In this paper, semiparametric methods are applied to estimate multivariate volatility functions, using a residual approach as in [J. Fan and Q. Yao, Efficient estimation of conditional variance functions in stochastic regression, Biometrika 85 (1998), pp. 645-660; F.A. Ziegelmann, Nonparametric estimation of volatility functions: The local exponential estimator, Econometric Theory 18 (2002), pp. 985-991; F.A. Ziegel-mann, A local linear least-absolute-deviations estimator of volatility, Comm. Statist. Simulation Comput. 37 (2008), pp. 1543-1564], among others. Our main goal here is two-fold: (1) describe and implement a number of semiparametric models, such as additive, single-index ae class of adaptive functional-coefficient models, choosing simultaneously the bandwidth, the number of covariates in the model and also the single-index smoothing variable. The modified cross-validation algorithm is able to tackle the computational burden caused by the model complexity, providing an important tool in semiparametric volatility estimation. We briefly discuss model identifiability when estimating volatility as well as nonnegativity of the resulting estimators. Furthermore, Monte Carlo simulations for several underlying generating models are implemented and applications to real data are provided
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