Nonparametric estimation of conditional VaR and expected shortfall

Weighted Nadaraya-Watson Estimation of Conditional Expected Shortfall

“…We compare the WNW estimator with the double kernel estimator of Cai and Wang (2008). As expected, the double kernel estimator contains an additional bias term carried over from the extra smoothing.…”

“…Actually, conditions (A1)-(A5) correspond to Conditions 1-3 in Masry and Fan (1997) except that we require that nh 3 → ∞. The condition that nh 3 → ∞ is not assumed in Cai (2001Cai ( , 2002 and Cai and Wang (2008) but is required for the asymptotics of the WNW method; see the proof of Lemma 1. Condition (A6) corresponds to condition (B7) in Cai (2001).…”

“…Specifically, he used the Nadaraya-Watson (NW) type double kernel estimator of the conditional density and the conditional quantile associated with the estimated conditional density. Later, based on the same approach, Cai and Wang (2008) proposed another nonparametric estimator of the CES. Instead of the NW double kernel estimator, they used the newly proposed weighted double kernel local linear estimator of the conditional density.…”

“…We refer to the proposed estimator as the WNW estimator of the CES. Unlike the double kernel estimators of Scaillet (2005) and Cai and Wang (2008), the WNW estimator contains only a single smoothing. We establish asymptotic normality of the WNW estimator under an α-mixing condition and show that it is design adaptive at interior points of the support of the design distribution.…”

“…In Section 3, we investigate sampling properties of the proposed estimator. In Section 4, we compare the proposed estimator with the double kernel estimator of Cai and Wang (2008). In Section 5, we report a simulation study.…”