Physically coherent probabilistic weather forecasts using multivariate discrete copula-based ensemble postprocessing methods

Being able to provide accurate forecasts of future quantities has always been a great human desire and is essential in numerous situations in daily life. Meanwhile, it has become routine to work with probabilistic forecasts in the form of full predictive distributions rather than with single deterministic point forecasts in many disciplines, with weather prediction acting as a key example. Nowadays, probabilistic weather forecasts are usually constructed from ensemble prediction systems, which consist of multiple runs of numerical weather prediction models differing in the initial conditions and/or the parameterized numerical representation of the atmosphere. The raw ensemble forecasts typically reveal biases and dispersion errors and thus call for statistical postprocessing to realize their full potential. Several ensemble postprocessing methods have been developed and are partly recapitulated in this thesis, yet many of them only apply to a single weather quantity at a single location and for a single prediction horizon. In many applications, however, there is a critical need to account for spatial, temporal and inter-variable dependencies. To address this, a tool called ensemble copula coupling (ECC) is introduced and examined. Essentially, ECC uses the empirical copula induced by the raw ensemble to aggregate samples from predictive distributions for each location, variable and look-ahead time separately, which are obtained via existing univariate postprocessing methods. The ECC ensemble inherits the multivariate rank dependence pattern from the raw ensemble, thereby capturing the flow dependence. Several variants and modifications of ECC are studied, and it is demonstrated that the ECC concept provides an overarching frame for existing techniques scattered in the literature. From a mathematical point of view, it is shown that ECC can be considered a copula approach by pointing out relationships to multivariate discrete copulas, which are introduced in this thesis and for which relevant mathematical properties are derived. A generalization of standard ECC is introduced, which aggregates samples from not necessarily univariate, but general predictive distributions obtained by low-dimensional postprocessing in an ECC-like manner. Finally, the SimSchaake approach, which combines the notion of similarity-based ensemble methods with that of the so-called Schaake shuffle, is presented as an alternative to ECC. In this technique, the dependence patterns are based on verifying observations rather than on raw ensemble forecasts as in ECC. The methods and concepts are illustrated and evaluated based on case studies, using real ensemble forecast data of the European Centre for Medium-Range Weather Forecasts. Essentially, the new multivariate approaches developed in this thesis reveal good predictive performances, thus contributing to improved probabilistic forecasts

A New Class of Quantile Processes with Applications in Risk Analysis and Valuation

This thesis presents a novel approach for the construction of quantile processes, governing the stochastic dynamics of quantiles in continuous time. Two constructions are proposed, one producing a function-valued quantile process and the second, a process with random quantile levels. The latter method employs a distortion map composed of a distribution function and a quantile function, similar to a transmutation map, applied to each marginal of a 'driving' process with cadlag paths. A multidimensional extension that utilises a copula is also presented. As a result, we obtain a one-step approach to constructing widely flexible classes of stochastic models, accommodating extensive ranges of higher-order moment behaviours (e.g., tail behaviours in the finite dimensional distributions, and asymmetry). Such features are parameterised in the composite map and are thus interpretable with respect to the driving process. Sub-classes of quantile processes are explored, with emphasis placed on the Tukey family of models whereby skewness and kurtosis are directly parameterised and thus the composite map is explicable with regard to such statistical behaviours. It is also shown that the quantile processes induce a distorted probability measure that is interpretable in its properties (which may be intentionally constructed), leading to the central application developed in this thesis. We propose a general, time-consistent, and dynamic risk valuation principle under the induced measures of quantile processes, allowing for pricing in incomplete markets and thus having application in insurance pricing. Here, the distorted measures are considered 'subjective' and are constructed in such a way to account for external market characteristics and investor risk attitudes, leading to a parametric system of risk-sensitive probability measures, indexed by such factors. The properties of the valuation principle based on the quantile process distortion measures are discussed with regard to stochastic ordering and risk-loadings, and a case study is presented where insurance instruments linked to greenhouse gas emissions are considered

Essays in partial identification and applications to treatment effects and policy evaluation

Dans cette thèse, je me suis interessé à l’identification partielle des effets de traitements dans différents modèles de choix discrets avec traitements endogènes. Les modèles d’effets de traitement ont pour but de mesurer l’impact de certaines interventions sur certaines variables d’intérêt. Le type de traitement et la variable d’intérêt peuvent être défini de manière générale afin de pouvoir être appliqué à plusieurs différents contextes. Il y a plusieurs exemples de traitement en économie du travail, de la santé, de l’éducation, ou en organisation industrielle telle que les programmes de formation à l’emploi, les techniques médicales, l’investissement en recherche et développement, ou l’appartenance à un syndicat. La décision d’être traité ou pas n’est généralement pas aléatoire mais est basée sur des choix et des préférences individuelles. Dans un tel contexte, mesurer l’effet du traitement devient problématique car il faut tenir compte du biais de sélection. Plusieurs versions paramétriques de ces modèles ont été largement étudiées dans la littérature, cependant dans les modèles à variation discrète, la paramétrisation est une source importante d’identification. Dans un tel contexte, il est donc difficile de savoir si les résultats empiriques obtenus sont guidés par les données ou par la paramétrisation imposée au modèle. Etant donné, que les formes paramétriques proposées pour ces types de modèles n’ont généralement pas de fondement économique, je propose dans cette thèse de regarder la version nonparamétrique de ces modèles. Ceci permettra donc de proposer des politiques économiques plus robustes. La principale difficulté dans l’identification nonparamétrique de fonctions structurelles, est le fait que la structure suggérée ne permet pas d’identifier un unique processus générateur des données et ceci peut être du soit à la présence d’équilibres multiples ou soit à des contraintes sur les observables. Dans de telles situations, les méthodes d’identifications traditionnelles deviennent inapplicable d’où le récent développement de la littérature sur l’identification dans les modèles incomplets. Cette littérature porte une attention particuliere à l’identification de l’ensemble des fonctions structurelles d’intérêt qui sont compatibles avec la vraie distribution des données, cet ensemble est appelé : l’ensemble identifié. Par conséquent, dans le premier chapitre de la thèse, je caractérise l’ensemble identifié pour les effets de traitements dans le modèle triangulaire binaire. Dans le second chapitre, je considère le modèle de Roy discret. Je caractérise l’ensemble identifié pour les effets de traitements dans un modèle de choix de secteur lorsque la variable d’intérêt est discrète. Les hypothèses de sélection du secteur comprennent le choix de sélection simple, étendu et généralisé de Roy. Dans le dernier chapitre, je considère un modèle à variable dépendante binaire avec plusieurs dimensions d’hétérogéneité, tels que les jeux d’entrées ou de participation. je caractérise l’ensemble identifié pour les fonctions de profits des firmes dans un jeux avec deux firmes et à information complète. Dans tout les chapitres, l’ensemble identifié des fonctions d’intérêt sont écrites sous formes de bornes et assez simple pour être estimées à partir des méthodes d’inférence existantes.In this thesis, I have been interested in the nonparametric (partial) identification of structural potential outcome functions and Average Treatment Effect (ATE) in various discrete models with endogenous selection and treatment. This topic of treatment effect concerns measuring the impact of an intervention on an outcome of interest. The type of treatments and outcomes may be broadly defined in order to be applied in many different contexts. There are many examples of treatment in economics (Labor, health, education, trade, industrial organization) such that Job training programs, surgical procedures, higher education level, research and development investment, being a member of a trade union etc. The decision to be treated or not, is usually not random but is based on individual choices or preferences. In such a context, determining the impact of the treatment becomes an important issue since we have to take into account the selectivity bias. The parametric version of such models has been widely studied in the literature, however in models with discrete variation, the parametrization is a strong source of identification. Then, we don’t know if the empirical results we obtain, are driven by the data or by the parametrization imposed on the model. I propose to look at a fully nonparametric version of those models, in order, to have more robust policy recommendations. The central challenge in this nonparametric structural identification is that the hypothesized structure fails to identify a single generating process for the data, either because of multiple equilibria or data observability constraints. In such cases, many traditional identification techniques become inapplicable and a framework for identification in incomplete models is developing, with an initial focus on identification of the set of structural functions of interest compatible with the true data distribution (hereafter identified set). Therefore, in the first chapter, I provide a full characterization of the identified set for the ATE in a binary triangular system. In the second chapter, I consider a model with sector specific unobserved heterogeneity. I provide the full characterization of the identified set for the structural potential outcome functions of an instrumental variables model of sectoral choice with discrete outcomes. Assumptions on selection include the simple, extended and generalized Roy models. In the last chapter, I consider a binary model with several unobserved heterogeneity dimensions, such as entry and participation games. I provide the full characterization of the identified set for the payoffs in 2 2 games with perfect information, including duopoly entry and coordination games. In all chapters, the identified set of the functions of interest are nonparametric intersection bounds and are simple enough to lend themselves to existing inference methods