Lecture 12 Flashcards
What is the main characteristic of multivariate distribtuion ?
Dependency parameter that measure strength of link between 2 series
How is dependecy measured for # of standard distribution and what familty of distribution ?
Elliptical family and by Person’s (or linear) correlation coefficient
On what are most asset allocations based ?
Use of correlation matrix computed over given sample period
By what could tail dependence be generated ?
- Dynamic correlations
* Distribution with different levels of dependence
How to test consistency of dependency parameter ?
• 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
In normal distribution, where does the dependency come from ?
Covariance matrix
In a multivariate normal distribution, when does the random vector Z ~ N(μ,Σ) ?
If Z = μ + AX with Σ = AA’
What are the two possibilities to compute the square root of covariance matrix ?
- Cholesky decomposition
* Spectral decomposition
When is the Cholesky more appropriate ?
When natural ranking of assets. In other cases, spectral safest approach
What is the main issue for the parameterizations for Σ(θ) ?
Dimensionality of parameter vector when # variables n increases
What are the trade offs of the main issue for the parameterization for Σ(θ) ?
- 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
In the Vech GARCH, what are each element of the covariance matrix ?
Linear function of most recent past cross-products of errors and conditional variances and covariances
What is the notation of a Vec GARCH(1,1) ?
Vech(Σt)=vech(Ω)+A vech[ϵ(t-1)ϵ(t-1)^’ ]+Bvech[Σ(t-1)]
What is the number of unknown parameters in a vech Garch ?
[n(n+1)]/2 ⋅ [1+(2n(n+1))/2]
What are the advantages and drawback of the VECH Garch ?
- Very flexible specification but # parameters increase n^4
* Difficult to verify and impose conditional covariance matrices positive definite
What is the Diagonal Vech Garch ?
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)
How do we get a PD Σ(t) ?
Parametrizing model with Cholesky matrices
How many parameters are there in the digonal VECH GARCH ?
3[n(n+1)/2]
What is the BEKK model and its main advantage ?
Σ(t)=Ω+A’[ϵ(t-1)ϵ(t-1)^’ ]A+B’Σ(t-1)B
Conditional covariance matrix PD if Ω PD
What if the models are estimated using the sample covariance matrix for the long-run matrix ?
Reduces # parameters and improves finite sample properties
What are the different parametrization possible with the constant term constraint ?
- VECH : vech(Ω)=(I-A-B)vech(S)
- Diagonal Vec : Ω=(ee^’-A-B)°S
- Bekk : Ω=S-ASA^’-BSB^’
→ S=1/T Σϵϵ’
What are the problems with 1st generation multivariate GACH model ?
- 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
What are the solutions to the problem of dimensionality ?
- Factor GARCH
* Flexible GARCH
What are the solution to dynamics of correlation ?
- CCC GARCH
* DCC GARCH
What is the idea behind the factor GARCH ?
Joint dynamics of vector of returns can be described using small # of observed factors.
What is the CCC model’s name ?
Constant conditional correlation model
What are the CCC model’s features ?
- 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
What does DCC stand for ?
Dynamic conditional correlation model
What is the basic idea for the DCC model ?
Conditional correlation matrix Γ = time varying → conditional covariance matrix : Σ = D^(1/2) Γ D^(1/2)
What are the estimation issues regarding GARCH ?
- Multivariate GARCH estimated ≈ univariate
- Log likelihood : logL(T)(θ) = Σlogl(θ)
- θ(ML) asymptotically normal : √T (θ-θo)~N[0,A^(-1)]
What if the distribution is correctly specified ?
Ao = outer product of gradients
→ practical since requires first-order derivatives = more stable than 2nd
On what is based the two-step estimation of DCC ?
Parameters of conditional variances (θv) and conditional correlations (θc) can be estimated separately → works with normal, not with a t
What is the second step regarding the estimation of DCC ?
Estimate parameters pertaining to correlation matrix, conditional on parameters estimated in first stage
On what relies the two-step estimation of DCC =
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)]
What is due to the structure of Ao in estimation of DCC ?
Asymptotic variances of GARCH parameter θv for each series = standard robust covariance matrix estimators
What is a normal mixture ?
Normal extension of normal distribution
What is the idea of a normal mixture ?
Introduce randomness into covariance matrix via positive mixing variable W
What do the class of normal mixture distribution show ?
Lack of correlation does not necessarily imply independence of components
What do normal mixture distributions allow ?
Introducing non-linear dependence between components
Why are multivariate distribution difficult to estimate ?
- Hard to find distribution describing properties of several series simultaneously
- Multivariate distribution involve lot of parameters
