Applications of state space models in finance

3.5 Introduction of explanatory variables 35
coefficients over time can be described as a random walk (RW):
ξ t+1 = ξ t + Rη t. (3.77)
The random walk is considered an essential scientific concept in the analysis of dynamics
across many disciplines. It had already been widely established in the natural sciences
when Samuelson (1965) introduced it to finance. Today, the random walk represents
one of the most important ideas in financial economics.
An illustrative example of the applicability of random walks to economics is the timevarying
parameter model for U.S. monetary growth presented by Kim and Nelson (1989).
Excellent surveys of the random walk and its link to financial markets and security prices
include Campbell et al. (1997, ¢ 2) and Lo and MacKinlay (1999).
3.5.2.3 The mean reverting model
Another important time-varying coefficient model is the mean reverting (MR) model,
proposed by Rosenberg (1973). The regression coefficients are modeled as a stationary
process:
ξ t+1 − ¯ ξ = T (ξ t − ¯ ξ) + Rη t, (3.78)
where ξ t is a vector of stochastic parameters with non-stochastic mean ¯ ξ in stochastic
equilibrium. For values of the roots of T inside the unit circle, stationarity of the
stochastic process is guaranteed and the coefficients move around the constant mean ¯ ξ.
The parameters to be estimated are ¯ ξ, T , Q and h. Following Collins et al. (1987) it is
assumed a priori that the major determinants of any time-varying regression coefficients
change only gradually over time. To assure smoothly developing coefficients from period
to period, the diagonal elements of T are restricted to values between zero and unity. For
T = I, Equation (3.78) reduces to a random walk. For T = 0, the model is equivalent
to the random coefficient model.
The state equation of the mean reverting model differs from the general state equation
in (3.74) by the incorporation of the parameter ¯ ξ. Substitution of (3.78) into (3.73)
brings the mean reverting model into the same form as (3.70)–(3.71):
� ξ ∗ t+1
¯ξ
� � ∗
ξt = T ¯ξ
� ∗
ξt ¯ξ
yt = zt
�
+ Rη t, (3.79)
�
+ ɛt, (3.80)
with ξ ∗ t = ξ t − ¯ ξ for all t (cf. Chow 1984). The diagonal elements of T that correspond to
¯ξ are equal to unity. As ¯ ξ is constant over time, the lower part of η t is set to zero. In this
representation, the state vector contains both time-varying as well as fixed regression
coefficients and can be estimated by the standard Kalman filter.
In the literature, the mean reverting model has been applied to the testing and modeling
of the time-variation of an asset’s systematic risk; see, for example, Bos and Newbold
(1984), Wells (1996, ¢ 5.3) and Brooks et al. (1998).