Choosing the Right Spatial Weighting Matrix in a Quantile Regression Model

This paper proposes computationally tractable methods for selecting the appropriate spatial weighting matrix in the context of a spatial quantile regression model. This selection is a notoriously difficult problem even in linear spatial models and is even more difficult in a quantile regression setup. The proposal is illustrated by an empirical example and manages to produce tractable models. One important feature of the proposed methodology is that by allowing different degrees and forms of spatial dependence across quantiles it further relaxes the usual quantile restriction attributable to the linear quantile regression. In this way we can obtain a more robust, with regard to potential functional misspecification, model, but nevertheless preserve the parametric rate of convergence and the established inferential apparatus associated with the linear quantile regression approach. 1. The Spatial Quantile Regression Model The spatial quantile regression model [1] is a straightforward quantile regression generalisation of the popular, in spatial econometrics, linear spatial lag model. More specifically it can be written as where is a spatially lagged dependent variable, specified via a predetermined spatial weighting matrix , is the design matrix containing the independent variables (covariates), and is a residuals vector. Here we only have one spatially lagged dependent variable but this is not an essential assumption, and more than one spatial weighting matrix can be easily incorporated. This representation is similar to the linear spatial lag regression model, but here coefficients are allowed to vary with the quantile, rather than being assumed fixed. This model has some attractive properties. First, the original motivation for Kostov’s [1] proposal is to alleviate the potential bias arising from inappropriate functional form assumptions in a spatial model. In simple terms the underlying logic is as follows. Omitting spatial dependence typically introduces estimation bias in the presence of spatial lag dependence when the wrong functional form specification is employed. Hence a natural way to circumvent the problem is to estimate the underlying function nonparametrically. The sample sizes used in many empirical studies are however often too small for efficient application of nonparametric methods. Semiparametric methods could then be used to alleviate the problem. The linear quantile regression is such a semi-parametric method. Although it cannot be guaranteed to entirely eliminate the adverse effects of functional form assumptions, such methods can