COST

“General” dependence

Various alternatives have been suggested and tested to different degrees including exceedence probabilities, rank correlations, Pearson correlations, and quantiles of conditional distributions. A key task of this theme is to investigate the strengths and weaknesses of each of these approaches and develop further approaches for eliciting dependence. A further challenge when eliciting dependency between different quantities is that of dimensionality. For n independent variables we can think of the elicitation process as specifying 3n qunatiles (3 quantile per variable) but if all of these variables are dependent there are a further n(n-1)/2 correlations needed in the elicitation session. The development of methods which can overcome this dimensionality issue in a practical and defensible way is another key task for this theme.

Tail dependence

Ignoring a particular type of dependence, e.g. tail dependence, might also lead to what some called a “Recipe for Disaster”  (see http://www.wired.com/techbiz/it/magazine/17-03/wp_quant?currentPage=all#). The tail dependence measures the probability of extreme values for one variable given that another assumes an extreme value too. The silent assumption of tail independence has been credited with excessive risk taking on Wall Street.

For tail independence,  the take home is "if something very bad happens to X, there's no reason for extra concern about Y, even if X and Y are positively correlated". Tail dependent shows a very different behaviour. This behaiviour may be interpreted as follows: If something bad happens to X, there is a good reason to fear something bad will happen to Y, and the reasons get better as the coupling between X and Y increases.

The dependence in the graph from Figure 1 was induced using the normal copula, this is the dependence structure of the joint normal distribution. The normal copula has tail independence  and most commercial packages use the normal copula.

Note what happens when we use the Gumbel copula to realize the pair wise micro correlation of r=0.01(see Figure 2) . The distribution of the sum of 100 standard normals acquires a longish right tail (note the scale on the horizontal axis).  This illustrates that micro correlations not only inflate the variance, they may also fatten the tails.  This fact is known, but the implications still are not fully understood by many specialists. 

The copula effect is usually smaller than the effect of neglecting dependence altogether. However, it becomes more important as it propagates through many variables. 

For information on possible ways to elicit dependence and tail dependence click here (link to Icesheet dependence elicitation.docx)

For the effects of dependence on ice sheet uncertainty click here (link to ESREL talk)