Page 142 - Special Topic Session (STS) - Volume 2

STS474 Takaki S. et al.
processes and unconditional variances in GARCH processes are constant.
Therefore, we regard volatilities in the model as constant and apply the quasi-
maximum likelihood method with a spatial panel model with heterogeneous
constant variances. GARCH parameters are scalar parameters for specifying
volatility behaviors. We calculate residuals by fitting the spatial panel model in
first step after that we apply GARCH models with the residuals of each asset.
In real data analysis, We apply the SARMA-GARCH model to daily returns
of the S&P 500 stock price data. We compare in-sample and out-sample
performances of the SARMA-GARCH model with those of CCC models which
is a multivariate volatility model and a baseline model in this study. First, we
check the in-sample performances based on log-likelihood. The results show
the log-likelihood of the CCC model is grater than that of the SARMA-GARCH
model. This means model fitting of the CCC model is better than the SARMA-
GARCH model because the number of parameters in CCC models is more than
five times of those of SARMA-GARCH models. Secondly, we compare out-
sample performances by using the quasi-likelihood loss function. The result
shows the quasi-likelihood loss function of the SARMA-GARCH model are
smaller than that of CCC models. Then, the out-sample performance of
SARMA-GARCH models is better. This is because the CCC model may be over-
fitting and it cause lower forecasting performances.
Moreover, stock prices in the U.S. market are volatile and correlation
structures between stocks may change over time, so the characteristic of the
SARMA-GARCH model which is that the model can capture dynamic
correlation between assets may play an important role in analysis.
The rest of paper proceeds as follows. Section 2 introduces SARMA-
GARCH models. The estimation procedures are described in section 3. Section
4 examines empirical properties of SARMA-GARCH models by applying the
models to real data such as stock price in the U.S. market. Section 5 discusses
some concluding remarks.

2. Methodology
Let { }, = 1, … , and = 1, … , , be return series of financial
,
instruments. We shall define SARMA-GARCH models to describe volatilities of
return series by
= + (1)

= + (2)

= , (3)
,
, ,
~ . . (0,1),
,
2
= + 2 + 2 ,
,−1

,
,−1
where = ( , … , ), = ( , … , , ), , is volatility, , is an

1,
,

1,
independent and identically distributed (i.i.d) random variable with mean zero
and variance 1. The matrix W, n by n matrix, is called a spatial weight matrix
131 | I S I W S C 2 0 1 9