Applications of state space models in finance

96 6 Time-varying market beta risk of pan-European sectors
null hypothesis of normality can be generally rejected, signs of autocorrelation in the
residuals are observed at the 5% level for twelve out of eighteen sectors. Even though the
ARCH-test is highly significant for all sectors, for eleven sectors the corresponding LMstatistics
are lower than in case of OLS. It can be concluded that the RW specification is
able to capture the heteroskedasticity in the resulting residuals at least partially. With
regard to the goodness of fit, the R 2 is always significantly higher than in case of OLS.
6.2.3.2 The mean reverting model
As an alternative to the random walk, the dynamic process of beta can be modeled as
being mean reverting. In the MR model, an autoregressive process of order one, AR(1),
with a constant mean is used for the evolution of beta:
ˆβ MR
i,t+1 − ¯ βi = φi(βi,t − ¯ βi) + ηi,t. (6.18)
In order to use the Kalman filter for estimating the MR model, the mean beta coefficient
is augmented into the state vector. This leads to the following state space model:
�
βi,t+1 − ¯ � � � �
βi φi 0 βi,t −
= ¯βi
0 1
¯ � � � � �
βi 1 0 ηi,t
+
, (6.19)
¯βi 0 1 0
yt = � �
xt xt
� βi,t − ¯ �
βi
+ ɛt, (6.20)
¯βi
where overall four parameters (σ 2 ɛi , σ2 ηi , ¯ βi, φi) have to be estimated. As outlined
in ¢ 3.5.2.3, the AR(1) parameter φi should be restricted to values between zero and
unity. This constraint is implemented in the estimation procedure by application of the
parameter transformation (3.67).
The estimated variances of observation and state disturbances are significant at the
1% level for every sector. Compared to the RW model above, the estimated variance
for the observation equation, σ2 i , is generally lower in case of the MR parameterization;
the opposite holds with respect to σ2 ηi , the variance of the dynamic process of the timevarying
beta. For the MR model, two additional parameters have been estimated. The
value for ¯ βi, which compares well to the estimated OLS betas as reported in Table 6.1, is
always significant at the 1% level. The estimates for the second extra parameter can be
clustered across all sectors into three groups. In the first group, φi is close to unity. The
closer to unity the transition parameter gets, the more the conditional beta resembles its
RW counterpart. In case of Food & Beverages, Healthcare and Personal & Household
Goods, the MR betas literally follow a random walk. In the second group with values
for φi around 0.5, the conditional betas return faster to their individual means, which
implies more noisy series of conditional betas. In the third group, where φi is close to
zero, the resulting beta series follow a random coefficient model as defined in ¢ 3.5.2.1.
Note that the estimated speed parameters in the last group (Industrials and Travel &
Leisure) are not statistically significant.
According to the fit statistics, the residuals are generally non-normal. With the exception
of Healthcare and Retail, the null of no autocorrelation can be rejected for all
sectors. According to the conducted ARCH-test, the null of no heteroskedasticity cannot