Page 140 - Special Topic Session (STS) - Volume 2

STS474 Takaki S. et al.

Spatial extension of GARCH Models for high-
dimensional financial time series
1
Takaki S. , Yasumasa Matsuda 2
1 Advanced Institute for Yotta Informatics, Tohoku University, Sendai, Japan
2 Graduate School of Economics and Management, Tohoku University, Sendai, Japan

Abstract
Autoregressive Conditional Heteroscedasticity (ARCH) models, which were
originally proposed by Engle (1982), have been playing major roles in
modeling volatilities in financial time series. The purpose of this study is a
multivariate extension of ARCH models to evaluate volatility matrices for high
dimensional multivariate financial time series. The critical difficulty in the
multivariate extension is in the so-called curse of dimension caused by a larger
number of parameters for a higher dimension of multivariate series. We
introduce financial distances among components of multivariate series, which
are different from the usual physical one but are based on the closeness of
financial conditions, and apply dynamic panel data models by spatial weight
matrices constructed by the financial distance. As a result, we propose spatial
autoregressive moving average models with generalized autoregressive
conditional heteroskedasticity processes (SARMA-GARCH models) that can
identify volatility matrices for high dimensional financial time series. We
conduct comparative studies by real financial time series and show empirical
features of the SARMA-GARCH models in terms of the forecast of volatilities.

Keywords
Volatility model; Spatial weight matrix; High-dimensional statistics

1. Introduction
Volatility which is a conditional variance in a model is one of the most
important concepts in financial econometrics because it is used in widely areas
such as risk management, option pricing and portfolio selection. Financial
market data often exhibits volatility clustering (i.e., volatility may be high for
certain time periods and low for other periods) This means time-varying
volatility is more common than constant volatility. Therefore, accurate
modeling of time-varying volatility is important in financial econometrics.
The seminal work of Engle (1982) proposes autoregressive conditional
heteroscedasticity (ARCH) models and the most important extension of the
model is generalized ARCH (GARCH) models proposed by Bollerslevv (1986).
The models have been widely used to identify volatilities. After that, many
extended GARCH models have been proposed. For example, integrated
GARCH models ( Engle and Bollerslev (1986)), exponential GARCH models
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