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We propose two types of Quantile Graphical Models (QGMs) --- ConditionalIndependence Quantile Graphical Models (CIQGMs) and Prediction QuantileGraphical Models (PQGMs). CIQGMs characterize the conditional independence ofdistributions by evaluating the distributional dependence structure at eachquantile index. As such, CIQGMs can be used for validation of the graphstructure in the causal graphical models (\cite{pearl2009causality,robins1986new, heckman2015causal}). One main advantage of these models is thatwe can apply them to large collections of variables driven by non-Gaussian andnon-separable shocks. PQGMs characterize the statistical dependencies throughthe graphs of the best linear predictors under asymmetric loss functions. PQGMsmake weaker assumptions than CIQGMs as they allow for misspecification. Becauseof QGMs' ability to handle large collections of variables and focus on specificparts of the distributions, we could apply them to quantify tailinterdependence. The resulting tail risk network can be used for measuringsystemic risk contributions that help make inroads in understandinginternational financial contagion and dependence structures of returns underdownside market movements.  We develop estimation and inference methods for QGMs focusing on thehigh-dimensional case, where the number of variables in the graph is largecompared to the number of observations. For CIQGMs, these methods and resultsinclude valid simultaneous choices of penalty functions, uniform rates ofconvergence, and confidence regions that are simultaneously valid. We alsoderive analogous results for PQGMs, which include new results for penalizedquantile regressions in high-dimensional settings to handle misspecification,many controls, and a continuum of additional conditioning events.

Quantile Graphical Models: Prediction and Conditional Independence with Applications to Systemic Risk