Renormalized Maximum Likelihood for Spatial Lag Models

Model selection is critical in spatial econometrics, particularly when specifying the degree of spatial dependence in regression models. Traditional criteria such as Akaike’s Information Criterion (AIC), Bayesian Information Criterion (BIC), and the Hannan-Quinn Criterion (HQC) are widely applied but do not adaptively account for the complexity introduced by spatial autocorrelation. This paper evaluates the performance of the Renormalized Maximum Likelihood (RNML) criterion, which incorporates a data-driven penalty derived from the Fisher Information Matrix to balance model fit and complexity. Through Monte Carlo simulations, we assess RNML’s capacity to accurately recover the true spatial dependence structure under varying degrees of spatial autocorrelation and sample sizes. We complement these simulations with three empirical applications: the spatial distribution of GDP per capita across European NUTS-2 regions, the spread of COVID-19 incidence across Italian provinces, and the share of foreign-born residents across Germany's NUTS-3 regions. These examples span diverse spatial scales and policy domains, economic performance, public health, and migration, providing a comprehensive evaluation of RNML in practice. Our findings consistently show that RNML tends to select higher spatial autoregressive coefficients compared to classical criteria, especially in settings with pronounced spatial dependence or higher spatial resolution. This adaptiveness enhances RNML’s reliability for robust inference in spatial modeling, particularly where spatial spillovers and feedback effects are substantively important.