Applications of state space models in finance

94 6 Time-varying market beta risk of pan-European sectors
cannot be estimated by a direct application of standard maximum likelihood techniques.
As outlined in ¢ 5.2.2, several procedures for estimating SV models have been proposed.
These include method of moments and QML estimators, Bayesian approaches based on
MCMC techniques, the MCL estimator and the efficient MCL procedure. A consensus
on how to estimate SV models is still missing.
As outlined in ¢ 5.2.3, in this thesis SV models are estimated via efficient MCL using
importance sampling techniques. Recalling the MSV model in (5.68) time-varying sector
betas can be constructed as
ˆβ SV
i,t = ρ0i
� hii,t
� . (6.14)
h00,t
In contrast to the GARCH based conditional betas defined in (6.10) smoothed rather
than filtered estimates of the conditional variance series of market and sector returns,
h00,t and hii,t, are employed. All the available information up to and including date T
can be relied upon for in-sample analysis purposes.
A summary of the estimation results of the considered univariate SV models for European
sectors over the full sample period is given in Table 6.5. The asymmetric 95%
confidence intervals for the persistence parameter φi are generally narrow. This indicates
a high level of significance. The degree of volatility persistence ranges from a low
for Travel & Leisure, to the highest level for Technology and Telecommunications. This
compares well to the GARCH results. For the two other parameters, σ2 ηi and σ2 ∗i , both
the asymmetric confidence intervals as well as the range of parameter estimates across
sectors, are wider. For the sectors Retail, Travel & Leisure and Utilities, the combination
of a low persistence parameter and a high value for σ2 ηi , which measures the variation of
the volatility process, implies that the process of volatility is less predictable for these
three sectors. The highest levels of volatility, as indicated by a high scaling parameter
σ2 ∗i , are found for Automobiles & Parts and the three sectors Telecommunications, Media
and Technology (TMT). This finding broadly corresponds to the calculated standard
deviations of weekly returns in Table 2.2. Notably, the null of normality cannot be rejected
at the 5% level for seven sectors. Compared to the GARCH models estimated in
the previous subsection, the estimated Jarque-Bera test statistics are significantly lower
for all sectors and also for the overall market. The reported Q-statistics indicate some
degree of autocorrelation at the 5% level for five sectors as well as the market index.
Figure 6.1 allows for a comparison of conditional volatility estimates based on a t-
GARCH(1,1) and a SV model. The graph displays the filtered and smoothed conditional
volatility estimates for the Telecommunications sector. Overall, the filtered conditional
volatility series show a very similar pattern. The range of the GARCH based series
is greater than in case of its SV counterpart. The smoothed estimate, which — per
definition — takes the full set of observations into account, reveals that both filtered
series tend to overstate the level of volatility.
6.2.3 Kalman filter based approaches
In contrast to volatility-based techniques, where the conditional beta series can only be
constructed indirectly after the conditional variances of the market and sector i have