12-FinEcon-Model Sel Page 347 Wednesday, February 4, 2004 12:59 PM
Financial Econometrics: Model Selection, Estimation, and Testing 347process. If the ARMA process for volatility is integrated (that is, it has unit
roots) then the GARCH process is called Integrated GARCH or IGARCH.
The ARCH technology is not restricted to univariate processes but
can be extended to multivariate processes. Multivariate GARCH pro-
cesses model the entire variance-covariance matrix as an autoregressive
process.
Multivariate models of the ARCH-GARCH type become rapidly
unmanageable as the number of parameters to estimate grows with the
fourth power of the number of assets. Dimensionality reduction is called
for. Different proposals have been made, in particular factor models for
the volatility process.
The random terms z might have arbitrary distributions. In practice,
normality is often assumed. However, though the conditional distribu-
tion is normal, the unconditional distribution of a GARCH process is
not normal but exhibits fat tails (see Chapter 13). This feature of
GARCH processes, in addition to the modeling of volatility clustering,
has made them attractive as models of returns. Returns at short time
horizons are, in fact, not normally distributed but exhibit fat tails.
However, fitting different families of GARCH processes to empirical
return data has shown that GARCH models cannot fit simultaneously
the volatility clustering and the fat-tailedness of returns. Distributions
of the shock z other than normal have been tried, for instance T-Student
distributions, but no good fit of volatility and returns has been reported
in the literature. GARCH models can be considered a useful economet-
ric tool, but not a firm theory of price processes.Markov Switching Models
Markov switching models belong to a vast family of models that have
found applications in many fields other than econometrics, such as
genomics and speech recognition. The economic idea behind Markov
switching models is that the economy undergoes discrete switches
between economic states at random times. To each state corresponds a
set of model parameters.
One of the first Markov switching models proposed is the Hamil-
ton^34 model. The Hamilton model is based on two states, a state of
“expansion” and a state of “recession.” Periods of recession are fol-
lowed by periods of expansion and vice versa. The time of transition
between states is governed by a two-state Markov chain. In each state,
price processes follow a random walk model.(^34) J.D. Hamilton, “A New Approach to the Economic Analysis of Nonstationary
Time Series and the Business Cycle,” Econometrica 57 (1989), pp. 357–384.