Applications of state space models in finance

18 3 Linear Gaussian state space models and the Kalman filter
The Kalman filter has originally been applied by engineers and physicists to estimate
the state of a noisy system. The classic Kalman filter application is the example of
tracking an orbiting satellites whose exact position and speed, which are not directly
measurable at any point of time, can be estimated using available data and well established
physical laws. A discussion of engineering-type applications of the Kalman
filter is provided by Anderson and Moore (1979). In economics and finance, we are
regularly confronted with similar situations: either the exact value of the variable of
interest is unobservable or the possibly time-varying relationship between two variables
is unknown. Nevertheless, the propagation of the Kalman filter among econometricians
and applied economists only really began with the introductory works of Harvey (1981)
and Meinhold and Singpurwella (1983).
In contrast to the Box-Jenkins methodology, which still plays an important role in
teaching and practicing time series analysis, the state space approach allows for a structural
analysis of univariate as well as multivariate problems. The different components
of a series, such as trend and seasonal terms, and the effects of explanatory variables
are modeled explicitly. They do not have to be removed prior to the main analysis as
is the case in the Box-Jenkins framework. Besides, state space models do not have to
be assumed to be homogeneous, which results in a high degree of flexibility. This allows
for time-varying regression coefficients, missing observations and calendar adjustments.
Transparency is another important feature of structural models as they allow for a visual
examination of the single components to check for derivations from expectations;
see Durbin and Koopman (2001, ¢ 3.5) for a comparison of the state space framework
and the Box-Jenkins approach.
Early applications of state space models and the Kalman filter to economics include
Fama and Gibbons (1982) who model the unobserved ex-ante real interest rate as a state
variable that follows an AR(1) process. Clark (1987) uses an unobserved-components
model to decompose quarterly real GNP data into the two independent components of a
stochastic trend component and a cyclical component. Another important contribution
is the work of Stock and Watson (1991) who define an unobserved variable, which represents
the state of the business cycle, to measure the common element of co-movements
in various macroeconomic variables. Surveys on the applicability of the state space approach
to economics and finance can be found in Hamilton (1994a) and Kim and Nelson
(1999).
The state space approach offers attractive features with respect to their generality,
flexibility and transparency. The lack of publicly available software to estimate these
models has been the main reason why only relatively few economic and finance related
problems have been analyzed in state space form so far. The subsequent sections aim at
providing a presentation of the Gaussian state space model that is as compact and intuitive
as possible, while being as comprehensive as necessary to render the employment
of this versatile framework by applied researchers possible. More detailed treatments
of state space models are given by Harvey (1989), Harvey and Shephard (1993) and
Hamilton (1994a), among others. An outline with a focus on applications can be found
at Kim and Nelson (1999). If not indicated otherwise, Durbin and Koopman (2001,
¢ 4–7) serve as standard reference for this chapter.