Applications of state space models in finance

3.7 Illustration: How to specify the MMR model for estimation using SsfPack 41
becomes a constant. For values of σ 2 ς close to zero, the model resembles a mean reverting
model with a constant mean.
In SsfPack, the initial state values are collected in the matrix Σ. In case of the
moving mean model, it is defined as
Σ =
⎡
⎢
⎣
−1 0 0
0 −1 0
0 0 −1
0 0 0
⎤
⎥
⎦ , (3.105)
where the value −1 indicates that the corresponding elements of the initial state vector
are diffuse. If OLS estimates were used as initial values, the diagonal elements of the first
three rows would be replaced by the corresponding elements of the estimated covariance
matrix; the last row would contain the estimated OLS regression coefficients as initial
values for the corresponding means of the state vector.
The lower part of the parameter matrix Φt = [T zt] ′ contains the time-varying
system element zt with the explanatory variables. In SsfPack, an index matrix J Φ,
whose index refers to the explanatory data in zt, is required to specify the time-varying
elements in Φt. The two matrices Φt and J Φ must be of same dimension. With the
exception of those elements for which time-variation in the corresponding elements of
Φt should be indicated, the elements of the index matrix take the value −1. An index
value greater than −1 refers to the column of zt with the time-varying values. For the
moving mean model, the index matrix is given by
J Φ =
⎡
⎢
⎣
−1 −1 −1
−1 −1 −1
−1 −1 −1
0 1 2
⎤
⎥
⎦ . (3.106)
Notice that indexing in Ox generally starts at zero. Therefore, the (4,1) element of J Φ
refers to the first column of zt.
As the corresponding elements in the last row of J Φ already indicate where the data
for the respective columns of zt come from, the values associated with zt in the last
row of Φt are set to zero when dealing with time-variation. In the current example Φt
becomes
Φt =
⎡
⎢
⎣
T11 0 0
0 T22 0
0 0 1
0 0 0
⎤
⎥
⎦ . (3.107)
Generally, SsfPack allows to assign time-varying elements to any system matrix by using
the index matrices J δ, J Ω and J Φ to indicate the time-varying elements of δt, Ωt and
Φt, respectively (cf. Zivot et al. 2002).