Applications of state space models in finance

104 6 Time-varying market beta risk of pan-European sectors
*
εt 10
5
0
−5
−10
(a) Auxiliary heteroskedastic residuals
1990 1995 2000 2005
w ^ t
(b) Annualized filtered conditional volatility
30
25
20
15
10
5
1990 1995 2000 2005
Figure 6.3: (a) Residuals from the auxiliary heteroskedastic regression model and
(b) GLS weighting factor for Automobiles and the overall market.
The described procedure to deal with volatility clusters and outliers shall be illustrated
at the example of the Automobiles sector. Figure 6.2 shows the weekly excess log-returns
on the sector and the DJ Stoxx Broad index. It is obvious that the volatility of both
series is not constant over time. It can be seen that some periods are characterized by
small absolute returns and others by large absolute returns. Especially in the second half
of the period, which is effected by the Asian crisis, the Russian crises and the boom and
bust of the new economy, some outlying absolute returns exceed the value of 10%. As
outlined above, the influence of heteroskedasticity can be removed by means of weighted
least squares. The weighting factor wt can be derived by an auxiliary heteroskedastic
regression model. Figure 6.3 displays the auxiliary residuals in Panel (a); Panel (b)
shows the weighting factor computed as the annualized conditional t-GARCH volatility
estimate.
The estimated conditional volatility confirms the impression of volatility clustering.
It is used to transform the return series according to (6.30) to yield the weighted return
series, which is plotted in Figure 6.4. As expected, the WLS transformation removed
the heteroskedasticity in the series. What remains are a few outliers defined as those
observations outside ˆµ ± 3ˆσ that can now be capped (floored) according to the “three
sigma”-rule. The trimmed generalized series can now be utilized as dependent and
independent variables to estimate time-varying GRW beta series. Figure 6.5 illustrates
the difference between the RW and GRW beta series for the Automobiles sector: the
proposed procedure to deal with heteroskedasticity and outliers leads to a smoother
conditional beta series whose major pattern remains intact.
The estimation results for all GRW models are summarized in Table 6.6. The estimated
variance of observation and state disturbances are significant at the 1% level
for all sectors. Although the null of normality can be rejected without exception, the
reported JB-statistics are all significantly lower than for the Kalman filter based models
considered above. The null of no autocorrelation can be rejected at the 5% level
for eleven sectors. According to the reported LM-tests, the weighted transformation
removed the volatility clusters for all sectors except for Personal & Household Goods.