Lecture 13

Question 1

What if a marginal distribution isn’t Gaussian ?

Answer 1

Impossible define joint distribution:

• Case when 2 variables = different marginal distribution
• Case for large number of marginal distribution → multivariate does not exist

→ use copula models

Question 2

What are copulas ?

Answer 2

Relate two marginal distributions instead 2 series directly:

• Able to relate any kind of margin
• Possible to generate non-linear dependence

→ Pearson’s correlation not appropriate measure of dependence

Question 3

What is the base of a copula ?

Answer 3

• 2 rv X and Y with marginal distributions Fx = Pr[X ≤ x] and Gy = Pr[Y ≤ y]
• Cdfs continuous
• Joint distribution H(x,y)= Pr[X ≤ x, Y ≤ y]
• All function’s range = [0,1]

→ H might not exist

Question 4

What is the definition of a bivariate copula ?

Answer 4

Function C : [0,1] x [0,1] → [0,1]

Question 5

What are the properties of a bivariate copula ?

Answer 5

• C(u,v) increases in u and v : if one marginal = cst, the other one increases

• C(u,0) = 0, C(u,1) = u , C(0,v) = 0, C(1,v) = v
o If one probability = 0 then joint also
o If one probability = 1 then joint determined by remaining one

• Pr[u1 ≤ U ≤ u2, v1 ≤ V ≤ v2] = C(u2,v2) – C(u2,v1) – C(u1,v2) + C(u1,v1) ≥ 0

Question 6

What is copula’s theorem ?

Answer 6

H = joint distribution of C and Y with marginal distribution F, G then :

• Exist copula C s.t. H(x,y) = C [F(x), G(y)]

→ if F and G continuous, C = unique

Question 7

What are the various measures of dependence and concordance ?

Answer 7

• Dependence = strength of relation between 2 variables
• Association = positive or negative relation
• Concordance = association where small values of one imply small values of the other and same for big values → X1 < X2 → Y1 < Y2 = (X1 -X2) (Y1 -Y2) > 0
• Discordant = inverse of concordant → (X1 -X2) (Y1 -Y2) < 0

Question 8

What is the measure of concordance between rv X and Y ? What are its properties ?

Answer 8

κ(x,y)

• Defined for every pair of rv = completeness
• Normalized measure -1≤ κ ≤1 and κ(x,x) =1 and κ(x,-x) = -1
• Symmetric : κ(y,x) = κ(x,y)
• If X and Y are independent → κ(x,y) =0
• κ(x,-y) = κ(-x,y) = - κ(x,y)

→ measure of concordance = invariant w.r. to linear increasing transformations

Question 9

Is Pearson’s correlation a measure of concordance ?

Answer 9

Only under normality, and is a measure of association

Question 10

What are Kendall’s Tau and Spearman’s rho ?

Answer 10

Measures of concordance

Question 11

What does it mean when two series are comonotonic ?

Answer 11

κ(x,y) = 1

Question 12

What does it mean when two series are counter-monotonic ?

Answer 12

κ(x,y) = -1

Question 13

What is Kendall’s tau for 2 rv ?

Answer 13

Probability of concordance - probability of discordance of two independent pairs

Question 14

What is Spearman’s rho ?

Answer 14

multiple of probability of concordance - probability of discordance of 2 independent pairs

Question 15

What happens if X2 and Y3 are independent in Spearman’s rho ?

Answer 15

ρs = distance between joint distribution of (X,Y) and independence

Question 16

How can Spearman’s rho also be viewed ?

Answer 16

Pearson’s correlation between F and G or ranks of X and Y

Question 17

What is the Pearson’s correlation ?

Answer 17

Natural scalar of linear dependence in elliptical distributions → misleading measure of dependence in more general situation

Question 18

What are Pearson’s correlation’s properties ?

Answer 18

• ρ[X,Y] = invariant under linear transformations only

• ρ[X,Y] = bounded: -1 ≤ ρL ≤ ρ[X,Y] ≤ ρU ≤ 1
o ρU = comonotonic and ρL = counter-monotonic

• ρ[X,Y] for comonotonic (counter-monotonic) can be different from 1 (-1)
• ρ[X,Y] = 0 does not imply independence between X and Y

Question 19

What is the first approach to modeling of non-linear dependence ?

Answer 19

estimating unrestricted joint density non-parametrically

→ deduce non-parametric estimate of associated unrestricted copula

Question 20

What are the advantages and disadvantages of empirical copulas ?

Answer 20

• Advantage
o Not require any additional assumption on non-linear-dependence

• Drawback
o Complicated interpretations of patterns of non-linear dependence
o Likely to provide inaccurate and erratic results

Question 21

What are the special cases of Elliptical copula

Answer 21

• Normal distribution
• T distribution
• Cauchy distribution
• Laplace distribution
• Uniform distribution

Question 22

What is the condition for an Elliptical copula ?

Answer 22

• Random vector X ϵ R^n has multivariate elliptical distribution if density∶f(x)=|Σ|^(-1/2) g[(X-μ) Σ^(-1) (X-μ)] for some g∶ R→R+where Σ=PD
• Contours of equal density form ellipsoids in R^n

Question 23

What are Archimedean Copulas ?

Answer 23

Copulas that are not derived from multivariate distribution functions

Question 24

What is the Archimedean’s theorem ?

Answer 24

φ = continuous, strictly decreasing function from [0,1] to [0,∞) s.t. φ(1) =0 and φ^(-1) = inverse of φ. Function fro [0,1]^2 to [0,1] : C(u,v) = φ^(-1) [φ(u) + φ(v)] = copula only if φ = convex

Question 25

What are Archimedean copulas’ advantage ?

Answer 25

Most of them have closed expressions

Question 26

What is φ for Archimedean Copulas ?

Answer 26

Generator of copula.

Question 27

What is C’s properties under archimedean copulas ?

Answer 27

• Symmetric: C(u,v) = C(v,u)
• Associative
• τ(C)=1+4∫φ(u)/φ(v) du

Question 28

What are Clayton copula’s properties ?

Answer 28

• φ(t)=(t^(-θ)-1)/θ for θϵ(0,∞)→C(u,v)=(u^(-θ)+v^(-θ)-1)^(-1/θ)
• Density of copula∶C(u,v)=(1+θ) (uv)^(-θ-1) (u^(-θ)+v^(-θ)-1)^(-2-(1/θ) )
• τ(C)=θ/(θ+2)

Question 29

What is on drawback of Clayton copula and the solution ?

Answer 29

Dependence only in lower tail → need rotated copula

Question 30

What is Clayton’s rotated copula )

Answer 30

Cr = (u.v) = u+v-1+C(1-u,1-v)

Question 31

What are Gumbel copula’s properties ?

Answer 31

• When φ(t)=[-logt]^θ for θ∈[1;∞)
• C(u,v) = exp{-[(-logu)^θ + (-logv)^θ]^(1/θ)}
• τ(C)=1-(1/θ)
• Rotated Gumbel same as rotated Clayton but has dependence in lower tail

Question 32

What are the various approaches to estimate the parameters of copula ?

Answer 32

• Standard ML estimation
• 2 steps estimation
• Semi-parametrically estimation
• Method of moments

Question 33

Why is MLE difficult to implement in practical application for copulas ?

Answer 33

• Dimension of optimization can be very large

* No analytical expression of gradient of likelihood

Question 34

What is the two-steps estimation procedure for copulas ?

Answer 34

Separation of vector of parameters into different parts → margin and copula

• Step 1 = estimation of margins
• Step 2 = estimation conditionally θγ of copula

• If model correctly specified, estimator consistent and asymptotically normal
o √T (θIFM-θo)~N(0,Ωo^((-1) ))
o Ωo=Ao^((-1) ) BoAo^(-1)

Question 35

What are the semi-parametric ML’s properties ?

Answer 35

• Avoids specifying margins → use marginal empirical cdf
• Obtain estimator of copula by maximizing pseudo-likelihood
• θγ = asymptotically normal with larger asymptotic variance than MLE (obtained assuming margins are known)

Question 36

What is convenient with the Method of moments while estimating copula ?

Answer 36

• Simpler estimator of copula’s parameter

* Equalize theoretical quantiles with empirical