Kernel density estimation for time series data

A time-varying probability density function, or the corresponding cumulative distribution function, may be estimated nonparametrically by using a kernel and weighting the observations using schemes derived from time series modelling. The parameters, including the bandwidth, may be estimated by maximum likelihood or cross-validation. Diagnostic checks may be carried out directly on residuals given by the predictive cumulative distribution function. Since tracking the distribution is only viable if it changes relatively slowly, the technique may need to be combined with a filter for scale and/or location. The methods are applied to data on the NASDAQ index and the Hong Kong and Korean stock market indices.     

Response to the Letter to the Editor on ‘On Quantile-based Asymmetric Family of Distributions: Properties and Inference’

Rubio (2020) points out an identification problem for the four-parameter family of two-piece asymmetric densities introduced by Nassiri & Loris (2013). This implies that statistical inference for that family is problematic. Establishing probabilistic properties for this four-parameter family however still makes sense. For the three-parameter family, there is no identification problem. The main contribution in Gijbels et al. (2019a) is to provide asymptotic results for maximum likelihood and method-of-moments estimators for all members of the three-parameter quantile-based asymmetric family of distributions.     

Heterogeneity dependence between oil prices and exchange rate: Evidence from a parametric test of Granger causality in quantiles

This paper re-examines the nexus between crude oil price and exchange rate by investigating their heterogeneity dependence structure within the framework of Granger causality in quantiles for a sample of developed and emerging economies (namely UK, Canada, Brazil, Russia, Mexico, Norway, India, Japan, South Africa, South Korea and European Union (EU)). The results indicate no distinct causality between the crude oil price changes and the real exchange rate returns for all countries besides Russia at the median of the conditional distribution. Besides, the crude oil price changes influence the exchange rate returns in all countries, except Norway and EU, particularly around the tails of the conditional distributions of exchange rate returns. This suggests that the oil price changes influence the real exchange rate returns when the real exchange rate returns are either in extreme appreciation or depreciation. Moreover, the crude oil price movement can be explained by the exchange rate returns for most oil importers only when the crude oil market is bearish or bullish. By contrast, the real exchange rate can permanently affect the crude oil price for most oil-importing countries irrespective of the crude oil market's state. Finally, our findings provide an essential reference for managing the extreme risk dependence between the exchange rate market and the crude oil market.     

Risk functionals with convex level sets

We analyze the “convex level sets” (CxLS) property of risk functionals, which is a necessary condition for the notions of elicitability, identifiability, and backtestability, popular in the recent statistics and risk management literature. We put the CxLS property in the multidimensional setting, with a special focus on signed Choquet integrals, a class of risk functionals that are generally not monotone or convex. We obtain two main analytical results in dimension one and dimension two, by characterizing the CxLS property of all one‐dimensional signed Choquet integrals, and that of all two‐dimensional signed Choquet integrals with a quantile component. Using these results, we proceed to show that under some continuity assumption, a comonotonic‐additive coherent risk measure is co‐elicitable with Value‐at‐Risk if and only if it is the corresponding Expected Shortfall. The new findings generalize several results in the recent literature, and partially answer an open question on the characterization of multidimensional elicitability.     

OPTIMAL DEMAND FOR CONTINGENT CLAIMS WHEN AGENTS HAVE LAW INVARIANT UTILITIES

We consider a class of law invariant utilities, which contains the rank‐dependent expected utility (RDU) and the cumulative prospect theory (CPT). We show that the computation of demand for a contingent claim when utilities are within that class, although not as simple as in the expected utility (EU) case, is still tractable. Specific attention is given to the RDU and to the CPT cases. Numerous examples are fully solved.