Lecture 13 Flashcards
What if a marginal distribution isn’t Gaussian ?
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
What are copulas ?
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
What is the base of a copula ?
- 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
What is the definition of a bivariate copula ?
Function C : [0,1] x [0,1] → [0,1]
What are the properties of a bivariate copula ?
• 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
What is copula’s theorem ?
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
What are the various measures of dependence and concordance ?
- 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
What is the measure of concordance between rv X and Y ? What are its properties ?
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
Is Pearson’s correlation a measure of concordance ?
Only under normality, and is a measure of association
What are Kendall’s Tau and Spearman’s rho ?
Measures of concordance
What does it mean when two series are comonotonic ?
κ(x,y) = 1
What does it mean when two series are counter-monotonic ?
κ(x,y) = -1
What is Kendall’s tau for 2 rv ?
Probability of concordance - probability of discordance of two independent pairs
What is Spearman’s rho ?
multiple of probability of concordance - probability of discordance of 2 independent pairs
What happens if X2 and Y3 are independent in Spearman’s rho ?
ρs = distance between joint distribution of (X,Y) and independence
How can Spearman’s rho also be viewed ?
Pearson’s correlation between F and G or ranks of X and Y
What is the Pearson’s correlation ?
Natural scalar of linear dependence in elliptical distributions → misleading measure of dependence in more general situation
What are Pearson’s correlation’s properties ?
• 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
What is the first approach to modeling of non-linear dependence ?
estimating unrestricted joint density non-parametrically
→ deduce non-parametric estimate of associated unrestricted copula
What are the advantages and disadvantages of empirical copulas ?
• 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
What are the special cases of Elliptical copula
- Normal distribution
- T distribution
- Cauchy distribution
- Laplace distribution
- Uniform distribution
What is the condition for an Elliptical copula ?
- 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
What are Archimedean Copulas ?
Copulas that are not derived from multivariate distribution functions
What is the Archimedean’s theorem ?
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
What are Archimedean copulas’ advantage ?
Most of them have closed expressions
What is φ for Archimedean Copulas ?
Generator of copula.
What is C’s properties under archimedean copulas ?
- Symmetric: C(u,v) = C(v,u)
- Associative
- τ(C)=1+4∫φ(u)/φ(v) du
What are Clayton copula’s properties ?
- 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)
What is on drawback of Clayton copula and the solution ?
Dependence only in lower tail → need rotated copula
What is Clayton’s rotated copula )
Cr = (u.v) = u+v-1+C(1-u,1-v)
What are Gumbel copula’s properties ?
- 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
What are the various approaches to estimate the parameters of copula ?
- Standard ML estimation
- 2 steps estimation
- Semi-parametrically estimation
- Method of moments
Why is MLE difficult to implement in practical application for copulas ?
- Dimension of optimization can be very large
* No analytical expression of gradient of likelihood
What is the two-steps estimation procedure for copulas ?
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)
What are the semi-parametric ML’s properties ?
- 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)
What is convenient with the Method of moments while estimating copula ?
- Simpler estimator of copula’s parameter
* Equalize theoretical quantiles with empirical
