The role of the information set for forecasting - with applications to risk management

Predictions are issued on the basis of certain information. If the forecasting mechanisms are correctly specified, a larger amount of available information should lead to better forecasts. For point forecasts, we show how the effect of increasing the information set can be quantified by using strictly consistent scoring functions, where it results in smaller average scores. Further, we show that the classical Diebold-Mariano test, based on strictly consistent scoring functions and asymptotically ideal forecasts, is a consistent test for the effect of an increase in a sequence of information sets on $h$-step point forecasts. For the value at risk (VaR), we show that the average score, which corresponds to the average quantile risk, directly relates to the expected shortfall. Thus, increasing the information set will result in VaR forecasts which lead on average to smaller expected shortfalls. We illustrate our results in simulations and applications to stock returns for unconditional versus conditional risk management as well as univariate modeling of portfolio returns versus multivariate modeling of individual risk factors. The role of the information set for evaluating probabilistic forecasts by using strictly proper scoring rules is also discussed.Comment: Published in at http://dx.doi.org/10.1214/13-AOAS709 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Statistical inference for inverse problems

In this paper we study statistical inference for certain inverse problems. We go beyond mere estimation purposes and review and develop the construction of confidence intervals and confidence bands in some inverse problems, including deconvolution and the backward heat equation. Further, we discuss the construction of certain hypothesis tests, in particular concerning the number of local maxima of the unknown function. The methods are illustrated in a case study, where we analyze the distribution of heliocentric escape velocities of galaxies in the Centaurus galaxy cluster, and provide statistical evidence for its bimodality. --Asymptotic normality,confidence interval,deconvolution,heat equation,modality,statistical inference,statistical inverse problem

Validating linear restrictions in linear regression models with general error structure

A new method for testing linear restrictions in linear regression models is suggested. It allows to validate the linear restriction, up to a specified approximation error and with a specified error probability. The test relies on asymptotic normality of the test statistic, and therefore normality of the errors in the regression model is not required. In a simulation study the performance of the suggested method for model selection purposes, as compared to standard model selection criteria and the t-test, is examined. As an illustration we analyze the US college spending data from 1994