Band 38, Heft 1-2

We consider a Lévy driven stochastic convolution, also called continuous time Lévy driven moving average model X ⁢ ( t ) = ∫ 0 t a ⁢ ( t - s ) ⁢ d Z ⁢ ( s ) X(t)=\int_{0}^{t}a(t-s)\,dZ(s) , where 𝑍 is a Lévy martingale and the kernel a ( . ) a(\,{.}\,) a deterministic function square integrable on R + \mathbb{R}^{+} . Given 𝑁 i.i.d. continuous time observations ( X i ⁢ ( t ) ) t ∈ [ 0 , T ] (X_{i}(t))_{t\in[0,T]} , i = 1 , … , N i=1,\dots,N , distributed like ( X ⁢ ( t ) ) t ∈ [ 0 , T ] (X(t))_{t\in[0,T]} , we propose two types of nonparametric projection estimators of a 2 a^{2} under different sets of assumptions. We bound the L 2 \mathbb{L}^{2} -risk of the estimators and propose a data driven procedure to select the dimension of the projection space, illustrated by a short simulation study.

The debate of which quantitative risk measure to choose in practice has mainly focused on the dichotomy between value at risk (VaR) and expected shortfall (ES). Range value at risk (RVaR) is a natural interpolation between VaR and ES, constituting a tradeoff between the sensitivity of ES and the robustness of VaR, turning it into a practically relevant risk measure on its own. Hence, there is a need to statistically assess, compare and rank the predictive performance of different RVaR models, tasks subsumed under the term “comparative backtesting” in finance. This is best done in terms of strictly consistent loss or scoring functions, i.e., functions which are minimized in expectation by the correct risk measure forecast. Much like ES, RVaR does not admit strictly consistent scoring functions, i.e., it is not elicitable. Mitigating this negative result, we show that a triplet of RVaR with two VaR-components is elicitable. We characterize all strictly consistent scoring functions for this triplet. Additional properties of these scoring functions are examined, including the diagnostic tool of Murphy diagrams. The results are illustrated with a simulation study, and we put our approach in perspective with respect to the classical approach of trimmed least squares regression.