Joint value at risk: a new conditional risk measure

In this PhD thesis a new firm level conditional risk measure is developed. It is named Joint Value at Risk (JVaR) and is defined as a quantile of a conditional distribution of interest, where the conditioning event is a latent upper tail event. It addresses the problem of how risk changes under extreme volatility scenarios. The properties of JVaR are studied based on a stochastic volatility representation of the underlying process. We prove that JVaR is leverage consistent, i.e. it is an increasing function of the dependence parameter in the stochastic representation. A feasible class of nonparametric M-estimators is introduced by exploiting the elicitability of quantiles and the stochastic ordering theory. Consistency and asymptotic normality of the two stage M-estimator are derived, and a simulation study is reported to illustrate its finite-sample properties. Parametric estimation methods are also discussed. The relation with the VaR is exploited to introduce a volatility contribution measure, and a tail risk measure is also proposed. The analysis of the dynamic JVaR is presented based on asymmetric stochastic volatility models. Empirical results with S&P500 data show that accounting for extreme volatility levels is relevant to better characterize the evolution of risk. The work is complemented by a review of the literature, where we provide an overview on quantile risk measures, elicitable functionals and several stochastic orderings.

Multilevel Analysis in Household Survey: An Application to Health Condition Data

The aim of this thesis is to apply multilevel regression model in context of household surveys. Hierarchical structure in this type of data is characterized by many small groups. In last years comparative and multilevel analysis in the field of perceived health have grown in size. The purpose of this thesis is to develop a multilevel analysis with three level of hierarchy for Physical Component Summary outcome to: evaluate magnitude of within and between variance at each level (individual, household and municipality); explore which covariates affect on perceived physical health at each level; compare model-based and design-based approach in order to establish informativeness of sampling design; estimate a quantile regression for hierarchical data. The target population are the Italian residents aged 18 years and older. Our study shows a high degree of homogeneity within level 1 units belonging from the same group, with an intraclass correlation of 27% in a level-2 null model. Almost all variance is explained by level 1 covariates. In fact, in our model the explanatory variables having more impact on the outcome are disability, unable to work, age and chronic diseases (18 pathologies). An additional analysis are performed by using novel procedure of analysis :"Linear Quantile Mixed Model", named "Multilevel Linear Quantile Regression", estimate. This give us the possibility to describe more generally the conditional distribution of the response through the estimation of its quantiles, while accounting for the dependence among the observations. This has represented a great advantage of our models with respect to classic multilevel regression. The median regression with random effects reveals to be more efficient than the mean regression in representation of the outcome central tendency. A more detailed analysis of the conditional distribution of the response on other quantiles highlighted a differential effect of some covariate along the distribution.