Applications of state space models in finance

3.6 Model diagnostics 37
3.6.1 Residuals
¢
Following the outline by Harvey (1989, 7.4.1) two sets of time series residuals can
be computed for a state space model with an observation equation as defined in (3.69).
While the generalized recursive residuals are calculated by standardization of the Kalman
filter innovations, the generalized least squares residuals can be obtained by standardization
of the Kalman smoother or disturbance smoother innovations.
3.6.1.1 Generalized recursive residuals
In accordance with the definition of an innovation term given in (3.15), T −d−k one-step
ahead prediction errors can be obtained by applying the Kalman filter to the state space
model defined in (3.70)–(3.71):
v †
t = yt − E(yt|Yt−1) = yt − ztat − xt ˜ β t|t−1 , t = d + k + 1, . . . , T, (3.83)
where ˜ βt|t−1 denotes the ML estimate of βt based on the set of observations up to time
t − 1. Standardizing the recursive residuals by dividing v †
t by √ ft as defined in (3.16)
yields the generalized recursive residuals, denoted as ˜v †
t :
t = v† t √
ft
˜v †
. (3.84)
Without any stochastic model components, the Kalman filter becomes equivalent to
standard OLS recursions such that ˜v †
t reduces to the recursive residuals. This property
and the fact that recursive residuals are independent over time can be used in procedures
to test for structural breaks.
3.6.1.2 Generalized least squares residuals
The application of the Kalman filter to the constructed series of observations yt−xt ˜ β t|T ,
where ˜ β t|T denotes the ML estimate of β t given the complete sample up to time T , yields
T − d least squares residuals
v + t = yt − ztat − xt ˜ β t|T , t = d + 1, . . . , T. (3.85)
In case where the coefficient vector β t is included in the state vector, overall two Kalman
filters are needed: one to obtain ˜ β t|T and another one that is applied to yt − xt ˜ β t|T =
ztξ t + ɛt (cf. Durbin and Koopman 2001, ¢ 6.2.4). Dividing the least squares residuals
by √ ft yields the generalized least squares (GLS) residuals
˜v + t = v+ t √ft . (3.86)
In the absence of any stochastic components, these residuals resemble standard OLS
residuals.
For a detailed description of applying the two types of residuals defined above to
major test procedures for models in state space form, the reader is referred to Harvey
(1989, ¢ 5 & ¢ 7.4).