Applications of state space models in finance

Chapter 5
Conditional heteroskedasticity models
It is a well established stylized fact of financial time series that the volatility of returns
on financial assets changes persistently over time and across assets (cf. ¢ 2.2.2). The
concept of conditional heteroskedasticity is key to many areas in finance and financial
econometrics, whether it is in asset allocation, risk management, asset pricing or duration
modeling (cf. Diebold 2004). In the context of this thesis, conditional heteroskedasticity
models are mainly used to model time-varying relationships in an indirect way. As a
simple regression coefficient is defined as a covariance-variance ratio, models of conditional
heteroskedasticity can be used to obtain conditional betas based on conditional
variance estimates.
The phenomenon of time-varying volatility, first recognized by Mandelbrot (1963) and
Fama (1965), is commonly referred to as volatility clustering: quiet periods with small
absolute returns are followed by volatile periods with large absolute returns, then quite
periods again, and so on. This chapter presents the basic ideas of the two most important
concepts of capturing time-varying volatility and excess kurtosis, which is usually
induced by volatility clusters. The ARCH/ GARCH framework by Engle (1982) and
Bollerslev (1986) represents practitioners’ preferred tool to model and forecast volatility.
An alternative way of modeling conditional heteroskedasticity is offered by the class
of stochastic volatility (SV) models, introduced by Taylor (1982, 1986). In contrast to
a GARCH model, the SV model adds an unobserved shock to the return variance. This
leads to a higher degree of flexibility in characterizing the dynamics related to volatility.
11 The major difference between these two approaches is that ARCH models are
observation-driven, while SV models are parameter-driven. In the context of an ARCH
model, the conditional variance is a function of lagged observations, with the corresponding
likelihood function being directly available. For SV models, the conditional variance
depends on a latent component. As no analytic one-step ahead forecast densities are
available, SV models can only be dealt with approximately or by employing numerically
11 In addition to these two concepts of historical volatility, alternative measurements such as
implied and realized volatility are available. As these procedures are based on option pricing
data and high-frequency data for single stocks, respectively, they will not be considered in
this thesis focusing on sectors. For details and an account on the available literature, see, for
example, Andersen et al. (2002), Koopman et al. (2004) or Andersen et al. (2005).