RESAPLE: An Approximate One-Step Restricted Likelihood Estimator of Spatial Dependence for Exploratory Spatial Analysis

Spatial autocorrelation diagnostics such as Moran's index and the approximate profile likelihood-based estimators (APLE) are widely used to assess spatial dependence in areal data.

Yet, although Moran's index is frequently applied to regression residuals and APLE is typically formulated for raw outcomes, neither is explicitly derived as an estimator of residual spatial dependence after adjusting for large-scale trends and covariates.

We propose RESAPLE, a one-step approximate restricted maximum likelihood (REML) estimator of the spatial error model's spatial dependence parameter $\rho$ based on REML residuals.

Because RESAPLE is a Rayleigh coefficient, it retains the interpretability and diagnostic convenience of exploratory indices, while also providing a computationally inexpensive and accurate estimator of $\rho$ under moderate spatial dependence.

We show theoretically that for small to medium sample sizes and well-specified trend models, RESAPLE is a better estimator of, and competitive test statistic for, residual spatial dependence relative to Moran's index and the APLE across a broad range of spatial configurations.

The theory we develop also yields a diagnostic for spatial weight selection, providing guidance towards resolving a longstanding challenge in spatial data analysis.

We illustrate the method through simulations on both regular and highly irregular lattices with a case study using American Community Survey tract-level data.