Lecture 12

Question 1

What is the main characteristic of multivariate distribtuion ?

Answer 1

Dependency parameter that measure strength of link between 2 series

Question 2

How is dependecy measured for # of standard distribution and what familty of distribution ?

Answer 2

Elliptical family and by Person’s (or linear) correlation coefficient

Question 3

On what are most asset allocations based ?

Answer 3

Use of correlation matrix computed over given sample period

Question 4

By what could tail dependence be generated ?

Answer 4

• Dynamic correlations

* Distribution with different levels of dependence

Question 5

How to test consistency of dependency parameter ?

Answer 5

• Test equality of linear combination coefficient computed before and after crash
o May be misleading because conditioning estimation of correlation coefficient on sample period induces estimator bias if variance changes over 2 subperiods

• Test in conditional model
o Estimate joint dynamics of stock mkt returns
o Describe how conditional correlation varies over time

• Need to model joint dynamics of a # of series
o Multivariate GARCH models
o Multivariate distributions or corpulas models

Question 6

In normal distribution, where does the dependency come from ?

Answer 6

Covariance matrix

Question 7

In a multivariate normal distribution, when does the random vector Z ~ N(μ,Σ) ?

Answer 7

If Z = μ + AX with Σ = AA’

Question 8

What are the two possibilities to compute the square root of covariance matrix ?

Answer 8

• Cholesky decomposition

* Spectral decomposition

Question 9

When is the Cholesky more appropriate ?

Answer 9

When natural ranking of assets. In other cases, spectral safest approach

Question 10

What is the main issue for the parameterizations for Σ(θ) ?

Answer 10

Dimensionality of parameter vector when # variables n increases

Question 11

What are the trade offs of the main issue for the parameterization for Σ(θ) ?

Answer 11

• Capturing main statistical features of distribution
• Estimating large # of parameters
• Incorporating additional constraints s.t. covariance matrix > 0 at each t

• Other issues
o Conditional correlation modelled instead of conditional covariance
o Conditional correlation time varying

Question 12

In the Vech GARCH, what are each element of the covariance matrix ?

Answer 12

Linear function of most recent past cross-products of errors and conditional variances and covariances

Question 13

What is the notation of a Vec GARCH(1,1) ?

Answer 13

Vech(Σt)=vech(Ω)+A vech[ϵ(t-1)ϵ(t-1)^’ ]+Bvech[Σ(t-1)]

Question 14

What is the number of unknown parameters in a vech Garch ?

Answer 14

[n(n+1)]/2 ⋅ [1+(2n(n+1))/2]

Question 15

What are the advantages and drawback of the VECH Garch ?

Answer 15

• Very flexible specification but # parameters increase n^4

* Difficult to verify and impose conditional covariance matrices positive definite

Question 16

What is the Diagonal Vech Garch ?

Answer 16

Each element of covariance matrix only depends on corresponding past elements
Σ(t)=Ω+A ° [ϵ(t-1)ϵ(t-1)’ ]+B°Σ(t-1)

Question 17

How do we get a PD Σ(t) ?

Answer 17

Parametrizing model with Cholesky matrices

Question 18

How many parameters are there in the digonal VECH GARCH ?

Answer 18

3[n(n+1)/2]

Question 19

What is the BEKK model and its main advantage ?

Answer 19

Σ(t)=Ω+A’[ϵ(t-1)ϵ(t-1)^’ ]A+B’Σ(t-1)B

Conditional covariance matrix PD if Ω PD

Question 20

What if the models are estimated using the sample covariance matrix for the long-run matrix ?

Answer 20

Reduces # parameters and improves finite sample properties

Question 21

What are the different parametrization possible with the constant term constraint ?

Answer 21

• VECH : vech(Ω)=(I-A-B)vech(S)
• Diagonal Vec : Ω=(ee^’-A-B)°S
• Bekk : Ω=S-ASA^’-BSB^’

→ S=1/T Σϵϵ’

Question 22

What are the problems with 1st generation multivariate GACH model ?

Answer 22

• Problem of dimensionality = difficult to handle with large-dimensional system
• Restrictions for positive definite matrix = often difficult to impose
• Dynamics of correlation : describe dynamics of covariance but not of correlations which are mostly interested in

Question 23

What are the solutions to the problem of dimensionality ?

Answer 23

• Factor GARCH

* Flexible GARCH

Question 24

What are the solution to dynamics of correlation ?

Answer 24

• CCC GARCH

* DCC GARCH

Question 25

What is the idea behind the factor GARCH ?

Answer 25

Joint dynamics of vector of returns can be described using small # of observed factors.

Question 26

What is the CCC model’s name ?

Answer 26

Constant conditional correlation model

Question 27

What are the CCC model’s features ?

Answer 27

• Time-varying conditional covariance parameterized and proportional to product of corresponding conditional std
• Temporal variation Σ(t) determined solely by conditional variance
• Correlation matrix Γ estimated in preliminary step using sample correlation matrix of residuals
• Model useful starting point for multivariate modelling but consistency of conditional correlation = unrealistic

Question 28

What does DCC stand for ?

Answer 28

Dynamic conditional correlation model

Question 29

What is the basic idea for the DCC model ?

Answer 29

Conditional correlation matrix Γ = time varying → conditional covariance matrix : Σ = D^(1/2) Γ D^(1/2)

Question 30

What are the estimation issues regarding GARCH ?

Answer 30

• Multivariate GARCH estimated ≈ univariate
• Log likelihood : logL(T)(θ) = Σlogl(θ)
• θ(ML) asymptotically normal : √T (θ-θo)~N[0,A^(-1)]

Question 31

What if the distribution is correctly specified ?

Answer 31

Ao = outer product of gradients

→ practical since requires first-order derivatives = more stable than 2nd

Question 32

On what is based the two-step estimation of DCC ?

Answer 32

Parameters of conditional variances (θv) and conditional correlations (θc) can be estimated separately → works with normal, not with a t

Question 33

What is the second step regarding the estimation of DCC ?

Answer 33

Estimate parameters pertaining to correlation matrix, conditional on parameters estimated in first stage

Question 34

On what relies the two-step estimation of DCC =

Answer 34

Maximizing loglikelhiood
• Estimate volatility parameters through : θv ϵ argmax logL(T)(θv)

• Estimate correlation parameters through : θc ϵ argmax logL(T)(θc)

→ consistent and asymptotically normal with distribution √T (θ-θo)~N[0,Ao^(-1)BoAo^(-1)]

Question 35

What is due to the structure of Ao in estimation of DCC ?

Answer 35

Asymptotic variances of GARCH parameter θv for each series = standard robust covariance matrix estimators

Question 36

What is a normal mixture ?

Answer 36

Normal extension of normal distribution

Question 37

What is the idea of a normal mixture ?

Answer 37

Introduce randomness into covariance matrix via positive mixing variable W

Question 38

What do the class of normal mixture distribution show ?

Answer 38

Lack of correlation does not necessarily imply independence of components

Question 39

What do normal mixture distributions allow ?

Answer 39

Introducing non-linear dependence between components

Question 40

Why are multivariate distribution difficult to estimate ?

Answer 40

• Hard to find distribution describing properties of several series simultaneously
• Multivariate distribution involve lot of parameters