Spatial Scalar‐on‐Function Quantile Regression Model

ABSTRACT This paper introduces a novel spatial scalar‐on‐function (SoF) quantile regression model that extends classical SoF models to account for spatial dependence and heterogeneous conditional distributions. The proposed model incorporates spatial autocorrelation through a spatially lagged response and characterises the entire conditional distribution of a scalar outcome given a functional predictor. To address endogeneity induced by the spatial lag term, we develop two robust estimation procedures based on instrumental variable (IV) strategies. The ‐consistency and asymptotic normality of the proposed estimators are established under mild regularity conditions. We demonstrate through extensive Monte Carlo simulations that the proposed estimators outperform existing mean‐based and robust alternatives, particularly in settings with strong spatial dependence and outlier contamination. We apply our method to high‐resolution environmental data from the Lombardy region in Italy, using daily ozone trajectories to predict daily mean concentrations of particulate matter with a diameter of less than 2.5 micrometers. The empirical results confirm the superiority of our approach in predictive accuracy, robustness and interpretability at various quantile levels. Our method has been implemented in the ssofqrm package.