Page 141 - Special Topic Session (STS) - Volume 2

STS474 Takaki S. et al.
(Nelson (1991)), threshold GARCH models (Glosten, etal (1993)), GARCH in the
mean models, and GJR-GARCH models are proposed.
Univariate volatility models are generalized to multivariate cases in many
ways. One important problem which multivariate volatility models contain is
the curse of dimensionality. We estimate a conditional covariance matrix which

has (+1) quantities for a n-dimensional time series in multivariate analysis.
2
However it is difficult to estimate all quantities. Thus, we attempt to give a
conditional covariance matrix some simple structures to reduce the number
of parameters. There are many ways for generalization. For example,
exponentially weighted moving average models, constant conditional
correlation models (Bollerslev (1990)), BEKK models (Engle and Kroner (1995)),
orthogonal GARCH models (Alexander (2001) ), dynamic conditional
correlation models (Tse and Tsui (2002)), dynamic orthogonal component
models, and factor GARCH models are proposed.
The ideas of spatial econometrics have been applied to volatility models
to reduce number of parameters in a covariance matrix and to extend the
models to spatial models in recent years. Caporin and Paruolo (2008) and
Borovkova and Lopuhaa (2012) have applied the ideas of spatial econometrics
to time series multivariate GARCH models. Yan (2007) and Robinson (2009)
have done spatial extensions of stochastic volatility models which are another
kind of volatility models. Sato and Matsuda (2017, 2018) have extend time
series GARCH models to spatial models.
This paper contributes to extend GARCH models to spatiotemporal models
for high dimensional financial time series which we call spatial autoregressive
moving average models with generalized autoregressive conditional
heteroskedasticity processes, namely SARMA-GARCH models by using spatial
econometrics ideas.
The model is characterized by a spatial weight matrix which express cross-
section correlations between assets and used to reduce the number of
parameters. A spatial weight matrix is usually determined by geographical
information of spatial data. However, financial data doesn't include
geographical information, therefore we need to consider a method to make
spatial weight matrix from financial data. Here, we apply the multiple linear
regression model to return series of assets to calculate spatial weight matrices
with stepwise model selection procedures for selecting subsets of explanatory
variables in the regression model.
Parameters in the SARMA-GARCH model are estimated by a two step
procedure. First step is the estimation of spatial parameters and second step
is the estimation of GARCH parameters. Spatial parameters which are scalar
parameters reflecting the strength of spatial dependence between assets are
estimated in first step. Conditional variances in the model follows GARCH

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