Applications of state space models in finance

34 3 Linear Gaussian state space models and the Kalman filter
Following Chow (1984) a general expression for the basic univariate time-varying
coefficient regression model with homoskedastic error terms can be formulated as
yt = ztξ t + ɛt, ɛt ∼ N(0, h), (3.73)
where zt denotes a 1 × m row vector with m = k fixed explanatory variables, of which
unity might be the first element. The column vector ξ t contains the regression coefficients
whose behavior over time is represented by
ξ t+1 = T ξ t + Rη t, η t ∼ N(0, Q), (3.74)
where the system matrices T and R are assumed to be time-homogeneous. To allow for a
meaningful economic interpretation, the state equation given in (3.74) can be rearranged
by introducing the mean state vector ¯ ξ and by replacing ξ t by ξ t − ¯ ξ. This leads to
a representation in which ¯ ξ can be interpreted as the mean coefficient over the entire
sample. The matrix T is referred to as the speed parameter, which measures how fast
the time-varying state vector returns to its mean:
ξ t+1 − ¯ ξ = T (ξ t − ¯ ξ) + Rη t. (3.75)
The literature has not arrived yet at a consensus on how to introduce time-variation
into the coefficients of explanatory variables. In the following, it will be demonstrated
how different processes for the time-varying state vector can be derived depending on
the chosen value for T .
3.5.2.1 The random coefficient model
Setting T to an m × m zero matrix yields a model that describes the time path of a
changing ξ t as random coefficients (RC):
ξ t+1 = ¯ ξ + Rη t. (3.76)
The random coefficient model, originally introduced by Hildreth and Houck (1968) in a
cross-sectional context, implies a long-run average coefficient around which the current
estimate fluctuates randomly. The parameters to be estimated are ¯ ξ and the variances
of the error terms. As it is not possible to distinguish between a randomly behaving
intercept and the observation disturbances, any intercept term has to be included as
being fixed in the observation equation. This reduces the state vector’s dimension by
one (cf. Wells 1996, ¢ 5.3).
Overall, the practical relevance of this model is limited: as the stochastic properties
of the underlying process are only reflected in the state disturbances, the average state
always represents the best forecast. For a review of the random coefficient model, see,
for example, Nicholls and Pagan (1985).
3.5.2.2 The random walk model
By setting the transition matrix to an m-dimensional identity matrix, one can derive
an important modeling class according to which the behavior of the changing regression