Micro Correlations and Tail Dependence

An elementary though seemingly underappreciated finding shows that small global correlations are amplified by aggregation. We observe this behavior in flood damage claims in the US. We also observe that upper tail dependence seems to be amplified by aggregation in these data. We seek to understand this behavior. For sums of exponential variables which are conditionally independent given a gamma-distributed rate, we derive explicit expressions for upper tail dependence and prove that it goes to one as the number of summands goes to infinity, and that the lower tail dependence is zero. We also study sums of events under a latent variable model, where each event occurs if a uniform variable exceeds a threshold, and all uniform variables are conditionally independent given a “latent variable”. We obtain a necessary and sufficient condition for strong asymptotic upper tail dependence as the number of summands goes to infinity. Curiously, the normal copula satisfies this condition, although it is not tail dependent via the usual definition. Thus, sums of events under the normal copula latent variable model have upper tail dependence increasing to 1. We also identify tail dependent-like behavior in finite sums of events with the latent variable model.