This thesis contributes to the modelling of multivariate rare events by combining extreme value theory and graphical modelling.
The central theme involves structural equation models (SEMs), which are powerful tools used to express causal extremal dependence on directed graphical structures. Using these models we address problems concerning hidden confounding, causal discovery and joint dependence modelling of multivariate extremes.

The first part of the thesis considers the recursive max-linear model (RMLM), a particular type of SEM supported on a directed acyclic graph (DAG).
For such model, we address issues that arise from hidden confounders, and provide necessary and sufficient conditions for the representation of the observed node variables as a (reduced) RMLM. In the framework of regularly varying innovations, or noise variables, we establish checkable criteria for the construction of such (sub)models via the extremal dependence measure of the transformed observations. We develop a statistical algorithm that detects bivariate RMLMs.

The second part of the thesis studies linear structural equation models supported on DAGs. In order to perform causal discovery we work with the extreme observations of the model under a regular variation framework. Our methodology is based on extremal scalings, and exploits asymmetries in the extremal dependence structure to identify causal directions between the node variables. We implement this procedure as an algorithm and show that the estimated causal orderings are consistent.

The last part of the thesis studies max-linear SEMs supported on directed, but not necessarily acyclic, graphs, giving rise to richer and more complex types of dependencies such as two-way causality. We work with the max-linear representation of these models, and establish an identifiability result for the corresponding dependence parameters which enables applications to causal discovery and clustering of extremal dependencies. For regularly varying innovations, we develop novel statistical methodology for the estimation of the extremal dependence structure from the empirical angular measure, and show consistency of the estimated dependence parameters. A simulation study and data application in high-dimensional settings show that our procedure performs satisfactorily.