Applications of state space models in finance

6.2 Modeling conditional betas 101
the two most important sources of non-Gaussianity are volatility clustering and outliers.
As neither of the Kalman filter based approaches above is capable of fully coping with
these issues, it might be sensible to remove these influences altogether. For this purpose,
a three-stage estimation procedure following Ghysels et al. (1996) is developed:
1. Correct the observations for heteroskedasticity.
2. Cut down the remaining outliers.
3. Apply the Kalman filter based random walk model to the transformed observations.
In order to account for ARCH effects, less weight should be placed on data points
with high conditional volatility. In the OLS context, this approach is referred to as
generalized least squares (GLS). Following the outline by Davidson and MacKinnon
(2004, ¢ 7), who give a general introduction to GLS and related topics, the concept of
GLS can be summarized as follows: in a standard linear regression model of the form
y = Xβ + ɛ, ɛ ∼ IID(0, Ω), (6.23)
the parameter vector β can only be estimated efficiently by least squares if the residuals
are uncorrelated and homoskedastic. Whenever these assumptions are violated, an
efficient GLS estimator of β can be found by appropriately transforming the regression
so that the Gauss-Markov conditions are satisfied. The corresponding transformation
depends on Ψ, a quadratic matrix that is usually triangular with
Ω −1 = ΨΨ ′ . (6.24)
Premultiplication of (6.23) by Ψ ′ yields the transformed regression model that can be
estimated by OLS to obtain efficient estimates:
The GLS estimator for β is given as
ˆβ GLS
Ψ ′ y = Ψ ′ Xβ + Ψ ′ ɛ. (6.25)
= (X ′ ΨΨ ′ X) −1 X ′ ΨΨ ′ y = (X ′ Ω −1 X) −1 X ′ Ω −1 y. (6.26)
In case of heteroskedastic but uncorrelated errors, the covariance matrix is diagonal and
a GLS estimator can be obtained by means of weighted least squares (WLS). Each observation
is weighted proportionally to the inverse of the nonconstant diagonal elements
of Ω. With w 2 t
denoting the t-th element of Ω and w−1
t
denoting the t-th element of Ψ,
for a typical observation at time t, the transformed regression model in (6.25) can be
written as
w −1
t yt = w −1
t Xtβ + w −1
t ɛt. (6.27)
The dependent and independent variables are simply multiplied by w −1
t , where the
weight observations depend negatively on the variance of the disturbance term. It can
be shown that the variance of the disturbances is equal to unity.
As the precise form of the covariance matrix is usually unknown in practice, a consistent
estimate of Ω can be employed to get feasible GLS estimators. A common way to