A preprint estimates that, under the realized pattern of Opportunity Zone designation in California, treated tracts had a 4.5-percentage-point direct housing-growth contrast. It also estimates a 1.4-point neighborhood-spillover contrast and a 5.9-point total contrast. The 90% credible interval for the direct estimate ran from 1.4 to 7.8 points, while the spillover contrast for untreated tracts was small and statistically insignificant.
The expansion scenario is more cautionary. In the paper’s policy scenarios, the estimated direct contrast declined, the spillover contrast rose and the total contrast eventually became negative. Those are posterior point estimates, however: the reported 90% credible intervals for both the direct and total contrasts included zero throughout the expansion range.
A model for connected decisions
The paper tackles settings in which treatment selection is endogenous and outcomes may spill over across a large network or spatial setting. Its framework estimates three policy-relevant quantities: a direct treatment contrast, a spillover contrast and a total contrast.
Identification rests on an excluded instrument—an input used in the selection part of the model but not the outcome part—and the paper assumes that it is both relevant and exogenous. In the California specification, that variable is partisan alignment between a tract’s state representative and the governor; poverty rate, median earnings and employment rate enter both equations.
The model handles unobserved variation with a finite mixture of multivariate normal components and fixes the first diagonal covariance element at one to set the scale of the binary selection equation. Its Bayesian estimation fills in a latent treatment index and a missing potential outcome, then cycles through regression, mixture-weight and covariance updates in a Gibbs sampler.
What the simulations showed
The authors first tested the framework on simulated network data. Their Monte Carlo exercise compared the Spillover Roy Model, or SRM, with a Non-Spillover Roy Model, or NSRM, at sample sizes of 500, 1,000 and 2,000 per replication. It ran 1,000 replications, used 11,000 MCMC iterations with the first 1,000 draws discarded as burn-in, and measured bias, root mean squared error and 95% interval coverage.
The model that included spillovers performed better in those tests. Across representative marginal treatment-effect grids—estimated gains at different levels of latent resistance and exposure—the SRM showed small bias, RMSE generally fell as the sample grew and coverage stayed close to the nominal 95% level. NSRM estimates remained substantially biased, with coverage worsening sharply at low and high exposure.
The California estimates
The empirical application covered 3,699 California census tracts eligible for the program: 727 designated Qualified Opportunity Zones and 2,972 eligible tracts that were not designated. The outcome was housing-unit growth from 2017 to 2022, and neighborhood exposure was the row-normalized share of neighboring designated tracts.
In the preferred specification, partisan alignment had a posterior mean of 0.162, with a 90% interval from 0.049 to 0.277. The estimated QOZ-exposure coefficient was 0.032, with an interval from 0.009 to 0.055. By contrast, the non-QOZ-exposure estimate was 0.009, with an interval from -0.006 to 0.024, and the treated-regime selection-correlation estimate was 0.182, with an interval from -0.063 to 0.412.
Those results describe a heterogeneous picture rather than one fixed effect. Estimated marginal treatment effects fell as latent resistance—the model’s hidden measure of resistance to participation—rose at every exposure level. Greater neighborhood exposure shifted them upward modestly, but estimated gains became negative toward the upper end of the resistance range.
Expansion shifts who benefits
That pattern helps explain the expansion results. Spillover estimates were largest for induced entrants and also positive for tracts that were always treated. Estimates for never-treated tracts were smaller, and their 90% credible intervals included zero at every reported policy shift.
Across the policy scenarios, the posterior mean direct contrast declined and eventually became negative under sufficiently large expansions. The spillover contrast rose, while the total contrast fell and eventually became negative as well. The paper presents this as a sign of diminishing point-estimate returns, not a settled finding: the 90% intervals for direct and total contrasts included zero across the reported range.
A conditional result
These findings are model-based, not randomized proof that designation caused housing-unit growth. The identification argument depends on partisan alignment being both relevant and exogenous, while the model assumes the finite-mixture disturbance structure and the scale restriction for the binary selection equation.
Simulation recovery shows how the method behaves in generated network settings; it does not establish that the identification assumptions hold in the California data. The evidence covers eligible California tracts and one outcome—housing-unit growth from 2017 to 2022—so the reported estimates remain conditional on this setting and on the model’s exposure mapping and policy counterfactuals.
Paper data and sources
Original title: Endogenous Selection and Spillovers: Bayesian Inference for Policy-Relevant Causal Effects
Authors: Duong Trinh
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text