Covariance Modeling

Question 1

Properties of Σ_t

Answer 1

• is spd k-by-k matrix
• has k(k+1)/2 “free parameters”

Question 2

Challenges of multivariate modeling

Answer 2

1. Curse of dimensionality
2. Cost of evaluating the likelihood
3. Ensuring well-defined dynamics

Question 3

Aims of multivariate covariance modeling

Answer 3

• Parameterising the dynamics cheaply
• Keeping the dynamics of the model realistic

Question 4

Alternative parameterisation of Σ_t

Answer 4

D_tR_tD_t

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)

Question 5

Does the correlation matrix change over time?

Answer 5

Yes.

A curse upon it.

Question 6

VEC/BEKK models

Answer 6

• Generalisations of GARCH models for multivariate data
• Flexible but impractical for large numbers of coefficients

Question 7

VEC(1,1)

Answer 7

h_t = c + An_t-1 + Gh_t-1

h_t = vech(Σ_t)

n_t = vech(r_tr_t’)

Question 8

VEC(1,1)

Number of parameters to estimate

Answer 8

k (k+1) (k( k+1) +1)/2

Like basically shitloads.

Question 9

DVEC

Answer 9

Is a VEC model where the matrices A and G must be diagonal

Has “only” k(k+5)/2 parameters

Question 10

In case of the DVEC model, conditions to ensure that the conditional covariance is positive definite are typically derived by

Answer 10

expressing the model in terms of Hadamard products

Question 11

the BEKK model was introduced to

Answer 11

make it easier to estimate Σ_t in such a way that it remained psd

Question 12

The BEKK model

Answer 12

Σt = C’C + A(r_t−1r_t−1‘)A + GΣ_t−1G

Question 13

Drawback of BEKK parameterisation

Answer 13

Coefficients are harder to interpret

Question 14

Answer 14

Question 15

Answer 15

Question 16

Answer 16

Question 17

Answer 17

Question 18

Pros of factor ARCH model

Answer 18

• 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

Question 19

Cons of factor ARCH model

Answer 19

• Conditionally on market, returns are independent
• Factor loading coefficient is not allowed to change over time