Applications of state space models in finance

2.2 Empirical properties 13
2.2.2 Volatility clustering
It appeared from Figure 2.2 that quiet periods, characterized by relatively small returns,
alternate with relatively volatile periods, where price changes are rather large. This can
be confirmed by looking at the first-order autocorrelation (AC) function of returns and
squared returns in Figure 2.3. While the autocorrelations of the return series only
show minor activity, the autocorrelation function of squared returns show significant
correlations up to an extended lag length. The corresponding Box-Ljung statistics,
reported in column Qsq of Table 2.2, confirm significant correlations for the squared
returns at the 1% level for all series.
The phenomenon of persistently changing volatility over time has been first recognized
by Mandelbrot (1963) and Fama (1965) who related volatility clustering to the observation
of fat-tailed returns. The two most important concepts to deal with time-varying
volatility are (i) the autoregressive conditional heteroskedasticity (ARCH) model by Engle
(1982) together with the generalized ARCH (GARCH) model by Bollerslev (1986),
and (ii) the stochastic volatility (SV) model by Taylor (1982, 1986). An alternative way
to deal with conditional volatility is to employ a Markov regime switching model: the
generation of excess returns under different volatility regimes is governed by different
states, each driven by a first-order Markov chain. Leading papers on the subject include
Hamilton and Susmel (1994) and Turner et al. (1989).
2.2.3 Leverage effects
Another common property of financial return series is the so-called leverage effect, where
future volatility depends negatively on the sign of past returns (Black 1976). Column lev
of Table 2.2 reports the estimated correlation coefficients between squared excess returns
at date t and excess returns at date t − 1 for all sectors and the overall market. With
the exception of Technology, all estimates are negative. This gives a first indication that
negative sector returns are followed by a pick-up in volatility.
2.2.4 Volatility co-movements
From the squared excess returns shown in the bottom row of Figure 2.2, it can be seen
that volatility peaks tend to occur at the same time for all three series (August 1990,
January 1999, period between July 2002 and March 2003). This is a common observation
as certain newsflow usually affects the volatility of different assets simultaneously.
This phenomenon might lead to a failure of joint stationarity of two return series even
though stationarity for the individual series holds. A possible solution to this problem is
to employ a multivariate GARCH or a multivariate SV framework to deal with volatility
co-movements by modeling conditional covariances. Important papers dealing with
multivariate conditional heteroskedasticity include Diebold and Nerlove (1989), Harvey
et al. (1992) and Harvey et al. (1994).