Publications by arthur charpentier
That damn R-squared !
Another post about the R-squared coefficient, and about why, after some years teaching econometrics, I still hate when students ask questions about it. Usually, it starts with “I have a _____ R-squared… isn’t it too low ?” Please, feel free to fill in the blanks with your favorite (low) number. Say 0.2. To make it simple, there are differ...
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Unit root, or not ? is it a big deal ?
Consider a time series, generated using set.seed(1) E=rnorm(240) X=rep(NA,240) rho=0.8 X[1]=0 for(t in 2:240){X[t]=rho*X[t-1]+E[t]} The idea is to assume that an autoregressive model can be considered, but we don’t know the value of the parameter. More precisely, we can’t choose if the parameter is either one (and the series is integrated), o...
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Association and concordance measures
Following the course, in order to define assocation measures (from Kruskal (1958)) or concordance measures (from Scarsini (1984)), define a concordance function as follows: let be a random pair with copula , and with copula . Then define the so-called concordance function. Thus As proved last week in class, Based on that function, several co...
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Kendall’s function for copulas
As mentioned in the course on copulas, a nice tool to describe dependence it Kendall’s cumulative function. Given a random pair with distribution , define random variable . Then Kendall’s cumulative function is Genest and Rivest (1993) introduced that function to choose among Archimedean copulas (we’ll get back to this point below). Fro...
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Copulas and tail dependence, part 1
As mentioned in the course last week Venter (2003) suggested nice functions to illustrate tail dependence (see also some slides used in Berlin a few years ago). Joe (1990)’s lambda Joe (1990) suggested a (strong) tail dependence index. For lower tails, for instance, consideri.e Upper and lower strong tail (empirical) dependence...
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Copulas and tail dependence, part 2
An alternative to describe tail dependence can be found in the Ledford & Tawn (1996) for instance. The intuition behind can be found in Fischer & Klein (2007)). Assume that and have the same distribution. Now, if we assume that those variables are (strictly) independent, But if we assume that those variables are (strictly) comonotonic (i.e. eq...
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Copulas and tail dependence, part 3
We have seen extreme value copulas in the section where we did consider general families of copulas. In the bivariate case, an extreme value can be writtenwhere is Pickands dependence function, which is a convex function satisfyingObserve that in this case, where is Kendall’tau, and can be writtenFor instance, if then, we obta...
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(nonparametric) Copula density estimation
Today, we will go further on the inference of copula functions. Some codes (and references) can be found on a previous post, on nonparametric estimators of copula densities (among other related things). Consider (as before) the loss-ALAE dataset (since we’ve been working a lot on that dataset) > library(MASS) > library(evd) > X=lossalae > U=c...
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Interactive 3d plot, in R
Following the course of this afternoon, I will just upload some codes to make interactive 3d plots, in R. > library(rgl) > library(evd); > data(lossalae) > U=rank(lossalae[,1]+rnorm(nrow(lossalae), + mean=0,sd=.001))/(nrow(lossalae)+1) > V=rank(lossalae[,2])/(nrow(lossalae)+1) > M=kde2d(qnorm(U),qnorm(V),n=35) > library(rgl) > persp3d(M$x,M$y,M$z...
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Maximum likelihood estimates for multivariate distributions
Consider our loss-ALAE dataset, and – as in Frees & Valdez (1998) – let us fit a parametric model, in order to price a reinsurance treaty. The dataset is the following, > library(evd) > data(lossalae) > Z=lossalae > X=Z[,1];Y=Z[,2] The first step can be to estimate marginal distributions, independently. Here, we consider lognormal distributio...
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