Yasser ABBAS will defend his thesis on Mathematics on December, Tuesday 17th 2024 at 9:30 am, building TSE (Auditorium 5).
To attend the conference, please contact the secretariat Christelle Fotso Tatchum
Supervisors : Anne Ruiz-Gazen & Abdelaati Daouia
Title: Contributions to heavy tail risk analysis: extreme expectiles and graph theory applications
Memberships are:
- Fabio Bellini, Professor, University of Milano Bicocca, Rapporteur
- Juan Juan Cai, Associate Professor, Vrije Universiteit Amsterdam, Examinatrice
- Abdelaati Daouia, Maitre de conférences, Toulouse School of Economics, Co-directeur de thèse
- Carlos B. Martins-Filho, Professor, University of Colorado at Boulder, Rapporteur
- Anne Ruiz-Gazen, Professeure, Toulouse School of Economics, Directrice de thèse
- Stéphane Villeneuve, Professeur, Toulouse School of Economics, Examinateur
Abstract:
The analysis of rare events at the heavy tails of Pareto-type distributions has received steadily increasing attention in extreme value theory literature, as said distributions are found to be ubiquitous in the study of financial returns, insurance payouts, and rainfall to name a few. Most attempts to quantify these events utilize (un)conditional quantiles, which has quickly become the most prominent risk measure in the field. While quantiles are normally praised for their intuitive definition, there’s been a recent pushback against their prolific use in heavy-tailed scenarios, where they are, by definition, insensitive to the magnitude of the recorded extremes. Paired with their lack of subadditivity, these drawbacks intensified the search for a more refined risk measure, allowing expectiles, quantiles’ least square analogs, to garner interest over the last two decades. Expectiles are the only coherent law-invariant risk measure that also offers elicitability, which facilitates backtesting and validation procedures. In this thesis, we present new innovative frameworks to improve the estimation of expectiles in both the unconditional and regression heavy-tailed cases. Following a detailed summary of the thesis’ contributions in Chapter 1, Chapter 2 tackles the unconditional case, where Weissman extrapolation techniques reign supreme. We challenge this dominance by introducing a completely novel estimation framework that relies on Generalized Pareto fitting. This approach incorporates efficient estimates of both the scale and shape extreme value parameters into the estimation paradigm, and offers two routes: direct asymmetric least squares, and indirect quantile-based estimation. We derive the asymptotic properties of both routes, then evaluate their performance through simulation testing. These frameworks are found to outperform the best available Weissman-type estimators for real-valued profit-loss distributions, while falling slightly behind in positive-valued cases. A rolling-window forecast verification exercise, applied on various sets of financial returns, finds that the Generalized Pareto approach yields more favorable scores across a wide range of target levels. Chapter 3 shifts focus to the regression case, where we utilize a flexible location-scale quantile regression model with heavy-tailed noise that excels in the estimation of both conditional quantiles and expectiles. We develop a general theory that relies on residual-based estimators of extremal regression quantiles and expectiles and explore their asymptotic behavior in generic settings. We then apply this framework to parametric and nonparametric regression scenarios. Simulations show the incredible potential of these methodologies for various distribution types, outperforming the best available conditional expectile estimators. A detailed application to concrete financial data further solidifies their dominance. Chapters 4 and 5 feature a foray into the world of graph theory, and an attempt at deploying the machinery of extreme value theory into these unchartered waters. Chapter 4 introduces novel network data, generated from cutting edge NLP algorithms and Bayesian networks applied to written media output. Using both existing and new graph theory metrics and clustering mechanisms, we extensively explore these dynamic networks and their community structure from January 2018 to January 2022. Paying close attention to the major structural shifts caused by the advent and spread of Covid-19, we manage to paint a clear narrative of the world economy’s evolution during the observed period. Chapter 5 then utilizes this dynamic data to create an early alert system that can automatically detect monumental events ahead of their arrival. Through suitable adjustments to the heavy-tailed estimation and backtesting paradigms explored in earlier chapters, we manage to identify Covid-19 as a major threat before the World Health Organization declared it a pandemic.
