Applications of state space models in finance

130 7 A Kalman filter based conditional multifactor pricing model
coefficients of the multifactor model in this chapter will be modeled as individual random
walks estimated via the Kalman filter.
7.2 Specification of a conditional multifactor risk model
Following the review of the standard unconditional multifactor pricing framework, this
subsection derives a conditional multifactor risk model, on which the subsequent empirical
analysis will be based. Subsection 7.2.1 outlines the conditional time series
representation. Subsection 7.2.2 summarizes how to estimate factor risk premia in a
cross-sectional setting and how to conduct inferences about them.
7.2.1 Time series representation
The starting point for modeling the time-varying impact of macroeconomics and fundamentals
on pan-European sector allocation is the general multifactor beta pricing
model as given by (7.3). A multifactor model for the realized excess returns Ri,t with
time-varying betas can be stated in state space form with observation equation
Ri,t = β ′ i,tf t + ɛi,t, ɛi,t ∼ N(0, σ 2 i ), (7.6)
for i = 1, . . . , N and t = 1, . . . , T ; f t and β i,t are the K × 1 vectors of risk factors
and corresponding factor loadings, respectively; ɛi,t is the vector of normally distributed
disturbances with unconditional variance σ2 i . The factor realizations are assumed to be
stationary with unconditional moments
and to be uncorrelated with the error terms:
E(f t) = 0, (7.7)
Cov(f t) = Ωf , (7.8)
Cov(fk,t, ɛi,t) = 0, (7.9)
for all i, k and t.
In accordance with (3.77) the evolution of the k factor loadings βik,t can be modeled
as individual random walks by the following set of K state equations:
βi1,t+1
βi2,t+1
βiK,t+1
=
=
.
=
βi1,t + ηi1,t,
βi2,t + ηi2,t,
βiK,t + ηiK,t,
ηi1,t ∼ N(0, σ 2 ηi1 ),
ηi2,t ∼ N(0, σ 2 ηi2 ),
ηiK,t ∼ N(0, σ 2 ηiK ).
(7.10)
The system of equations (7.6)–(7.10) is a special case of the general state space framework
presented in ¢ 3.2. The assumptions made in (3.3)–(3.7) apply. The constant
variances σ2 i and σ2 ηik represent the K + 1 unknown hyperparameters of the system,
which can be estimated by means of maximum likelihood as discussed in ¢ 3.4. The path
of time-varying factor sensitivities can be tracked and predicted by the Kalman filter
and smoother as outlined in ¢ 3.3.