MODIFICATIONS OF CONDITIONAL QUANTILES OF DEPENDENCE MODELS AS SYSTEMIC RISK MEASURES WITH GRAPH REPRESENTATION - Rahman Hakim, Arief
As one of the statistical properties of a random variable, the quantile plays a crucial role in many statistical problems, including in the computation of the Value-at-Risk (VaR) risk measure in financial and actuarial statistics. The quantile can be determined for a target random variable, given (quantile) values of one or more other random variables. This conditional quantile is frequently constructed through a multiple quantile regression model or classical multivariate dependence models (e.g., normal and t models). As an alternative, this study aims to construct the conditional quantile of a multivariate dependence model following Johnson’s system of unbounded distributions (SU). This model is skewed and heavy-tailed and has finite moments for all orders; thus, its conditional quantile can be modified through the Cornish–Fisher expansion by involving these moments. The results of a numerical simulation using generated data reveal that an estimator for such a modified conditional quantile has higher accuracy with a better conditional coverage probability. To capture complex dependence structures (e.g., tail and asymmetric dependence), multivariate dependence models based on copulas (e.g., elliptical and Archimedean copulas) can be utilized. Accordingly, this study also aims to construct the conditional quantile of these models and then modify it through the Cornish–Fisher expansion. Based on the results of a numerical simulation using generated data, an estimator for the copula-based modified conditional quantile involving estimators for higher-order conditional moments tends to be more accurate if the estimation is performed using the ratio method (compared to the naive method). Its accuracy is much higher if higher-order unconditional moments are taken into consideration. Because the conditional quantile is a value minimizing the conditional expectation of a piecewise-linear loss function, it depends only on the probability of the values of the random variables and disregards their magnitudes. Therefore, this study also aims to construct a conditional generalized quantile by minimizing the conditional expectation of a piecewise-power loss function. Such a conditional generalized quantile is determined for multivariate dependence models based on vine copulas. These copulas can be built from different bivariate copulas and can be represented through tree graphs. Using generated data, the results of a numerical simulation demonstrate that an estimator for the vine copula-based conditional generalized quantile may possess the highest accuracy at some real power between one and two. Its accuracy tends to be better if it is modified by accounting for higher-order conditional moment estimators. Afterward, the results of the conditional (generalized) quantile construction and modification are employed to determine risk measures for systemic risk, i.e., the risk that can propagate from one or more financial entities to the others in a financial system and can threaten its stability. These systemic risk measures include the (multiple) conditional VaR [(M)CoVaR] based on conditional quantile, the (multiple) conditional VaR based on conditional expectile [(M)CoEVaR], and their generalizations. The superiority of the Cornish–Fisher expansion in modifying these systemic risk measures is confirmed based on the results of their estimation using data that consist of the returns of assets from several financial markets. Furthermore, financial systems/markets are represented by financial graphs, whose vertex denotes a financial entity and whose edge denotes (undirected) dependence or (directed) risk propagation. In addition to centrality measures, their characteristics can be explained using a clustering coefficient, which measures the tendency of a vertex and its neighbors (making up path subgraphs of length two) to cluster together and form triangle/cycle subgraphs of length three. In this study, generalizations of the clustering coefficient are proposed by considering cycle and path subgraphs of higher length. These generalizations are applied to a dependence measure-based undirected (un)weighted financial graph or a systemic risk measure-based directed (un)weighted financial graph. The generalization of a continuous weighted clustering coefficient can better explain the clustering tendency of financial entities to be adjacent to each other or propagate risk through longer propagation channels. The results of the conditional (generalized) quantile construction and modification derived in this dissertation research may add contributions to the field of statistics. Their applications as risk measures for systemic risk in several financial systems with graph representation are expected to be beneficial in an effort to quantitatively manage systemic risk and maintain their stability.