Learning which places move together could improve economic error bars

How much independent information does a dataset really contain? Thousands of counties do not necessarily provide thousands of independent observations. Their economic outcomes can move together, and ignoring those connections can make an estimated relationship look more precise than it is.

In an arXiv preprint, Stefano DellaVigna, Guido Imbens, Woojin Kim and David M. Ritzwoller propose learning some of those connections from additional outcomes measured in the same locations. Their method, Thresholding Multiple Outcomes, or TMO, adjusts regression standard errors: the quantities used to describe an estimate’s statistical uncertainty.

Connections need not follow borders

Common corrections allow errors to correlate within predefined groups, such as states, or account for dependence using geographic distance. But economic similarity does not necessarily stop at a state border or diminish neatly with miles.

The researchers illustrate this using 50 outcomes for 3,060 U.S. counties. Across standardized outcome levels, San Francisco resembles other densely populated counties, including distant New York County. Rural Modoc County in California instead resembles some distant, less populated counties more closely than nearby urban ones. These comparisons motivate looking beyond geography; they do not establish causal connections between counties.

The key proposal is to use auxiliary outcomes—other measured variables—to learn which pairs of places may have correlated errors in the regression being studied.

How the correction works

A regression’s residual is the difference between an observed outcome and the value predicted by the fitted model. TMO begins with residuals for the main outcome and each auxiliary outcome, accounting for the regression’s explanatory variables.

The procedure standardizes auxiliary residuals and removes their fitted relationship with the main outcome’s residuals. This second step helps limit selection effects that could otherwise arise when the auxiliary outcomes closely track the main outcome.

It then estimates a correlation for each pair of locations across the processed auxiliary outcomes. Pairs whose estimated correlation is sufficiently large in absolute value are allowed to contribute cross-location dependence to the variance calculation. Importantly, auxiliary outcomes select the pairs; products of the main regression’s residuals supply their covariance contributions. The original regression coefficient is unchanged.

Choosing the cutoff is crucial. Too high a threshold misses real connections; too low a threshold admits too many noisy ones. The authors develop a selection rule that penalizes unstable variance estimates, rather than simply choosing the threshold producing the largest estimate. In their applications, the search permits at most 5% of eligible location pairs to be selected.

A substantial improvement in a favorable simulation

The team built a simulation using county correlations estimated from their outcome collection. They organized counties into 99 groups, retained estimated correlations within those groups and set correlations between groups to zero. They then generated artificial outcomes with a known zero regression coefficient and repeated the experiment 1,000 times.

In the main comparison, ordinary heteroskedasticity-robust standard errors averaged only half the true standard error. State clustering and two geographic corrections produced averages around 54–55% of the truth. TMO reached 91%.

The corresponding tests falsely rejected the zero-effect hypothesis in 34% of simulations with the ordinary robust calculation, 29% with state clustering and 9% with TMO. That is an improvement, not perfect calibration.

The authors explicitly acknowledge that this simulation favors TMO: auxiliary outcomes share the target outcome’s correlation structure, and connections are sparse. More diffuse geographic dependence may favor other methods.

Real studies show sensitivity, not a known answer

The researchers also reanalyzed seven county-level papers from a survey of economics research published in 2023, plus studies using commuting zones and countries. Eight other county-level papers could not be reproduced because essential variables were confidential.

Across the seven county applications, TMO increased standard errors by a median of 39% relative to a baseline without cross-location adjustment. That baseline used ordinary robust errors for cross-sectional data and location-clustered errors for panels. It was not necessarily the correction originally published in each paper.

Larger error bars are not automatically more accurate. In these real datasets, the true standard errors are unknown.

The assumptions remain consequential

TMO assumes that auxiliary outcomes approximately share the main outcome’s residual correlation structure, and that most location pairs lack meaningful dependence. Its formal threshold analysis also uses a simplified Gaussian, equal-variance setting with additional restrictions.

Outcome choice matters: near-copies of the main outcome can lose much of their useful variation during processing, while unrelated outcomes may have the wrong spatial structure. The authors therefore present TMO as a complement to geographic corrections, not a universal replacement. Its contribution is a way to use existing measurements to question familiar uncertainty estimates—without claiming that more variables eliminate uncertainty.