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

Abstract

The understanding of co-movements, dependence, and influence between variables of interest is key in many applications. Broadly speaking such understanding can lead to better predictions and decision making in many settings. We propose Quantile Graphical Models (QGMs) to characterize prediction and conditional independence relationships within a set of random variables of interest. Although those models are of interest in a variety of applications, we draw our motivation and contribute to the financial risk management literature. Importantly, the proposed framework is intended to be applied to non-Gaussian settings, which are ubiquitous in many real applications, and to handle a large number of variables and conditioning events.We propose two distinct QGMs. First, Condition Independence Quantile Graphical Models (CIQGMs)characterize conditional independence at each quantile index revealing the distributional dependence structure. Second, Prediction Quantile Graphical Models (PQGMs) characterize the best linear predictor under asymmetric loss functions. A key difference between those models is the (non-vanishing) misspecification between the best linear predictor and the conditional quantile functions. We also propose estimators for those QGMs. Due to high-dimensionality, the two distinct QGMs require different estimators. The estimators are based on high-dimensional techniques including (a continuum of) 1 -penalized quantile regressions (and low biased equations), which allow us to handle the potential large number of variables. We build upon a recent literature to obtain new results for valid choice of the penalty parameters, rates of convergence, and confidence regions that are simultaneously valid.We illustrate how to use QGMs to quantify tail interdependence (instead of mean dependence) between a large set of variables which is relevant in applications concerning with extreme events. We show that the associated tail risk network can be used for measuring systemic risk contributions. We also apply the framework to study international financial contagion and the impact of market downside movement on the dependence structure of assets' returns.

Mixed Effect Modeling and Variable Selection for Quantile Regression

“…It was critical in the model selection procedure to use different quantiles (q=0.25, 0.5, 0.75) rather than a single fixed level. There has been a lot of recent interest in multiple quantile graphical model (Ali et al, 2016;Belloni and Chernozhukov, 2011;Belloni et al, 2016;Karpman and Basu, 2018). These models are essentially fitted using a quantile version of the neighborhood selection approach of Meinshausen and Bühlmann (2006) for learning sparse graphical models, which is equivalent to variable selection for penalized quantile regression models.…”

Non-separable models with high-dimensional data

“…Therefore, by averaging observations only when their treatment levels are close to the one of interest, the convergence rates of our estimators are nonparametric, which is in contrast with the √ n-rate obtained in Belloni et al (2017a) and Farrell (2015). Albeit motivated by distinct models, Belloni, Chen, and Chernozhukov (2016) also estimate the irregular identified parameters in the high-dimensional setting. However, the irregularity faced by Belloni et al (2016) is not due to the continuity of the variable of interest.…”

“…Consequently, Belloni et al (2016) do not study the regularized estimator with localization as we do in this paper.…”

“…Albeit motivated by distinct models, Belloni, Chen, and Chernozhukov (2016) also estimate the irregular identified parameters in the high-dimensional setting. However, the irregularity faced by Belloni et al (2016) is not due to the continuity of the variable of interest.…”

LASSO-Driven Inference in Time and Space

“…regressing every factor in the system on a large subset of other factors. Examples include analysis of financial systemic risk by quantile predictive graphical models with LASSO (Hautsch et al, 2015;Härdle et al, 2016;Belloni et al, 2016), limit order book network modeling via the penalized vector autoregressive approach (Härdle et al, 2018), analysis of psychology data with temporal and cross-sectional dependencies (Epskamp et al (2016)). Another example is quantifying the spillover effects or externalities for a social network, especially when the social interactions (or the interconnectedness) is not obvious (Manresa, 2013).…”