  We study the estimation of the parametric components of single and multiple
index volatility models. Using the first- and second-order Stein's identities,
we develop methods that are applicable for the estimation of the variance index
in the high-dimensional setting requiring finite moment condition, which allows
for heavy-tailed data. Our approach complements the existing literature in the
low-dimensional setting, while relaxing the conditions on estimation, and
provides a novel approach in the high-dimensional setting. We prove that the
statistical rate of convergence of our variance index estimators consists of a
parametric rate and a nonparametric rate, where the latter appears from the
estimation of the mean link function. However, under standard assumptions, the
parametric rate dominates the rate of convergence and our results match the
minimax optimal rate for the mean index estimation. Simulation results
illustrate finite sample properties of our methodology and back our theoretical
conclusions.
