Applications of state space models in finance

7.4 Empirical results 147
Table 7.7: Average out-of-sample errors across sectors for multiple factor models.
Mean absolute error (×10 2 ) Mean squared error (×10 4 )
KF RLS RR5 RR1 KF RLS RR5 RR1
Average error 1.270 1.401 1.371 1.345 3.143 4.051 3.767 3.505
Average rank 1.00 3.67 3.00 2.33 1.00 3.89 3.06 2.06
for t = 164, . . . , 683 with T = 520 where ˆ Ri,t represents the series of return forecasts for
sector i defined as
ˆRi,t = ˆ β ′
i,t|t−1f t , (7.23)
The betas employed to generate the return forecasts for date t are based only on information
available at time t − 1. For all specifications, betas are predicted naïvely with
the respective latest estimate of beta being used as the corresponding one-step ahead
prediction. The series of factor realizations, f t , are assumed to be known with perfect
foresight, which represents a common modus operandi in empirical research to isolate
the impact of conditionality.
The resulting mean error measures are summarized in Table 7.7 which reports the
respective averages together with the corresponding average ranks each across all sectors
for all four employed specifications; for a detailed sectoral breakdown, see Table C.8 in
the appendix. For the sample under consideration, the Kalman filter based betas yield
the relatively best out-of-sample forecasts. For both error measures, the Kalman filter
based specification ranks first for all eighteen sectors. Compared to the least squares
alternatives, the mean absolute error is between 5.9% and 10.4% lower. When using the
mean squared error as evaluation criterion, the superiority of the Kalman filter based
betas becomes even more obvious: compared to the RLS, RR5 and RR1 alternatives,
the average MSE is 28.9%, 19.9% and 11.5% lower, respectively.
7.4.3 Practical relevance of time-variation in factor loadings
The results above indicate that the proposed dynamic specification, where conditional
betas are modeled as individual random walks estimated via the Kalman filter, produces
the relatively most accurate estimates of time-varying factor loadings. This subsection
examines whether the statistical superiority of the proposed methodology can
be exploited in practice, either in the pricing of risk or from a portfolio management
perspective.
7.4.3.1 Risk pricing
As outlined in Subsection 7.2.2, a common procedure to analyze how well the chosen
set of risk factors explains the cross-section of assets is to employ the Fama-MacBeth
cross-sectional regression approach. Based on the proposed set of explanatory variables,
(7.11) implies that the cross-section of sector returns follows a linear factor model of the