Applications of state space models in finance

148 7 A Kalman filter based conditional multifactor pricing model
following form:
Rt = λ1,t ˆβ 1,t−1 + λ2,t ˆβ 2,t−1 + λ3,t ˆβ 3,t−1 + λ4,t ˆβ 4,t−1 + λ5,t ˆβ 5,t−1
λ6,t ˆ β 6,t−1 + νt, for (7.24)
for t = 164, . . . , 683, with Rt and ˆ β 1,t−1, . . . , ˆ β 6,t−1 denoting 18 × 1 vectors of excess
sector returns and factor loadings, respectively. The factor loadings used as independent
variables have been estimated from one of the alternative time series regressions (KF,
RLS, RR5 or RR1) of excess sector returns on the set of explanatory variables given
by (7.18)–(7.20). Estimation of (7.24) for each date t of the out-of-sample period yields
a time series of associated risk premia λk,t, k = 1, . . . , 6, each of length 520. 24 In
accordance with (7.12)–(7.14) the significance of the chosen risk factors is tested by
employing a t-test to the time series means of estimated risk premia.
The cross-sectional regression results are summarized in Table 7.8, where the coefficient
averages together with the corresponding Fama-MacBeth t-statistics are reported.
To ascertain whether the proposed Kalman filter based conditional factor loadings are
relatively better in explaining sector pricing, the cross-sectional regressions have also
been conducted for the betas estimated by recursive least squares and by rolling OLS.
Table 7.8 is divided into four panels: Panel A covers the complete out-of-sample period
from 22 February 1995 to 2 February 2005; Panels B to D refer to the three subperiods of
observed bull market, bear market and market recovery regimes. Overall, there is only
little evidence that the multifactor model is better able to describe the cross-section of
returns when conditional Kalman filter based factor loadings are used as independent
variables. Over the entire estimation period, the average ¯ R 2 is only slightly higher than
in case of the three alternative least squares specifications. The four different models
explain between 28.9% (KF ) and 28.0% (RR1) of the total cross-sectional variation of
excess sector returns. The only risk premium that is significantly different from zero
over the complete sample is estimated for the Kalman filter based T S factor.
Dividing the sample under consideration into three market-regime dependent subperiods
reveals that both the average ability to explain the cross-sectional variability and
the significance of estimated risk premia differ across subperiods. For the bull market
period (Panel B) the risk premium of the SIZ factor is significantly positive for all models.
The sign on SIZ reflects the outperformance of large caps over small caps during
the dotcom bubble. Besides, for the KF and RR1 based factor loadings the risk premium
of the T S factor is significantly negative at the 10% level. This means that during
24 For each cross-sectional regression at a given date t, six parameters have to be estimated
with the dependent variable consisting of 18 data points only. Similar settings with comparable
degrees of freedom are commonly employed in the literature, see, for example, Chen et al.
(1986) who use 20 portfolios as dependent variables to estimate up to seven parameters in
the cross-section. Nevertheless, in order to check whether the conclusion to be drawn in this
subsection is sensitive to the dimension of the vector of dependent variables, (7.24) has also
been estimated using 37 sectors instead of the 18 supersectors employed throughout this thesis.
As the employment of a finer market segmentation leads to the same conclusion with respect
to the practical relevance of time-varying factor loadings, only the case with 18 supersectors
is discussed here; results of the Fama-MacBeth procedure using 37 sectors are available on
request.