Applications of state space models in finance

32 3 Linear Gaussian state space models and the Kalman filter
Similarly, whenever a parameter θ should be restricted to the range [−a; a] with a > 0,
it is possible to use the transformation
ψθ =
where the restricted parameter can be obtained as
ˆθ =
θ
√ a 2 − θ 2 , −∞ < ψθ < ∞, (3.65)
a ˆ ψθ
�
1 + ˆ ψ2 , −a <
θ
ˆ θ < a. (3.66)
In case where a parameter θ is only allowed to take on values between zero and unity,
the transformed parameter
�
θ
ψθ =
1 − θ , −∞ < ψθ < ∞, (3.67)
will be used with θ being computed as
ˆθ = ˆ ψ2 θ
1 + ˆ ψ2 , 0 ≤
θ
ˆ θ < 1. (3.68)
With these transformations, the ML estimates are obtained in two steps. At first, the
likelihood is maximized with respect to the unconstrained but transformed parameters.
In a second step, the estimates for the transformed parameters are converted back to
the constrained original parameters.
3.5 Introduction of explanatory variables
The general state space model can be extended to include explanatory variables that are
able to partially explain the movements of the vector of observations. Subsection 3.5.1
introduces regression effects to the state space framework. Subsection 3.5.2 discusses
the estimation of regression models with time-varying parameters and develops various
dynamic processes for changing regression coefficients. The section closes with a
brief discussion of how to set the initial values needed to initialize the Kalman filter in
Subsection 3.5.3.
3.5.1 Incorporation of regression effects
A state space model can be allowed for regression effects by setting dt in the observation
equation (3.2) equal to the linear combination xtβ t. Given an expanded set of information
with k observable variables, which are assumed to be weakly exogenous 8 and to be
linearly related to the univariate vector of observations yt, the observation equation can
be stated as
yt = ztξ t + xtβ t + ɛt, t = 1, . . . , T, (3.69)
8 According to the definition provided by Harvey (1989, 7.1.4) explanatory variables are
weakly exogenous if they can be dealt with as though they do not change in repeated samples.
A formal treatment of the concept of exogeneity can be found in Engle et al. (1983).