Elicitability and backtesting: Perspectives for banking regulation

Conditional forecasts of risk measures play an important role in internal risk management of financial institutions as well as in regulatory capital calculations. In order to assess forecasting performance of a risk measurement procedure, risk measure forecasts are compared to the realized financial losses over a period of time and a statistical test of correctness of the procedure is conducted. This process is known as backtesting. Such traditional backtests are concerned with assessing some optimality property of a set of risk measure estimates. However, they are not suited to compare different risk estimation procedures. We investigate the proposal of comparative backtests, which are better suited for method comparisons on the basis of forecasting accuracy, but necessitate an elicitable risk measure. We argue that supplementing traditional backtests with comparative backtests will enhance the existing trading book regulatory framework for banks by providing the correct incentive for accuracy of risk measure forecasts. In addition, the comparative backtesting framework could be used by banks internally as well as by researchers to guide selection of forecasting methods. The discussion focuses on three risk measures, Value-at-Risk, expected shortfall and expectiles, and is supported by a simulation study and data analysis

Bayesian inference for the half-normal and half-t distributions

In this article we consider approaches to Bayesian inference for the half-normal and half-t distributions. We show that a generalized version of the normal-gamma distribution is conjugate to the half-normal likelihood and give the moments of this new distribution. The bias and coverage of the Bayesian posterior mean estimator of the halfnormal location parameter are compared with those of maximum likelihood based estimators. Inference for the half-t distribution is performed using Gibbs sampling and model comparison is carried out using Bayes factors. A real data example is presented which demonstrates the fitting of the half-normal and half-t models

The Reduced Form of Litigation Models and the Plaintiff\u27s Win Rate

In this paper I introduce what I call the reduced form approach to studying the plaintiff\u27s win rate in litigation selection models. A reduced form comprises a joint distribution of plaintiff\u27s and defendant\u27s beliefs concerning the probability that the plaintiff would win in the event a dispute were litigated; a conditional win rate function that tells us the actual probability of a plaintiff win in the event of litigation, given the parties\u27 subjective beliefs; and a litigation rule that provides the probability that a case will be litigated given the two parties\u27 beliefs. I show how models with very different-looking structure can be understood in common reduced form terms, and I then use the reduced form to prove several general results. First, a generalized version of the Priest-Klein model can be used to represent any other model\u27s reduced form, even though the Priest-Klein model uses the Landes-Posner-Gould ( LPG ) litigation rule while some other models do not. Second, Shavell\u27s famous any-win-rate result holds generally, even in models with party belief distributions that are both highly accurate and identical across plaintiffs and defendants. Third, there are only limited conditions under which the LPG litigation rule can be rejected empirically; this result undermines the case against the LPG rules\u27 admittedly non-optimizing approach to modeling litigation selection. Finally, I use the reduced form approach to clarify how selection effects complicate the use of data on the plaintiff\u27s win rate to measure changes in legal rules. The result, I suggest, is that recent work by Klerman & Lee advocating the use of such data is unduly optimistic

The Reduced Form of Litigation Models and the Plaintiff\u27s Win Rate

In this paper I introduce what I call the reduced form approach to studying the plaintiff\u27s win rate in litigation selection models. A reduced form comprises a joint distribution of plaintiff\u27s and defendant\u27s beliefs concerning the probability that the plaintiff would win in the event a dispute were litigated; a conditional win rate function that tells us the actual probability of a plaintiff win in the event of litigation, given the parties\u27 subjective beliefs; and a litigation rule that provides the probability that a case will be litigated given the two parties\u27 beliefs. I show how models with very different-looking structure can be understood in common reduced form terms, and I then use the reduced form to prove several general results. First, a generalized version of the Priest-Klein model can be used to represent any other model\u27s reduced form, even though the Priest-Klein model uses the Landes-Posner-Gould ( LPG ) litigation rule while some other models do not. Second, Shavell\u27s famous any-win-rate result holds generally, even in models with party belief distributions that are both highly accurate and identical across plaintiffs and defendants. Third, there are only limited conditions under which the LPG litigation rule can be rejected empirically; this result undermines the case against the LPG rules\u27 admittedly non-optimizing approach to modeling litigation selection. Finally, I use the reduced form approach to clarify how selection effects complicate the use of data on the plaintiff\u27s win rate to measure changes in legal rules. The result, I suggest, is that recent work by Klerman & Lee advocating the use of such data is unduly optimistic

A new flexible family of continuous distributions: the additive Odd-G family

This paper introduces a new family of distributions based on the additive model structure. Three submodels of the proposed family are studied in detail. Two simulation studies were performed to discuss the maximum likelihood estimators of the model parameters. The log location-scale regression model based on a new generalization of the Weibull distribution is introduced. Three datasets were used to show the importance of the proposed family. Based on the empirical results, we concluded that the proposed family is quite competitive compared to other models