Applications of state space models in finance

38 3 Linear Gaussian state space models and the Kalman filter
3.6.2 Goodness of fit
Measures of the goodness of fit are usually computed on the basis of forecast errors.
The basic goodness of fit measure for time series models is the prediction error variance,
which also serves as an input to calculate the coefficient of determination. A comparison
between models with different numbers of parameters is commonly made on the basis
of information criteria.
3.6.2.1 Prediction error variance
In case of time-homogeneity, the prediction error variance (p.e.v.) is defined as
σ 2 = σ 2 ∗ ¯ f, (3.87)
where ¯ f represents the value to which ft converges in steady-state. When a concentrated
likelihood function is used with σ 2 ∗ being estimated by (3.61), σ2 can be estimated as
˜σ 2 = ˜σ 2 ∗ ¯ f. (3.88)
The incorporation of regression effects into a model slightly changes the definition of the
. When dealing with time-varying regression models, the ML estimator
estimator of σ2 ∗
of σ2 ∗ for stochastic βt is a function of the generalized recursive residuals:
s 2 ∗
= (T − d − k)−1
T�
t=d+k+1
˜v †2
t , (3.89)
where k is equal to the dimension of the vector of explanatory variables. For fixed β t ,
the ML estimator of σ 2 ∗ depends on the GLS residuals:
˜σ +2
∗
= (T − d)−1
T�
t=d+1
˜v +2
t , (3.90)
where the sums of squares are identical in both cases. Taking (3.89) and (3.90) into
account, the prediction error variance for models that contain explanatory variables can
be estimated as
or as
s 2 = s 2 ∗ ¯ f, (3.91)
˜σ +2 = ˜σ +2
∗ ¯ f. (3.92)
According to Harvey (1989, ¢ 7.4.3) the prediction error variance can be approximated
for large samples in terms of the unstandardized GLS residuals as defined in (3.85):
˜σ 2 =
¯ f
T − d
T�
t=d+1
˜v +2
t
= 1
T − d
T�
t=d+1
v +2
t
ft
¯f ≈ 1
T − d
T�
t=d+1
v +2
t . (3.93)