Properties of Σt
- is spd k-by-k matrix
- has k(k+1)/2 “free parameters”
Challenges of multivariate modeling
- Curse of dimensionality
- Cost of evaluating the likelihood
- Ensuring well-defined dynamics
Aims of multivariate covariance modeling
- Parameterising the dynamics cheaply
- Keeping the dynamics of the model realistic
Alternative parameterisation of Σt
DtRtDt
Where: D is a diagonal matrix that contains the conditional standard deviations (k free parameters)
R is a conditional correlation matrix (k(k-1)/2 free parameters)
Does the correlation matrix change over time?
Yes.
A curse upon it.
VEC/BEKK models
- Generalisations of GARCH models for multivariate data
- Flexible but impractical for large numbers of coefficients
VEC(1,1)
ht = c + Ant-1 + Ght-1
ht = vech(Σt)
nt = vech(rtrt’)
VEC(1,1)
Number of parameters to estimate
k (k+1) (k( k+1) +1)/2
Like basically shitloads.
DVEC
Is a VEC model where the matrices A and G must be diagonal
Has “only” k(k+5)/2 parameters
In case of the DVEC model, conditions to ensure that the conditional covariance is positive definite are typically derived by
expressing the model in terms of Hadamard products
the BEKK model was introduced to
make it easier to estimate Σt in such a way that it remained psd
The BEKK model
Σt = C’C + A(rt−1rt−1‘)A + GΣt−1G
Drawback of BEKK parameterisation
Coefficients are harder to interpret
Pros of factor ARCH model
- easy to interpret
- consistent with financial theory
- number of parameters increases linearly with number of assets
- Estimation can be decomposed in a series of univariate estimations
Cons of factor ARCH model
- Conditionally on market, returns are independent
- Factor loading coefficient is not allowed to change over time
