A simultaneous spatial autoregressive model for compositional data

Abstract

In an election, the vote shares by party on a given subdivision of a territory form a vector with positive components adding up to 1 called a composition. Using a conventional multiple linear regression model to explain this vector by some factors is not adapted for at least two reasons: the existence of the constraint on the sum of the components and the assumption of statistical independence across territorial units questionable due to potential spatial autocorrelation. We develop a simultaneous spatial autoregressive model for compositional data which allows for both spatial correlation and correlations across equations. We propose an estimation method based on two-stage and three-stage least squares. We illustrate the method with simulations and with a data set from the 2015 French departmental election.

Spatial Simultaneous Autoregressive Models for Compositional Data: Application to Land Use

“…In this section, after recalling some classical facts about compositional data analysis, we recall the models from [Chakir and Lungarska, 2017] and [Nguyen et al, 2019] with a common notation and refer to the first one as to the UniSEM model and to the second one as to the MultiLAG model.…”

“…As detailed in [Pawlowsky- Glahn et al, 2015], the previous expression comes from the usual product matrix definition in the coordinate space for an ilr transformation: B u = ilr −1 (B * ilr (u)) where B * = V BV. We also have…”

“…Among land use models, we restrict attention to econometric formulations based on simultaneous spatial autoregressive models (SAR) because they are common in the econometric land use literature (see for instance [LeSage and Pace, 2009]) and easy to fit. There are alternative approaches based on conditional autoregressive models such as [Pirzamanbein et al, 2018] and [Leininger et al, 2013] but they require MCMC techniques, see [Nguyen et al, 2019] for further details.…”

“…Our starting point is the application to land use in [Chakir and Lungarska, 2017] and [Lungarska and which is based on a spatial error model (SEM) in alr coordinates. The objective, using [Nguyen et al, 2019], is to propose and compare alternative models based on alternative coordinate systems (ilr instead of alr) and alternative model formulations in the same family (LAG instead of SEM as we will explain later on). Using the same dataset, we estimate both models and question model specification and interpretation.…”

“…First of all, the first approach specifies the model in an alr coordinate space whereas the second one uses an ilr coordinate space, see e.g. [Pawlowsky- Glahn et al, 2015] for the definition of alr and ilr transformations. Secondly the first approach uses a spatial error model formulation for each coordinate separately while the second approach uses a joint multivariate spatial lag model.…”