An Econometric Analysis of Volatility Discovery

Figures & data

Table 1 Cofractional rank test and estimates of the cointegrating vector.

	Nasdaq-Arca			Nasdaq-Nyse			Arca-Nyse			Nasdaq-Arca-Nyse
	LR	d	β2	LR	d	β2	LR	d	β2	LR	d	${\beta }_{3,1}$	${\beta }_{3,2}$
AA	0.38	0.52***	−1.00	0.36	0.52***	−1.00	0.42	0.53***	−1.00	0.35	0.52***	−1.00	−1.00
AAPL	0.99	0.52***	−1.00	–	–	–	–	–	–	–	–	–	–
BAC	0.70	0.60***	−1.00	0.59	0.60***	−1.00	0.71	0.59***	−1.00	0.57	0.59***	−1.00	−1.00
BRKB	0.04	0.52***	−1.01	–	–	–	–	–	–	–	–	–	–
C	–	–	–	–	–	–	0.94	0.59**	−1.00	–	–	–	–
CSCO	0.72	0.54***	−1.00	–	–	–	–	–	–	–	–	–	–
F	0.83	0.56***	−1.00	0.80	0.56***	−1.01	0.80	0.56***	−1.01	0.78	0.56***	−1.01	−1.01
GE	0.51	0.55***	−1.00	0.26	0.54***	−1.00	0.22	0.53***	−1.00	0.15	0.54***	−1.00	−1.00
GM	0.71	0.54***	−1.00	0.73	0.54***	−1.01	0.60	0.54***	−1.01	0.61	0.54***	−1.01	−1.01
GOOG	0.71	0.51***	−1.00	–	–	–	–	–	–	–	–	–	–
HPQ	0.88	0.51***	−1.00	0.89	0.51***	−1.00	0.96	0.51***	−1.00	0.92	0.50***	−1.00	−1.00
IBM	0.99	0.62***	−1.00	0.95	0.60***	−1.00	0.95	0.64***	−1.01	0.46	0.53***	−1.00	−1.00
JCP	0.02	0.49***	−1.01	0.01	0.49***	−1.00	0.01	0.47***	−0.99	0.00	0.48***	−1.00	−0.99
JNJ	0.70	0.53***	−1.00	0.57	0.52***	−1.01	0.46	0.52***	−1.01	0.48	0.52***	−1.01	−1.01
JPM	0.76	0.61***	−1.00	0.74	0.60***	−1.00	0.81	0.59***	−1.00	0.67	0.60***	−1.00	−1.00
KO	0.85	0.67***	−0.99	0.87	0.51***	−1.01	0.85	0.51***	−1.01	0.86	0.50***	−1.01	−1.02
MO	0.16	0.49***	−1.00	0.14	0.49***	−1.02	0.04	0.48***	−1.02	0.07	0.49***	−1.02	−1.02
MRK	0.72	0.51***	−1.00	0.57	0.51***	−1.02	0.56	0.51***	−1.01	0.66	0.50***	−1.02	−1.01
MRVL	0.00	0.44***	−1.00	–	–	–	–	–	–	–	–	–	–
MS	0.86	0.61***	−1.00	–	–	–	–	–	–	–	–	–	–
MSFT	0.92	0.51***	−1.00	–	–	–	–	–	–	–	–	–	–
NOK	0.64	0.62***	−1.00	0.34	0.51***	−1.00	0.69	0.51***	−1.00	0.99	0.71***	−1.00	−1.00
ORCL	0.58	0.51***	−1.00	–	–	–	–	–	–	–	–	–	–
PFE	0.53	0.52***	−1.00	0.48	0.51***	−1.00	0.49	0.51***	−1.00	0.29	0.51***	−1.00	−1.00
PG	0.57	0.51***	−1.00	0.62	0.52***	−1.00	0.47	0.51***	−1.00	0.46	0.51***	−1.00	−1.01
VZ	0.44	0.51***	−1.00	0.39	0.52***	−1.01	0.24	0.51***	−1.00	0.31	0.51***	−1.01	−1.00
WFC	–	–	–	–	–	–	0.58	0.59***	−1.00	–	–	–	–
WMT	0.04	0.48***	−1.00	0.06	0.49***	−1.00	0.01	0.48**	−1.00	0.05	0.49**	−1.00	−1.00
XOM	0.45	0.73***	−1.00	0.44	0.74***	−1.00	0.89	0.71***	−1.00	0.97	0.57***	−1.00	−1.00
YHOO	0.15	0.47***	−1.00	–	–	–	–	–	–	–	–	–	–

NOTE: We report results for 30 assets considering the Nasdaq and Arca, Nasdaq and Nyse, Arca and Nyse, and Nasdaq, Arca and Nyse market combinations. For each two-markets set of results, the first column ($\text{rk}(\alpha {\beta }^{\prime })$) displays the number of cointegrating relations used in the estimation of the FCVAR model. Considering the two-market setting, the first column has the p-value of the likelihood-ratio (LR) cofractional rank test when the null hypothesis is $\text{rk}(\alpha {\beta }^{\prime })\le 1$. The second column displays the estimates of d and the last column presents the second element of the cointegrating vector, such that $\beta =(1,{\beta }_2)\prime$. As for the three-markets set of results, the first column presents the p-value of the likelihood-ratio (LR) cofractional rank test when the null hypothesis is $\text{rk}(\alpha {\beta }^{\prime })\le 2$. The second column displays the estimates of d. The third and fourth columns are the free-varying estimates of the normalized cointegrating vector, such that $\beta =(I_2,-{\beta }_3)\prime$ with ${\beta }_3=({\beta }_{3,1},{\beta }_{3,2})\prime$. *, $\ast \ast$, and $\ast \ast \ast$ denote significance at 1%, 5%, and 10% significance levels.“-” implies that the stock is not traded in at least one of the two trading venues during the entire sample.

Table 2 Volatility discovery measures.

	Nasdaq-Arca		Nasdaq-Nyse		Arca-Nyse		Nasdaq-Arca-Nyse
	${\alpha }_{\perp ,T}$	${\alpha }_{\perp ,P}$	${\alpha }_{\perp ,T}$	${\alpha }_{\perp ,N}$	${\alpha }_{\perp ,P}$	${\alpha }_{\perp ,N}$	${\alpha }_{\perp ,T}$	${\alpha }_{\perp ,P}$	${\alpha }_{\perp ,N}$	${\mathcal{H}}_{T,P}$	${\mathcal{H}}_{T,N}$	${\mathcal{H}}_{P,N}$
AA	0.57*	0.43	0.45*	0.55**	0.52**	0.48**	0.25	0.29	0.47*	[0.26]	[0.12]	[0.18]
AAPL	0.85*	0.15	–	–	–	–	–	–	–	–	–	–
BAC	0.74*	0.26	0.88***	0.12	0.73**	0.27	0.72*	0.24	0.04	[0.05]	[0.29]	[0.85]
BRKB	0.34	0.66***					–	–	–	–	–	–
C	–	–	–	–	1.20***	−0.20	–	–	–	–	–	–
CSCO	0.68***	0.32	–	–	–	–	–	–	–	–	–	–
F	−0.10	1.10**	1.08***	−0.08	1.43***	−0.43	0.09	1.34**	−0.43	[0.01]	[0.64]	[0.14]
GE	0.00	1.00**	0.87**	0.13	1.27***	−0.27	0.07	1.29**	−0.36	[0.03]	[0.78]	[0.15]
GM	−0.11	1.11***	0.64***	0.36*	0.91***	0.09	−0.12	1.08**	0.05	[0.02]	[0.96]	[0.04]
GOOG	0.26	0.74**	–	–	–	–	–	–	–	–	–	–
HPQ	−0.09	1.09***	0.06	0.94***	0.36	0.64**	−0.31	0.51	0.80**	[0.56]	[0.09]	[0.00]
IBM	0.43	0.57*	0.45	0.55	0.76*	0.24	0.20	0.28	0.52*	[0.33]	[0.05]	[0.04]
JCP	0.41*	0.59**	0.14	0.86***	0.24	0.76**	0.05	0.21	0.74**	[0.78]	[0.06]	[0.01]
JNJ	0.45**	0.55**	0.29	0.71***	0.24	0.76***	0.20	0.15	0.65***	[0.43]	[0.01]	[0.00]
JPM	0.66**	0.34	0.83***	0.17	0.65**	0.35	0.66**	0.32	0.03	[0.02]	[0.21]	[0.66]
KO	0.18	0.82*	0.26	0.74***	0.20	0.80***	0.10	0.20	0.70***	[0.46]	[0.00]	[0.00]
MO	0.44**	0.56**	0.27	0.73***	0.24	0.76***	0.13	0.20	0.67**	[0.52]	[0.01]	[0.01]
MRK	0.44**	0.56**	0.02	0.98***	0.04	0.96***	−0.04	0.08	0.96***	[0.96]	[0.00]	[0.00]
MRVL	0.22	0.78**	–	–	–	–	–	–	–	–	–	–
MS	0.63*	0.37	–	–	–	–	–	–	–	–	–	–
MSFT	1.15***	−0.15	–	–	–	–	–	–	–	–	–	–
NOK	−0.31	1.31*	0.57**	0.43*	0.55	0.45	−0.39	1.08	0.31	[0.71]	[0.87]	[0.32]
ORCL	0.93***	0.07	–	–	–	–	–	–	–	–	–	–
PFE	0.13	0.87**	0.31	0.69***	0.35	0.65**	−0.06	0.51	0.56**	[0.35]	[0.22]	[0.02]
PG	0.69***	0.31	0.37**	0.63***	0.25	0.75***	0.37	0.00	0.63***	[0.20]	[0.00]	[0.01]
VZ	0.40**	0.60***	0.30*	0.70***	0.24	0.76***	0.17	0.24	0.59**	[0.34]	[0.02]	[0.00]
WFC	–	–	–	–	0.47**	0.53**	–	–	–	–	–	–
WMT	0.29	0.71**	0.49**	0.51**	0.63**	0.37*	0.14	0.57*	0.28	[0.08]	[0.50]	[0.06]
XOM	−1.64	2.64***	0.03	0.97*	1.77**	−0.77	0.00	−0.25	1.25***	[0.80]	[0.00]	[0.00]
YHOO	−0.01	1.01***	–	–	–	–	–	–	–	–	–	–

NOTE: We report the volatility discovery measures for 30 assets considering the Nasdaq and Arca, Nasdaq and Nyse, Arca and Nyse, and Nasdaq, Arca and Nyse market combinations. T, P, and N denote Nasdaq, Arca, and Nyse, respectively. The FCVAR parameters are computed using the MLE estimator of Johansen and Nielsen (2012) where $\text{rk}(\alpha {\beta }^{\prime })=1$ and $\text{rk}(\alpha {\beta }^{\prime })=2$ for the two- and three-market settings, respectively, κ is chosen according to Table S.4, and d = b. For each set of results, we report the elements of the orthogonal complement of α, where the subscript identifies the market, for example, ${\alpha }_{\perp ,T}$ denotes the orthogonal complement associated with the Nasdaq market. As for the Nasdaq-Arca-Nyse set of results, the three last columns report the p-values associated with the null hypothesis that volatility discovery measures of two markets are equal to zero, for example, $H_{T,P}$ reports the p-value of the null given by ${\alpha }_{\perp ,T}={\alpha }_{\perp ,P}=0$. *, $\ast \ast$, and $\ast \ast \ast$ denote significance at 1%, 5%, and 10% significance levels. “-” implies that the stock is not traded in at least one of the two trading venues during the entire sample.

Table 3 Price discovery measures.

	Nasdaq-Arca		Nasdaq-Nyse		Arca-Nyse		Nasdaq-Arca-Nyse
	${\gamma }_{\perp ,T}$	${\gamma }_{\perp ,P}$	${\gamma }_{\perp ,T}$	${\gamma }_{\perp ,N}$	${\gamma }_{\perp ,P}$	${\gamma }_{\perp ,N}$	${\gamma }_{\perp ,T}$	${\gamma }_{\perp ,P}$	${\gamma }_{\perp ,N}$	${\mathcal{H}}_{T,P}$	${\mathcal{H}}_{T,N}$	${\mathcal{H}}_{P,N}$
AA	0.57***	0.43***	0.60***	0.40***	0.54***	0.46***	0.42***	0.27***	0.32***	[0.00]	[0.00]	[0.00]
AAPL	0.51***	0.49***	–	–	–	–	–	–	–	–	–	–
BAC	0.63***	0.37***	0.67***	0.33***	0.62***	0.38***	0.53***	0.21***	0.26***	[0.00]	[0.00]	[0.00]
BRKB	0.71***	0.29***	–	–	–	–	–	–	–	–	–	–
C	–	–	–	–	0.53***	0.47***	–	–	–	–	–	–
CSCO	0.67***	0.33***	–	–	–	–	–	–	–	–	–	–
F	0.55***	0.45***	0.69***	0.31***	0.65***	0.35***	0.41***	0.33***	0.26***	[0.00]	[0.00]	[0.00]
GE	0.70***	0.30***	0.71***	0.29***	0.60***	0.40***	0.58***	0.17***	0.24***	[0.00]	[0.00]	[0.00]
GM	0.50***	0.50***	0.53***	0.47***	0.53***	0.47***	0.32***	0.27***	0.40***	[0.00]	[0.00]	[0.00]
GOOG	0.68***	0.32***	–	–	–	–	–	–	–	–	–	–
HPQ	0.53***	0.47***	0.60***	0.40***	0.59***	0.41***	0.39***	0.33***	0.28***	[0.00]	[0.00]	[0.00]
IBM	0.37**	0.63***	0.37**	0.63**	0.49***	0.51***	0.22***	0.37***	0.41***	[0.00]	[0.00]	[0.00]
JCP	0.45***	0.55***	0.44***	0.56***	0.48***	0.52***	0.27***	0.34***	0.39***	[0.00]	[0.00]	[0.00]
JNJ	0.69***	0.31***	0.58***	0.42***	0.40***	0.60***	0.52***	0.13***	0.35***	[0.00]	[0.00]	[0.00]
JPM	0.59***	0.41***	0.75***	0.25***	0.73***	0.27***	0.52***	0.31***	0.17***	[0.00]	[0.00]	[0.00]
KO	0.52***	0.48***	0.59***	0.41***	0.57***	0.43***	0.37***	0.34***	0.29***	[0.00]	[0.00]	[0.00]
MO	0.50***	0.50***	0.53***	0.47***	0.53***	0.47***	0.34***	0.28***	0.38***	[0.00]	[0.00]	[0.00]
MRK	0.56***	0.44***	0.63***	0.37***	0.59***	0.41***	0.48***	0.32***	0.20***	[0.00]	[0.00]	[0.00]
MRVL	0.60***	0.40***	–	–	–	–	–	–	–	–	–	–
MS	0.51***	0.49***	–	–	–	–	–	–	–	–	–	–
MSFT	0.56***	0.44***	–	–	–	–	–	–	–	–	–	–
NOK	0.42***	0.58***	0.41***	0.59***	0.39***	0.61***	0.27***	0.23***	0.51***	[0.00]	[0.00]	[0.00]
ORCL	0.57***	0.43***	–	–	–	–	–	–	–	–	–	–
PFE	0.51***	0.49***	0.42***	0.58***	0.38***	0.62***	0.27***	0.20***	0.53***	[0.00]	[0.00]	[0.00]
PG	0.22	0.78**	0.47***	0.53**	0.46***	0.54***	0.15***	0.38***	0.47***	[0.00]	[0.00]	[0.00]
VZ	0.70***	0.30***	0.63***	0.37***	0.40***	0.60***	0.54***	0.11***	0.35***	[0.00]	[0.00]	[0.00]
WFC	–	–	–	–	0.44***	0.56***	–	–	–	–	–	–
WMT	0.64***	0.36***	0.64***	0.36***	0.52***	0.48***	0.48***	0.24***	0.28***	[0.00]	[0.00]	[0.00]
XOM	0.45*	0.55**	0.67**	0.33**	0.73***	0.27***	0.42***	0.40***	0.18***	[0.00]	[0.00]	[0.00]
YHOO	0.60***	0.40***	–	–	–	–	–	–	–	–	–	–

NOTE: We report price discovery results for 30 assets considering Nasdaq and Arca, Nasdaq and Nyse, Arca and Nyse, and Nasdaq, Arca, and Nyse market combinations. The VEC parameters are computed using the OLS estimator where $\text{rk}(\gamma {\delta }^{\prime })=S-1,\delta ={(I_{S-1},-{\iota }_{S-1})}^{\prime }$ with S denoting the number of markets, and the lag length is chosen as the minimum value which makes the LM test for serial correlation on the residuals at lag 15 to be insignificant at the 5% significance level. For the first three sets of results, the first and second column show the estimate of the CS measure associated with the first and second market, respectively. As for the Nasdaq-Arca-Nyse set of results, the first three columns display the price discovery measures for Nasdaq, Arca, and Nyse, respectively. The last three columns report the p-values associated with the null hypothesis that volatility discovery measures of two markets are equal to zero, for example, $H_{T,P}$ reports the p-value of the null given by ${\gamma }_{\perp ,T}={\gamma }_{\perp ,P}=0$. *, $\ast \ast$, and $\ast \ast \ast$ denote significance at 1%, 5%, and 10% significance levels. “-” implies that the stock is not traded in at least one of the two trading venues during the entire sample.

Johansen, S., and Nielsen, M. Ø. (2012), “Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model,” Econometrica, 80, 2667–2732.

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