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Main advantage of copula fn over joint distr

- In many situation, when MD aren't Gaussian, it is impossible to define a joint distr

- This is the case when 2 variables have diff MD

- In addition, a lot of MD don't have a multivariate extension

- Sol: copula fns: they have the property to relate 2 MD instead 2 series directly


If non-normal returns, when is MV criterion good?

If MV investor, nothing has to be changed in case of non-normality


Advantages and drawbacks of elliptical vs Archimedean copulas

- Adv of Archimedean: most of them have closed form expression

- Drawback of Archimedean: their multivariate extensions are difficult to establish


Why Pearson's corr coeff not good to measure concordance?

- Measure of association but not necessarily of concordance

- Concordance only under normality

- Pearson's corr is a natural scalar measure of linear dependence in elliptical distributions

- BUT may be misleading in more general situations


Relation btw cdf of joint distr fn and corresponding copula (Sklar theorem)

- Sklar: let H be joint distr fn of X,Y with MDs F,G

- Then, there exists a copula C:(0,1)*(0,1) -> (0,1) s.t. H(x,y)=C(F(x),G(y))

- Furthermore, if F and G are continuous, C is unique


Why is 2-step estimation consistent with copula?

- Since MLE may be difficult to implement, in some cases, the vector of parameters can be split in 2 diff parts: those associated with the margins and with the copula:

1) Estimation of the margins
2) Estimation of the parameters teta_gamma of the copula model conditionally on the margin parameter


How use method of moments for estimating copula parameters?

- Let consider a concordance measure (Kendall's tau or Spearman's rho) which can be easily estimated as a fn of the copula parameters

- Then, an estimation of the parameter is obtained by equalizing the theoretical and empirical quantities


Why Kendall's tau better measure of dependence than Pearson's corr coeff

- Pearson's corr is a valid measure for linear dependence within the elliptical family of distr

- Kendall's tau also captures non-linear associations



Used to describe deviations from normality of innovations in à GARCH framework


Types of distr converge to

- uniform -> Weibull

- normal -> Gumbel

- Student, GARCH -> Fréchet