Applications of state space models in finance

7.4 Empirical results 139
Table 7.4: Summary statistics for the set of risk factors (8.1.1992–2.2.2005).
µ a
σ b
ρ1 ρ2 ρ3 ρ4 ρ5 ρ12 Q(12) c
BMR 0.03 (0.68) 13.14 0.14 0.13 0.02 0.02 0.03 −0.08 33.45 (0.01)
V GS 0.03 (0.52) 9.73 0.03 0.10 0.13 0.03 0.02 0.04 81.54 (0.00)
SIZ 0.11 (0.04) 9.80 −0.11 0.11 0.15 0.02 −0.01 0.06 46.70 (0.00)
T S 0.71 (0.14) 90.29 0.00 0.03 0.11 0.00 0.02 0.02 25.09 (0.01)
OIL 0.14 (0.39) 29.51 −0.02 0.04 0.03 −0.03 0.03 0.03 12.98 (0.37)
F X −0.01 (0.82) 10.06 0.01 −0.02 0.00 0.05 0.08 −0.08 14.39 (0.28)
a The mean is expressed in percentage terms. Figures in parentheses denote p-values for a
simple t-test of the null hypothesis of a zero mean.
b The standard deviation is expressed in annualized percentage terms.
c Q(l) is the test statistic of the Ljung-Box portmanteau test for the null hypothesis of no
autocorrelation in the errors up to order l. The Q-statistic is asymptotically χ 2 distributed
with l degrees of freedom. Figures in parentheses denote p-values.
the full sample, the highest correlations occur between SIZ and F X (−0.204) and
between SIZ and V GS (−0.227). This indicates that the outperformance of small caps is
associated (i) with a falling dollar, and (ii) with value outperforming growth. The highest
correlation coefficients are measured for Panel E, which is based on fewer observations
than the other subperiods. During this time, the exchange rate factor’s correlation with
the value-growth-spread (−0.370) and the term structure (−0.385) is elevated. Overall,
the observed pairwise correlations indicate that the selected risk factors are far from
being excessively collinear, which suggests that none is dispensable.
7.4 Empirical results
Employing the risk factors defined above together with the conditional multifactor model
given by (7.6)–(7.10) implies the following data-generating process for the set of excess
sector returns, Ri,t. The observation equation can be written as
Ri,t = βi1,tBMRt + βi2,tV GSt + βi3,tSIZt + βi4,tT St + βi5,tOILt
+ βi6,tF Xt + ɛi,t, ɛi,t ∼ N(0, σ2 i ), (7.18)
for i = 1, . . . , 18. The state equation is given as
β i,t+1 = β i,t + η i,t, η i,t ∼ N(0, Q), (7.19)
where βi,t+1 and ηi,t are 6 × 1 vectors of states and state disturbances, respectively;
Q = diag(σ2 ηi1 , . . . , σ2 ηi6 ) is the variance term of state errors. With reference to (7.11)
we are ultimately interested in the factor loadings β i,t−1, which are based only on
information available at date t − 1. Hence, filtered state estimates as defined in (3.13)
serve as time series estimates of time-varying factor loadings.
In order to assess the relative superiority of the proposed dynamic specification with
conditional factor loadings modeled as individual random walks, three alternative models