🤖 AI Summary
This study addresses the complex nonlinear influence of high-dimensional exogenous variables on the conditional variance of financial volatility by proposing the first nonparametric significance test and consistent variable selection method tailored for ARCH-X type models. The approach models covariate effects as unknown nonparametric functions, constructs a test statistic via an artificial one-way ANOVA framework, and controls the false discovery rate using the Benjamini–Yekutieli procedure. Theoretically, the test statistic is shown to be asymptotically standard normal, and the selection procedure consistently includes all truly relevant variables with probability approaching one. Extensive simulations demonstrate superior performance over existing methods, and an empirical application to S&P 500 volatility confirms its practical utility, offering both theoretical rigor and real-world applicability.
📝 Abstract
We introduce the ARCH-m(X) model, a semiparametric extension of the ARCH-X framework in which the effect of a multivariate exogenous covariate vector X on the conditional variance is modeled through an unknown nonparametric function m(), accommodating complex nonlinear relationships between external predictors and financial volatility. Within this model, we develop a novel hypothesis test for the significance of covariates constructed with an artificial one-way ANOVA. Under some regularity conditions, the test statistic is shown to converge in distribution to the standard Normal. Another key contribution of this paper is the construction of a variable selection procedure based on the Benjamini-Yekutieli false discovery rate correction applied to covariate-level p-values. We show that the resulting index set coincides with the true set of relevant covariates with probability tending to one as n goes to infinity. Extensive simulations confirm that the proposed methods outperform existing competitors, and an empirical application to SP500 return volatility illustrates the practical utility of the proposed variable selection framework.