where d
k,t − 1
is the time-varying price of factor risk. Eq. 7 is the conditional three-factor asset pricing model derived from the
intertemporal consumption-investment optimization problem which will be esti- mated and tested via pricing kernel approach in Section 6.
Alternatively the conditional three-factor asset pricing model can also be derived based on arbitrage arguments, such as Arbitrage Pricing Theory APT by Ross
1976. Substituting the factor model Eq. 3 into the right hand side of Eq. 6 and assuming that Co6o
i,t
; M
t + 1
= 0 implies E[r
i,t
V
t − 1
] =
k
b
ik
Co6F
k,t
− M
t
V
t − 1
E[M
t
V
t − 1
]
n
=
k
b
ik
l
k,t − 1
Ö k
= W, INT, FX
8 where l
k,t − 1
is the time-varying risk premium per unit of beta risk. Assuming F
k,t
= E[F
k,t
V
t − 1
] + o
k,t
, where o
k,t
is the factor innovation with E[o
k,t
V
t − 1
] = 0, and E[o
k,t
o
j,t
V
t − 1
] Ö
k j, then Eq. 3 can be rewritten as: r
i,t
= a
i
+
k
b
ik
E[F
k,t
V
t − 1
] +
k
b
ik
o
k,t
+ o
i,t
Ö i = 1 ······ N; Ök = W, INT, FX
9 Taking conditional expectation on both sides of Eq. 9 and compare it with Eq.
8, then under the null hypothesis of a
i
= 0 obtaining:
E[F
k,t
V
t − 1
] = l
k,t − 1
Ö k = W, INT, FX
10 Substituting Eq. 10 into Eq. 9, and assuming a
i
= 0, Eq. 9 becomes
r
i,t
=
k
b
ik
l
k,t − 1
+ o
k,t
+ o
i,t
Ö i = 1 ······N; Ök = W, INT, FX
11 Eq. 11 is the conditional three-factor asset pricing model based on the arbitrage
arguments which will be estimated and tested using MGRACH-M methodology in Section 6.
To compare with previous studies in testing an unconditional multi-factor model, the unconditional version of Eq. 11 where expected factor risk premia, l
k,t − 1
’s are restricted to be time invariant will also be estimated and tested in Section 6.
4. Econometric methodologies
4
.
1
. Pricing kernel approach The ‘‘Pricing Kernel’’ approach, initiated by Hansen and Jagannathan 1991,
was generalized by Dumas and Solnik 1995, and Tai 1999a to test asset pricing models and will be used in this paper. The M
t
for conditional three-factor asset pricing model in Eq. 7 has the following form:
M
t
= 1 − d
0,t − 1
−
k
d
k,t − 1
F
k,t
n,
1 + r
0,t − 1
Ö k = W, INT, FX
12 where d
0,t − 1
= −
k
d
l,t − 1
E[F
k,t
V
t − 1
] and R
0,t − 1
= 1 + r
0,t − 1
is the risk-free return.
The new time varying term, d
0,t − 1
, appears as a way of ensuring Eq. 4 holds. For econometric purposes, following Dumas and Solnik 1995 two auxiliary
assumptions are needed:
Assumption 1: the information set V
t − 1
is generated by a vector of instrumental variables Z
t − 1
. Z
t − 1
is a 1 × l vector of predetermined instrumental variables that reflect every- thing that is known to investors at time t − 1.
Assumption 2:
d
0,t − 1
= − Z
t − 1
8 d
k,t − 1
= Z
t − 1
8
k
, Ö
k = W, INT, FX Here the 8
and 8
k
’s are the time-invariant row vectors of weights for the instruments for each of the risk factors.
Based on Eq. 4, defining the innovation u
t
: M
t
1 + r
0,t − 1
= 1 − u
t
13 and given assumption 2 and the definition of M
t
in Eq. 13, u
t
can be written as: u
t
= 1 − M
t
1 + r
0,t − 1
= − Z
t − 1
8 +
k
Z
t − 1
8
k
F,t Ö
k = W, INT, FX 14
with u
t
satisfying: E[u
t
V
t − 1
] = 0 15
Next, based on Eq. 5 defining the innovation h
it
: E[M
t
r
it
V
t − 1
] = E 1 − u
t
1 + r
o,t − 1
r
it
V
t − 1
n
= 0 [ h
it
= r
it
− r
it
u
t
Ö i
= 1 ······ N
16 with h
it
satisfying: E[h
it
V
t − 1
] = 0 17
One can form the 1 + N vector of residuals e
t
= u
t
, h
t
. Combining Eq. 15 and Eq. 17 and using Assumption 1 yields:
E[e
t
Z
t − 1
] = 0 18
It implies the following unconditional condition: E[m
t
b ] = E[e
t
Z
t − 1
] = 0 Ö
t = 1,2 ··· T 19
The sample version of this population moment restriction is the moment condition:
Ze = 0 20
where Z is a T × l matrix and e is a T × 1 + N matrix, with T being the number of observations over time. We test restrictions implied by the theory using Hansen’s
test of the orthogonality conditions used in estimation Hansen, 1982. He shows that the minimizer quadratic criterion function is asymptotically central chi-
square distributed with N − 3 × l degrees of freedom under the null hypothesis that the model is correctly specified.
3
4
.
2
. Multi6ariate GARCH in mean MGARCH-M
approach Theoretical work by Merton 1973 relate the expected risk premium of factor k,
l
k,t − 1
, in Eq. 11 to its volatility and a constant proportionality factor. In supporting these theoretical results, Merton 1980 tested a single-beta market
model and found that the expected risk premium on the stock market is positively correlated with the predictable volatility of stock returns. As a result, the following
relationship is postulated for l
k,t − 1
Ö k = W, INT, FX:
l
W,t − 1
= EF
W,t
V
t − 1
= w +
w
1
h
w,t
21 l
INT,t − 1
= EF
INT,t
V
t − 1
= l +
l
1
h
INT,t
22 l
FX,t − 1
= EF
FX,t
V
t − 1
= c +
c
1
h
FX,t
23 where h
k,t
Ök = W, INT, FX is factor k’s conditional volatility. To complete the conditional three-factor model with time-varying risk premia, Eq. 11 can be
rewritten as: r
i,t
= w
+ w
1
h
w,t
+ o
W,t
b
iW
+ l
+ l
1
h
INT,t
+ o
INT,t
b
iINT
+ c
+ c
1
h
FX,t
+ o
FX,t
b
iFX
+ o
i,t
Ö i = 1 ······ N
24 Eq. 24 is the expanded three-factor asset pricing system, which can be used to
test whether the predictable volatilities of the market-wide risk factors are signifi- cant sources of risk. This model allows for a test of the null hypothesis of the
existence of one or more significant risk premia, and for a test of the hypothesis that the risk premia are jointly time-varying. Similarly the null hypothesis of the
existence of one or more significant factor sensitivities can also be tested based on Eq. 24.
To estimate and test dynamic factor models similar to Eq. 24, Engle et al. 1990, Ng et al. 1992, and Flannery et al. 1997 utilize a factor GARCH model
because it provides a plausible and parsimonious parameterization of time-varying variance-covariance structure of asset returns. However, they employ a two-step
3
There are l parameters 8 , and l × 3 parameters 8
k
’s so the total number of parameters is 4 × l. From Eq. 20 the number of moment conditions is l × 1 + N. Thus, we have N − 3 × l degrees of
freedom.
procedure to estimate the model.
4
In the first step, the time-varying factor risk premia are estimated via an univariate GARCH-M model; in the second step, the
estimated premia and conditional variance are then taken as the data series in the estimation of the conditional means and variances of each individual asset return
series Engle et al., 1990 and Ng et al., 1992 or of the conditional means but constant variances of a set of portfolio returns Flannery et al., 1997. The obvious
advantage of this procedure is that an arbitrarily large system can be estimated without much difficulty. The disadvantage is that cross-asset correlations and
parameter restrictions are ignored so that efficiency is sacrificed.
Given the computational difficulties in estimating a larger system of asset returns, parsimony becomes an important factor in choosing different parameterizations. A
popular parameterization of the dynamics of the conditional second moments is BEKK, proposed by Engle and Kroner 1995. The major feature of this parame-
terization is that it guarantees that the covariance matrices in the system are positive definite. However, it still requires researchers to estimate a larger number
of parameters. Instead of using BEKK specification, this paper employs a parsimo- nious GARCH process proposed by Ding and Engle 1994 to parameterize the
conditional variance-covariance structure of asset returns. This specification allows one to reduce the number of parameters to be estimated significantly if the
conditional second moments are assumed to follow a diagonal process and the system is covariance stationary.
5
Consequently, the process for the conditional variance-covariance matrix of asset returns can be written as:
H
t
= H
ii − aa − bb + aao
t − 1
o
t − 1
+ bbH
t − 1
25 where H
t
is N + 3 × N + 3 time-varying variance-covariance matrix of asset returns and risk factors. N + 3 is the number of equations where the first N
equations are those for the bank portfolios, the N + 1th equation is for the interest rate risk factor; the N + 2th equation is for the exchange rate risk factor,
and the N + 3th equation is for the world market risk factor. The elements on the diagonal of H
t
are given by Eq. 24 for the individual bank portfolios, and by Eq. 21, Eq. 22 and Eq. 23 for three risk factors. H
is the unconditional variance- covariance matrix of innovations, o
t
·i is a N + 3 × 1 vector of ones, a and b are N + 3 × 1 vectors of unknown parameters, and denotes element by element
matrix product. The H is unobservable and has to be estimated. As suggested by
De Santis and Gerard 1997, 1998, it can be consistently estimated using iterative procedure. In particular, H
is set equal to the sample covariance matrix of the excess return in the first iteration, and then it is updated using the covariance
matrix of the estimated residual at the end of each iteration. Under the assumption of conditional normality, the log-likelihood to be maximized can be written as:
4
Koutmos 1997 applies a multivariate factor GARCH model to test if market portfolio is a dynamic factor, but he only considers market risk.
5
In a diagonal system with N assets, the number of unknown parameters in the conditional variance equation is reduced from 2N
2
+ NN + 12 under BEKK specification to 2N under the Ding-Engle
specification.
ln Lu = − TN
2 ln 2p −
1 2
T t = 1
ln H
t
u −
1 2
o
t
uH
t
u
− 1
o
t
u 26
where u is the vector of unknown parameters in the model and T is the number of observations over time. Since the normality assumption is often violated in
financial time series, a quasi-maximum likelihood estimation QML proposed by Bollerslev and Wooldridge 1992 which allows inference in the presence of depar-
tures from conditional normality is used. Under standard regularity conditions, the QML estimator is consistent and asymptotically normal and statistical inferences
can be carried out by computing robust Wald statistics. The QML estimates can be obtained by maximizing Eq. 26, and calculating a robust estimate of the covari-
ance of the parameter estimates using the matrix of second derivatives and the average of the period-by-period outer products of the gradient. Optimization is
performed using the Broyden, Fletcher, Goldfarb and Shanno BFGS algorithm, and the robust variance-covariance matrix of the estimated parameters is computed
from the last BFGS iteration.
Given the computational complexity of the multivariate approach, its application is restricted to three bank portfolios, which are simultaneously modeled with the
three risk factors. Thus, the dimension of o
t
is 6 and that of the variance-covariance matrix is 6 × 6. Even with this low dimensional system the number of parameters
to be estimated is 27.
5. The data and summary statistics
