The paper provides detailed guidance on selecting suitable collections of auxiliary outcomes and combining TMO with existing spatial standard errors. STATA code for implementing TMO is available here:
github.com/wjnkim/tmo
arxiv.org/abs/2504.13295

Applying TMO to nine recent papers, we find significant impacts on estimated standard errors, with a median increase of 37% compared to the published estimates.

(2) Determine a correlation threshold from these estimates; pairs exceeding this threshold are modeled as correlated.
(3) Compute standard errors by accounting only for correlations above the threshold.

Our proposed method, Thresholding Multiple Outcomes (TMO), has three steps:
(1) Estimate pairwise correlations across locations using multiple outcomes.

The main idea of this paper is to use collections of outcomes, of this form, to identify which location pairs should be allowed to correlate when constructing standard errors in regression problems.

This suggests geographic proximity alone inadequately captures spatial dependence. Even adding population as a covariate doesn’t fully resolve the issue. While several covariates predict high correlations, no single factor completely captures the dependence structure.

Here's a correlogram for counties in CA, NY, and ND, sorted by state and population. Urban counties in CA correlate more strongly with urban counties in NY than with rural counties in CA. Rural CA counties correlate more closely with ND counties than with urban areas within CA.

Are these methods appropriate for the types of dependence that we might expect for economic data? We assess this by collecting 91 U.S. county-level outcomes (unemployment, income, etc) and computing the correlation, across outcomes, between each pair of counties

About half of the papers in top-5 economics journals in 2023 analyze data indexed by geographic locations. Typically, these papers handle spatial dependence by clustering SEs at a higher aggregation level or by modeling dependence based on geographic distance (e.g., Conley SEs).

Thanks Cameron!