2(24)$	39.895^b	34.632^a	36.185^a	38.410^b	39.963^b	38.722^b
Panel G. The settings for the pre- and post-SB periods
Pre-SB	2003/10/22- 2008/5/21	2003/10/22- 2008/6/11	2003/10/22- 2008/7/02	2003/10/22- 2010/11/10	2003/10/22- 2010/8/04	2003/10/22- 2009/11/04
Post-SB	2008/05/21- 2015/08/05	2008/06/11- 2015/08/05	2008/07/02- 2015/08/05	2010/11/10- 2015/08/05	2010/08/04- 2015/08/05	2009/11/04- 2015/08/05

Note: 1. Please see the notes 1–4 in Table 2.

Source: Author.

Table 4. The summary results of return spillover for 15 pairs of data.

Panel A. The Oil-stock paired market data
	Wti- Dj	Wti- Nasdaq	Wti- Sp500	GasNyh- Dj	GasNyh- Nasdaq	GasNyh- Sp500
Pre-SB	←(+)	←(+)	←(+)	←(+)	←(+)	←(+)
Post-SB	×	×	×	×	→(+)	→(+)
	Heat- Dj	Heat- Nasdaq	Heat- Sp500
Pre-SB	×	←(+) →(-)	←(+)
Post-SB	×	×	×
Panel B. The Oil-FX and Stock-FX types of paired market data
	Wti- UDI	GasNyh- UDI	Heat- UDI	Dj- UDI	Nasdaq- UDI	Sp500- UDI
Pre-SB	→(-)	→(-)	→(-)	→(-)	→(-)	→(-)
Post-SB	→(-,↓)	×	←(-)	→(-,↓)	→(-,↓)	→(-,↓)
Panel C. The summary of return spillover for three types of paired market data
	Oil-stock		Oil-FX		Stock-FX
Pre-SB	Oil$\stackrel{\leftharpoonup }{(+)}\mathrm{Stock}$ $$		$\mathrm{Oil}\stackrel{\rightharpoonup }{(-)}\mathrm{F}\mathrm{X}$		$\mathrm{Stock}\stackrel{\rightharpoonup }{(-)}\mathrm{F}\mathrm{X}$
Post-SB	×		?		$\mathrm{Stock}\stackrel{\rightharpoonup }{(¯,\downarrow )}\mathrm{F}\mathrm{X}$
Panel D. The messages of significant situations for three types of paired market data
	Total	Oil-stock		Oil-FX		Stock-FX
	(o,×)[ o%,S]	(o,×) [o%,S]		(o,×) [o%,S]		(o,×) [o%,S]
Pre-SB	(14,1)[93.3%,⊕]	(8,1)[88.9%,⊕]		(3,0) [100%,⊕]		(3,0) [100%,⊕]
Post-SB	(7,8) [46.7%,⦶]	(2,7) [22.2%,⊗]		(2,1) [66.7%,⦶]		(3,0) [100%,⊕]
difference	−46.6%	−66.7%		−33.3%		0%

Note: 1. The symbol × represents that the return spillover does not exist. 2. The symbol → denotes that the return spillover from the first asset to the second asset significantly exists because the value of parameter ${\varphi }_{21}^B$ or ${\varphi }_{21}^A$ in Tables 2 and 3 is significant. 3. The symbol ← denotes that the return spillover from the second asset to the first asset significantly exists because the value of parameter ${\varphi }_{12}^B$ or ${\varphi }_{12}^A$ in Tables 2 and 3 is significant. 4. The symbol + (respectively, -) inside the bracket beside the symbol → or ← denotes that the return spillover is significantly positive (respectively, negative). 5. The symbol ↑ (respectively, ↓) inside the bracket beside the symbol → or ← corresponding to row ‘Post-SB’ denotes that, based on having the same direction of return spillover for the pre- and post-SB periods, the return spillover exhibits greater (respectively, smaller) after the structural break. 6. The numbers in columns ‘o’ and ‘×’ underneath a specific paired market data (or ‘Total’) at panel D respectively represent the total numbers of significant and insignificant cases for this specific paired market data (or the entire US market). 7. The numbers or symbols in columns ‘o%’ and ‘S’ underneath a specific paired market data (or ‘Total’) at panel D respectively represent the significant ratio and the significant degree for this specific paired market data (or the entire US market). Moreover, the significant ratio equals to the total numbers of significant cases divided by the sum value of the total numbers of significant and insignificant cases. 8. There are three types of significant degree: ‘⊕’, ‘⦶’, and ‘⊗’. The significant degree is determined by its corresponding value of significant ratio. For example, the symbols ‘⊕’ and ‘⊗’ denote the values of significant ratio are greater than 2/3 and smaller than 1/3, respectively. On the other hand, the symbol ‘⦶’denotes the value of significant ratio is between 1/3 and 2/3. Hence, the symbols ‘⊕’ and ‘⦶’ denote the discussed financial feature significantly and slightly exist, respectively. Conversely, the symbol ‘⊗’ denotes the discussed financial feature does not exist. 9. The number at the row ‘difference’ and the column corresponding to 'a specific paired market data' (or ‘Total’) at panel D denotes the difference of two significant ratios corresponding to the pre- and post-SB periods for this specific paired market data (or the entire US market).

Source: Author.

Table 5. The summary results of volatility spillover effect for 15 pairs of data.

Panel A. The Oil-stock paired market data
	Wti- Dj	Wti- Nasdaq	Wti- Sp500	GasNyh- Dj	GasNyh- Nasdaq	GasNyh- Sp500
Pre-SB	←(+) →(+)	←(+)	←(+)	←(+)	←(+)	←(+)
Post-SB	×	×	×	×	×	×
	Heat- Dj	Heat- Nasdaq	Heat- Sp500
Pre-SB	×	←(+)	×
Post-SB	×	×	×
Panel B. The Oil-FX and Stock-FX types of paired market data
	Wti- UDI	GasNyh- UDI	Heat- UDI	Dj- UDI	Nasdaq- UDI	Sp500- UDI
Pre-SB	←(+) →(0,+)	←(+)	←(+)	←(+) →(0,+)	←(+) →(0,+)	←(+) →(0,+)
Post-SB	←(+,↓) →(0,+,↑)	←(+,↓) →(0,+)	→(+)	←(+,↑)	←(+,↑)	←(+,↑)
Panel C. The summary of volatility spillover for three types of paired market data
	Oil-stock		Oil-FX		Stock-FX
Pre-SB	Oil$\stackrel{\leftharpoonup }{(+)}\mathrm{Stock}$		Oil$\stackrel{\leftharpoonup }{(+)}\mathrm{F}\mathrm{X}$		Stock$\stackrel{\leftharpoonup }{(+)}\mathrm{F}\mathrm{X}$
Post-SB	×		$\mathrm{Oil}\stackrel{\leftharpoonup }{(+,\downarrow )}\mathrm{F}\mathrm{X}$		$\mathrm{Stock}\stackrel{\leftharpoonup }{(+,\uparrow )}\mathrm{F}\mathrm{X}$
Panel D. The messages of significant situations for three types of paired market data
	Total	Oil-stock		Oil-FX		Stock-FX
	(o,×)[ o%,S]	(o,×) [ o%,S]		(o,×) [ o%,S]		(o,×) [ o%,S]
Pre-SB	(13,2) [86.7%,⊕]	(7,2) [77.8%,⊕]		(3,0) [100%,⊕]		(3,0) [100%,⊕]
Post-SB	(6,9) [40%,⦶]	(0,9) [0%,⊗]		(3,0) [100%,⊕]		(3,0) [100%,⊕]
difference	−46.7%	−77.8%		0%		0%

Note: 1. The symbol × represents that the volatility spillover does not exist. 2. The symbol → denotes that the volatility spillover from the first asset to the second asset significantly exists because the value of parameter ${\theta }_{21}^B$ or ${\theta }_{21}^A$ in Tables 2 and 3 is significant. 3. The symbol ← denotes the volatility spillover from the second asset to the first asset significantly exists because the value of parameter ${\theta }_{12}^B\mathrm{ }{\text{or θ}}_{12}^{\mathrm{A}}$ in Tables 2 and 3 is significant. 4. The symbol + (respectively, −) inside the bracket beside the symbol → or ← denotes that the volatility spillover is significantly positive (respectively, negative). 5. The symbol ↑(respectively, ↓) inside the bracket beside the symbol → or ← corresponding to row ‘Post-SB’ denotes that, based on having the same direction of volatility spillover for the pre- and post-SB periods, the volatility spillover exhibits greater (respectively, smaller) after the structural break. 6. The number '0' inside the bracket beside the symbol → or ← denotes that the volatility spillover is very small because the value of the corresponding parameter is smaller than 0.001. 7. As to the others notes, please see the notes 6–9 in Table 4.

Source: Author.

Table 6. The summary results of correlation for 15 pairs of data.

Panel A. The Oil-stock paired market data
	Wti- Dj	Wti- Nasdaq	Wti- Sp500	GasNyh- Dj	GasNyh- Nasdaq	GasNyh- Sp500
Pre-SB	–	×(-)	×(-)	×(-)	×(-)	×(-)
Post-SB	+	+	+	+	+	+
	Heat- Dj	Heat- Nasdaq	Heat- Sp500
Pre-SB	–	×(-)	×(-)
Post-SB	+	+	+
Panel B. The Oil-FX and Stock-FX types of paired market data
	Wti- UDI	GasNyh- UDI	Heat- UDI	Dj- UDI	Nasdaq- UDI	Sp500- UDI
Pre-SB	×(0,-)	×(0,-)	×(0,-)	×(0,-)	×(0,+)	×(0,-)
Post-SB	×(0,-)	×(0,-)	×(0,-)	×(0,-)	×(0,+)	×(0,-)
Panel C. The summary of correlation for three types of paired market data
	Oil-stock		Oil-FX		Stock-FX
Pre-SB	×(-)		×(0,-)		×(0,-)
Post-SB	+		×(0,-)		×(0,-)
Panel D. The messages of significant situations for three types of paired market data
	Total	Oil-stock		Oil-FX		Stock-FX
	(o,×)[ o%,S]	(o,×) [ o%,S]		(o,×) [ o%,S]		(o,×) [ o%,S]
Pre-SB	(2,13)[13.3%,⊗]	(2,7) [22.2%,⊗]		(0,3) [0%,⊗]		(0,3) [0%,⊗]
Post-SB	(9,6) [60%,⦶]	(9,0) [100%,⊕]		(0,3) [0%,⊗]		(0,3) [0%,⊗]
difference	46.7%	77.8%		0%		0%

Note: 1. In this table, I use the conditional covariance to represent the correlation. 2. The symbol + (respectively, −) denotes that the correlation is significantly positive (respectively, negative) because the value of parameter ${\omega }_{12}^B$ or ${\omega }_{12}^A$ in Tables 2 and 3 is significantly positive (respectively, negative). 3. The symbol × represents that the correlation does not exist. 4. The symbol + (respectively, −) inside the bracket beside the symbol × denotes that the value of correlation is positive (respectively, negative) but isn’t significant. 5. The number ‘0’ inside the bracket beside the symbol × denotes that the value of correlation is smaller than 0.001. 6. As to the others notes, please see the notes 6–9 in Table 4.

Source: Author.

Table 7. The summary results of risk premium for 15 pairs of data.

Panel A. The Oil-stock paired market data
	Wti- Dj		Wti- Nasdaq		Wti- Sp500		GasNyh- Dj		GasNyh- Nasdaq		GasNyh- Sp500
	OM	SM	OM	SM	OM	SM	OM	SM	OM	SM	OM	SM
Pre-SB	o	o	o	×	o	o	×	×	×	×	×	o
Post-SB	×	×	×	o	×	o(↓)	×	×	×	×	×	o(↓)
	Heat- Dj		Heat- Nasdaq		Heat- Sp500
	OM	SM	OM	SM	OM	SM
Pre-SB	o	×	o	×	o	o
Post-SB	×	×	×	×	×	o(↓)
Panel B. The Oil-FX and Stock-FX types of paired market data
	Wti- UDI		GasNyh- UDI		Heat- UDI		Dj- UDI		Nasdaq- UDI		Sp500- UDI
	OM	FX	OM	FX	OM	FX	SM	FX	SM	FX	SM	FX
Pre-SB	o	×	×	×	o	×	×	×	×	×	o	×
Post-SB	×	×	×	×	×	×	×	×	×	×	×	×
Panel C. The messages of significant situations for three markets data
	Total			Oil			Stock			FX
	(o,×)[ o%,S]			(o,×) [ o%,S]			(o,×) [ o%,S]			(o,×) [ o%,S]
Pre-SB	(13,17)[ 43.3%,⦶]			(8,4) [66.7%,⦶]			(5,7) [41.7%,⦶]			(0,6) [0%,⊗]
Post-SB	(4,26) [13.3%,⊗]			(0,12) [0%,⊗]			(4,8) [33.3%,⦶]			(0,6) [0%,⊗]
difference	−30%			−66.7%			−8.4%			0%

Note: 1.SM (respectively, OM) denotes the stock (respectively, oil) market whereas FX denotes the exchange rate market. 2. The symbol × represents that the risk premium does not exist. 3. The symbol o represents that the risk premium significantly exists in a specified market because the value of parameter ${\tau }_1^B,$ ${\tau }_1^A,$ ${\tau }_2^B,$ or ${\tau }_2^A$ in Tables 2-3 is positive significantly. 4. The symbol ↑(respectively, ↓) inside the bracket beside the symbol o corresponding to row ‘Post-SB’ denotes that the risk premium exhibits greater (respectively, smaller) after the structural break. The above comparison is under the condition of risk premium significantly existing at the pre- and post-SB periods simultaneously. 5. The numbers in columns ‘o’ and ‘×’ underneath a specific market data (or ‘Total’) at panel C respectively represent the total numbers of significant and insignificant cases for this specific market data (or the entire US market). 6. The numbers or symbols in columns ‘o%’ and ‘S’ underneath a specific market data (or ‘Total’) at panel C respectively represent the significant ratio and the significant degree for this specific market data (or the entire US market). As to the significant degree, it is defined in the note 8 in Table 4. 7. The number at the row ‘difference’ and the column corresponding to 'a specific market data' (or ‘Total’) at panel C denotes the difference of two significant ratios corresponding to the pre- and post-SB periods for this specific market data (or the entire US market).

Source: Author.

Table 8. The summary results of leverage effect for 15 pairs of data.

Panel A. The Oil-stock paired market data
	Wti- Dj		Wti- Nasdaq		Wti- Sp500		GasNyh- Dj		GasNyh- Nasdaq		GasNyh- Sp500
	OM	SM	OM	SM	OM	SM	OM	SM	OM	SM	OM	SM
Pre-SB	o	o	o	o	o	o	×	o	×	o	×	o
Post-SB	o(↓)	o(↓)	o(↑)	o(↓)	o(↓)	o(↑)	×	o(↑)	×	o(↓)	×	o(↑)
	Heat- Dj		Heat- Nasdaq		Heat- Sp500
	OM	SM	OM	SM	OM	SM
Pre-SB	o	o	o	o	o	o
Post-SB	×	o(↓)	×	o(↓)	×	o(↓)
Panel B. The Oil-FX and Stock-FX types of paired market data
	Wti- UDI		GasNyh- UDI		Heat- UDI		Dj- UDI		Nasdaq- UDI		Sp500- UDI
	OM	FX	OM	FX	OM	FX	SM	FX	SM	FX	SM	FX
Pre-SB	o	×	×	×	×	×	o	×	o	×	o	×
Post-SB	o(↑)	×	×	×	o	×	o(↓)	×	o(↓)	×	o(↓)	×
Panel C. The messages of significant situations for three markets data
	Total			Oil			Stock			FX
	(o,×)[ o%,S]			(o,×) [ o%,S]			(o,×) [ o%,S]			(o,×) [ o%,S]
Pre-SB	(19,11)[ 63.3%,⦶]			(7,5)[ 58.3%,⦶]			(12,0)[ 100%,⊕]			(0,6)[ 0%,⊗]
Post-SB	(17{↓11; ↑5},13)[56.7%,⦶]			(5{↓2; ↑2},7)[ 41.7%,⦶]			(12{↓9; ↑3},0)[ 100%,⊕]			(0,6)[ 0%,⊗]
difference	−6.6%			−16.6%			0%			0%

Note: 1. SM (respectively, OM) denotes the stock (respectively, oil) market whereas FX denotes the exchange rate market. 2. The symbol × represents that the leverage effect does not exist. 3. The symbol o represents that the leverage effect significantly exists in a specified market because the value of parameter ${\zeta }_1^B,$ ${\zeta }_1^A,$ ${\zeta }_2^B,$ or ${\zeta }_2^A$ in Tables 2 and 3 is positive significantly. 4. The symbol ↑(respectively, ↓) inside the bracket beside the symbol o corresponding to row ‘Post-SB’ denotes that the leverage effect exhibits greater (respectively, smaller) after the structural break. The above comparison is under the condition of leverage effect significantly existing at the pre- and post-SB periods simultaneously. 5. As to the others notes, please see the notes 5–7 in Table 7.

Source: Author.

Jarque, C. M., & Bera, A. K. (1987). A test for normality of observations and regression residuals. International Statistical Review / Revue Internationale de Statistique, 55(2), 163–172. https://doi.org/10.2307/1403192

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