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Abstract:
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Nonparametric quantile regression with multivariate covariates is a difficult estimation problem due to the "curse of dimensionality." To reduce the dimensionality while still retaining the flexibility of a nonparametric model, we propose modeling the conditional quantile by a single-index function $g(x\gamma)$, where a univariate link function $g(.)$ is applied to a linear combination of covariates $x\gamma$, often called index. We introduce a practical algorithm where the unknown link function $g(.)$ is estimated by local linear quantile regression and the parametric index is estimated through linear quantile regression. A simple rule-of-thumb bandwidth is adopted. Large sample properties of estimators are studied, which facilitate further inference. Both the modeling and estimation approaches are demonstrated by simulation studies and real data applications.
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