High-dimensional Index Volatility Models via Stein's Identity
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In this paper, we consider estimating the parametric components of index volatility models, whose variance function has semiparametric form with two common index structures: single index and multiple index. Our approach applies the first- and second-order Stein's identities on the empirical mean squared error (MSE) to extract the direction of true signals. We study both low-dimensional setting and high-dimensional setting under finite moment condition, which is weaker than existing literature and makes our estimators applicable even for some heavy-tailed data. From our theoretical analysis, we prove that the statistical rate of convergence has two components: parametric rate and nonparametric rate. For the parametric rate, we achieve √(n)-consistency for low-dimensional setting and optimal/sub-optimal rate for high-dimensional setting. For the nonparametric rate, we show it's asymptotically bounded by n^-4/5 under both settings when the mean function has bounded second derivative, so it only contributes high-order terms. Simulation results also back our theoretical conclusions.
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