Sparse-smooth spatially varying coefficient quantile regression

doi:10.1016/j.spasta.2026.100983

Sparse-smooth spatially varying coefficient quantile regression

2026-08-01

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Haerdle, Wolfgang Karl

DOI：10.1016/j.spasta.2026.100983

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Spatial nonstationarity is commonly studied with local regression methods, but spatial quantile regression estimators can be unstable under irregular sampling and heavy-tailed noise, and they often produce coefficient surfaces that vary everywhere. We propose a sparse-smooth spatially varying coefficient quantile regression (SS-SVCQR) estimator that decomposes each coefficient into a global baseline and a spatial deviation. A group l(2) penalty shrinks an entire deviation field to zero, yielding an automatic global-versus-local decision for each covariate. For covariates selected as local, deviations are stabilized by a normalized graph Laplacian penalty defined on a proximity graph of the sampling locations. The resulting objective is convex; we develop scalable solvers based on ADMM and a Moreau-smoothed proximal-gradient scheme to handle the non-differentiable check loss. Under regularity conditions, global coefficients are root-n consistent and asymptotically normal, and the global/local selection is consistent. Simulations and an application to Lucas County house prices illustrate improved quantile prediction and an interpretable global-local structure compared with kernel-and spline-based alternatives, particularly under heavy-tailed errors.

Keywords:

Quantile regression

Spatially varying coefficients

Graph Laplacian

Variable selection

Group lasso

ADMM

Sparse-smooth spatially varying coefficient quantile regression

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