Statistical models are at best approximations to reality; in practice, we have to presume that all models are misspecified. Since quantile regression (QR) belongs to the family of robust statistical models and requires only mild assumptions on the error distribution, it seems to be a natural candidate to be analyzed under misspecification. Yet, with few exceptions research on QR has focused on correct specification of the model only.
 
We analyze linear QR models under misspecification and present two new results: First, we prove an estimation consistency result about high-dimensional misspecified QR problems. Second, we derive a generalized Akaike information criterion (AIC) for model selection in smoothed QR problems. For the first time, we propose an AIC-type model selection criterion for QR that has a quantile dependent penalty term which penalizes model complexity and model misspecification.
 
Along with our methodological developments for misspecified QR models, we also make several technical and theoretical contributions: We prove a new implicit function theorem for Hadamard differentiable functionals in general Banach spaces, derive a Bahadur-type representation for misspecified quantile regression models, and propose a second-order asymptotic representations for an approximate solution to the QR problem.