would expect price movements to be similar in the two markets and this might result in similar or identical orders of integration in the two processes but without
fractional cointegration. A second possibility is that changes in volatility in one market causes changes in volatility in the other. The causal links may be
unidirectional or bidirectional. It is often supposed that NYMEX is the more important oil futures market, both because it has historical priority, and because
trading NYMEX volumes are typically higher than IPE volumes. That view might suggest that NYMEX volatility drives IPE volatility but not vice versa.
These hypotheses are not mutually inconsistent. If the two volatility processes are highly correlated, perhaps because of a common information arrival process,
differences in volatility across the two markets will motivate straddle trading in which similar options are sold on the higher volatility market and bought on the
lower volatility market. Strategies of this sort will tend to drive the two volatilities together, but will be risky if the two processes are not cointegrated. The question
of whether or not the volatility processes are cointegrated is therefore of some importance in evaluation of the risk profiles of institutions that engage in volatility
arbitrage on the two markets.
2. The GARCH and FIGARCH classes of volatility models
2.1. UniÕariate models In order to capture the long-memory component in the mean, Granger and
Ž .
Ž .
Joyeux 1980 and Hosking 1981 separately formulated the Fractionally Inte- grated ARMA process or ARFIMA
d
w L 1 y L
y y m s q L ´ 1
Ž . Ž . Ž
. Ž .
Ž .
t t
Ž .
s j
Ž .
k j
where w L s 1 y Ý w L , q L s 1 q Ý
q L , m is the mean of y and ´
js 1 j
js1 j
t t
Ž .
is white noise. In the same way, Baillie et al. 1996 , introduced the Fractionally Ž
. Integrated GARCH FIGARCH model thereby generalizing the GARCH and
w x
IGARCH specifications. Define the conditional variance h s Var ´ V where
t t
ty1
V is the information set in period t y 1. The GARCH model may be written as
ty 1 2
1 y b L h s v q a L ´
2
Ž . Ž .
Ž .
t t
Ž . Ž .
where a L and b L are lag polynomials of order q and p, respectively. By defining the skedastic innovation as
Õ s ´
2
y h 3
Ž .
t t
t
Ž .
2
this may be recognized as an ARMA m, p process in ´
t 2
1 y b L y a L ´ s v q 1 y b L
Õ 4
Ž . Ž .
Ž . Ž .
t t
Ž .
Ž .
where m s max q, p . The corresponding FIGARCH p, d, r representation is
given by
d 2
f L 1 y L
´ s v q 1 y b L Õ
5
Ž . Ž .
Ž . Ž .
t t
Ž . w
Ž . Ž .xŽ
.
y d
Ž . where f L s 1 y b L y a L
1 y L . Note that for d 0, f L
is in principle of infinite order. The FIGARCH model reduces to a GARCH model
Ž .
when d s 0 and to an Integrated GARCH IGARCH model when d s 1. The conditional variance of the FIGARCH process may be written as
v
2
h s q l L ´
6
Ž . Ž .
t t
1 y b 1
Ž .
Ž . w Ž .Ž
.
d
x w Ž .x4
where l L s 1 y f L 1 y L r 1 y b L
. 2.2. MultiÕariate models
The initial generalization of univariate to multivariate GARCH is due to Ž
. Bollerslev et al. 1988 . As in univariate GARCH, the conditional variance–co-
variance matrix of the n-dimensional error term ´
in the multivariate
t
Ž .
GARCH p, q model is a function of the information set at time t y 1. Therefore, the elements of the covariance matrix follow a vector ARMA process in the
squares and cross-products of the innovations. Restricting attention to the bivariate Ž
.
1r2
case, setting q s p s 1 and writing the covariance h s r h
h , Boller-
12, t 11, t
22, t
Ž .
Ž .
slev 1990 developed the constant correlation bivariate GARCH 1,1 representa- tion
h s a ´
2
q b h q v
11 , t 11
1 , ty1 11
11 , ty1 1
h s a ´
2
q b h q v
22 , t 22
2 , ty1 22
22 , ty1 2
1 2
h s r h
h .
7
Ž .
Ž .
12 , t 11 , t
22 , t
This specification is very parsimonious. Positive definiteness is guaranteed Ž
. provided r - 1 and a , b
and v j s 1,2 are such that h
and h are
j j j j
j 11, t
22, t
always positive.
4
Ž .
Brunetti and Gilbert 1998
extended the multivariate GARCH model to multivariate FIGARCH using the constant correlation parameterization. This
choice is motivated by three considerations: 1. It is the most parsimonious of the available specifications;
2. The variance–covariance matrix is positive definite under weak conditions Ž
. see below ;
3. Stationarity is ensured by restrictions only on the diagonal elements of the variance–covariance parameters matrices.
5
4
A general factor-based solution to the problem of ensuring positive definite variance–covariance Ž
. matrices was provided by Baba et al. 1989 . We do not pursue this.
5
Ž .
This is true for all diagonal models—see Engle and Kroner 1995 .
Ž .
The bivariate constant correlation FIGARCH 1, d,1 model may be represented as
v
j 2
h s l
L ´ q 8
Ž . Ž .
j j, t j j
j, t
1 y b 1
Ž .
j j 1
2
w x
h s r h
h 9
Ž .
12 , t 11 , t
22 , t
wŽ Ž ..Ž
.
d
j
x w Ž .x4
where l s 1 y f
L 1 y L
r 1 y b L
, and j s 1, 2. It follows from
j j j j
j j
Ž .
the results in Bollerslev and Mikkelsen 1996 that positive definiteness in the Ž
. bivariate diagonal FIGARCH 1, d,1
model is ensured if r - 1, b y d F
j j j
Ž .Ž
. w
Ž .x
Ž .
1r3 2 y d , and d f y 1r2 1 y d F b
f y b qd . The conditional
j j
j j j
j j j j
j j
variance defined by this process is stationary for all 0 F d F 1. In what follows,
j
Ž . we use a general representation of Eq. 8 , which may be represented in the
ARFIMA form as
d
1
1 y L
Ž .
2
F L ´ s v q I y B L
v 10
Ž . Ž .
Ž .
Ž .
t t
d
2
ž
1 y L
Ž .
where F s I. It is straightforward to test the hypothesis d s d within this
1 2
framework.
3. Fractional cointegraiton
