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
The sample consists of 31 commercial bank stocks traded on the New York and American stock exchanges. The excess return on a bank stock is the log first
difference of total return index in excess of 7-day Eurodollar deposit rate. The sample is disaggerated by size into three equally weighted bank portfolios-the
money center bank portfolio seven banks, the large bank portfolio 11 banks and the regional bank portfolio 13 banks.
Three economic risk variables are a world market risk F
W
measured as the US dollar return of the Morgan Stanley Capital International MSCI world equity
market in excess of 7-day Eurodollar deposit rate, an interest rate risk F
INT
measured as the log first difference in the 10-year US Treasury Composite yield, and an exchange rate risk F
FX
measured as the log first difference in the trade-weighted US dollar price of the currencies of 10 industrialized countries. A
positive change F
FX
\ 0 indicates a depreciation of the dollar.
The instruments used in the GMM estimations include the world excess equity return F
W,t − 1
, a dividend yield on SP 500 index in excess of the 7-day Eurodollar deposit rate SPDIV, a change in the US term premium, measured by
the yield on the 10-year US treasury note in excess of the 7-day Eurodollar deposit rate DUSTP, a change in the US default premium, measured by the difference
between Moody’s Baa-rated and Aaa-rated US corporate bond yields DUSDP, and a constant.
Observations are sampled at weekly intervals. The weekly data ranges from November 6, 1987 to August 28, 1998, which is a 565-data-point series. However,
this paper works with rates of return and use the first difference of information variables, and finally all the instruments are used with a one-week lag, relative to
the excess return series; that leaves 562 observations expanding from November 27, 1987 to August 28, 1998. Table 1 describes the variables and their symbols used in
this paper. All the data are extracted from DATASTREAM.
Summary statistics for bank stock returns, risk factors, and instruments used in this paper are presented in Table 2. The mean excess returns for three bank stock
portfolios are 0.2551 for Money Center bank, 0.2372 for Large bank, and 0.2452 for Regional bank. These mean excess returns are all greater than
0.1065, the mean excess return for MSCI world equity index. However, their standard deviations are also greater than that of MSCI world equity index,
indicating that investors are compensated for a higher risk premium when holding bank stocks. The positive change in the exchange rate reflects the depreciation of
the US dollar against the currencies of ten industrialized countries. The coefficients of skewness and excess kurtosis reveal nonnormality in the data. The last two
columns in Table 2 report the Ljung-Box portmanteau test statistics for indepen- dence in the return and squared return series up to 24 lags, denoted by
Q
24 and Q
2
24 respectively. The Ljung-Box portmanteau test statistics for independence in the standardized residuals are calculated using autocorrelations up to 24 lags, and
they follow a x
2
distribution with 24 degrees of freedom.
6
The hypothesis of linear independence is rejected at 5 level for Money Center bank and 1 level for
Regional bank. Independence of the squared return series is rejected at 1 level for all three bank portfolio returns, MSCI world equity returns, and the exchange rate
changes. Clearly, the nonlinear dependencies are much prevalent than the linear dependencies found in the data and it is consistent with the volatility clustering
observed in most financial data: Large small changes in prices tend to be followed by large small changes of either sign. The GARCH model used in this study is
well known to capture this property.
6. Empirical results
