Directional spatial autoregressive dependence in the conditional first- and second-order moments

Abstract

In contrast to classical econometric approaches which are based on prespecified isotropic weighting schemes, we suggest that the spatial weighting matrix in the presence of directional dependencies should be estimated. We identify this direction based on different candidate neighbourhood sets. In this paper, we consider two different types of processes – namely spatial autoregressive and spatial autoregressive conditional heteroscedastic processes – and derive the consistency of the corresponding maximum likelihood estimates in the presence of directional dependencies. Moreover, Monte Carlo simulation results indicate that the model’s performance improves with sample size and with smaller neighbourhood subset sizes. Finally, we apply this approach to aerosol observations over the North Atlantic region and show that their spatial dependence matches the direction of the trade winds in this area.

Introduction

Spatial dependence can occur in various forms when analysing georeferenced data. Depending on whether geostatistical or spatial econometric methods are used, the spatial dependence structure can be characterised by the covariance between the observations at different locations, or by a spatial weighting matrix that connects each location to all other locations. In the latter case, it is often assumed that the dependence structure is known and can be adequately represented by a priori fixed specifications such as $q$-nearest neighbours, inverse-distance relations or binary contiguity matrices. More precisely, these dependence structures are typically assumed to be isotropic — i.e. they are only subject to the distances between the observations. However, real-world processes are often characterised by irregular patterns, such as anisotropic or directional dependence structures. In contrast to isotropic processes that are invariant under spatial rotation, anisotropy exhibits properties that vary with direction. In this paper, we refer particularly to directional processes as a special case of anisotropic pattern with purely one-sided dependence, similar to a temporal process where the present is solely determined by past events and not by future ones. Directional processes in space may involve various meteorological response variables (e.g., Bárdossy and Hörning, 2017), fluxes of pollutants or chemical compounds in a river (e.g., Cressie et al., 2006, Ver Hoef et al., 2006, Peterson et al., 2007, Álvarez-Cabria et al., 2016), the dispersion of aerosols due to currents or wind direction (e.g., Tai et al., 2010, Merk and Otto, 2020) and species distribution (Blanchet et al., 2008), among others.

Several procedures have been suggested to account for directional spatial autocorrelation in the literature regarding spatial econometrics. These include two-dimensional correlograms (Oden and Sokal, 1986) or bearing correlograms (Rosenberg, 2000). Moreover, the concept of asymmetrical spatial effects has been addressed as part of origin–destination flows (LeSage and Pace, 2008) to model migration, knowledge (Ertur and Koch, 2007) or journey-to-work flows (Griffith and Jones, 1980). However, to the best of our knowledge, there have been few studies regarding the estimation of the unknown connections of a weighting matrix in the likely presence of directional spatial dependencies. This concept, therefore, is the focus of this paper.

For spatiotemporal data, different data-driven approaches have been suggested to estimate the spatial weighting matrix (e.g., Bhattacharjee and Jensen-Butler, 2013, Ahrens and Bhattacharjee, 2015, Otto and Steinert, 2018, Lam and Souza, 2019). Basak et al. (2018) suggested identifying a lower triangular spatial weighting matrix from the covariance matrix under the assumption of causal ordering in a recursive network.

Fewer studies have considered the recovery of the spatial dependence structure for purely spatial data. Zhu et al. (2010) suggested penalised maximum likelihood (ML) estimation for selecting a neighbourhood structure from a linear combination of different weighting matrices reflecting symmetric neighbourhood sets of different orders. Bhattacharjee et al. (2012) proposed symmetry as a structural assumption for estimating a spatial weighting matrix in hedonic pricing models. In contrast to the assumption of symmetry, this paper considers the opposite case: that of asymmetrical, directional dependence. We suggest recovering the spatial dependence in a purely spatial setting by specifying competing directional neighbourhood sets, which can be estimated and compared using the ML approach.

The distinction between directional and non-directional processes is important for a variety of spatial applications. In the following sections, we cover two different processes. First, spatial autoregressive (SAR) models have received much attention in the literature of spatial econometrics (e.g., Anselin, 1988, LeSage and Pace, 2009). For these kinds of processes, it is generally assumed that neighbouring observations resemble each other, and that spatial dependence can be modelled through conditional first-order moments. In contrast, the second model covers spatial autoregressive conditional heteroscedastic (ARCH) processes exhibiting spatial dependence in conditional variances resulting in clusters of high (or low) volatility. ARCH processes were originally introduced to model time series data (Engle, 1982) but have been extended to spatial settings by Otto et al. (2018).

In the following sections, we provide the framework for SAR and spatial ARCH models, introduce an approach for identifying directional dependencies (Section 2.1) and elaborate on the ML estimation of directional spatial AR and ARCH models (Section 2.2). Section 3 reports the results of a Monte Carlo simulation emphasising the performance of these estimation approaches in the presence of directional spatial dependencies. In Section 4, we evaluate the dispersion of aerosols in the North Atlantic region that is characterised by trade winds, before providing a conclusion in Section 5.