Design-Based Inference for Spatial Experiments under Unknown Interference

Abstract

We consider design-based causal inference for spatial experiments in which treatments may have effects that bleed out and feed back in complex ways. Such spatial spillover effects violate the standard ``no interference'' assumption for standard causal inference methods. The complexity of spatial spillover effects also raises the risk of misspecification and bias in model-based analyses. We offer an approach for robust inference in such settings without having to specify a parametric outcome model. We define a spatial ``average marginalized effect'' (AME) that characterizes how, in expectation, units of observation that are a specified distance from an intervention location are affected by treatment at that location, averaging over effects emanating from other intervention nodes. We show that randomization is sufficient for non-parametric identification of the AME even if the nature of interference is unknown. Under mild restrictions on the extent of interference, we establish asymptotic distributions of estimators and provide methods for both sample-theoretic and randomization-based inference. We show conditions under which the AME recovers a structural effect. We illustrate our approach with a simulation study. Then we re-analyze a randomized field experiment and a quasi-experiment on forest conservation, showing how our approach offers robust inference on policy-relevant spillover effects.

Figures (23)

• Figure 1: Illustrations of hypothetical spatial experiments in which interventions are applied to points (left) or polygons (right). The background raster captures the geographic outcome data. Effects may bleed out in space, as illustrated by the concentric dashed lines.
• Figure 2: Left: Illustration of a "null raster," with $N=4$ intervention nodes (points), none of which are assigned to treatment. Raster cells are colored according to outcome levels. White circles around the nodes are where circle averages are computed. Lighter colors represent larger outcome values. Right: a possible effect function that is non-monotonic in distance.
• Figure 3: Illustration of how outcomes are affected given different treatment allocations given the effect function in Figure \ref{['fig:null-raster']}. Treated intervention points are white, while non-treated intervention points are black.
• Figure 4: Illustration of condition C3. White circles around the nodes are circle averages defined at the distance value $d$. Black circles with a radius of $d+\bar{d}$ depict the maximal range that interference can happen for the $d$-circle averages. As the distance between node 1 and node 3 is larger than $2\bar{d}+2d$, the circle averages of the two nodes do not depend on each other, even when a third node lies between them.
• Figure 5: This figure displays the AME curves in solid lines for both the additive case (\ref{['additive_effect']}) and the interactive case (\ref{['interactive_effect']}). For the interactive case, the dashed lines are the marginalized effect curves when the nearest neighbor is treated and when it is not.

Theorems & Definitions (52)

• Proposition 1: Unbiasedness
• Proposition 2: Asymptotic Distribution for the Horvitz-Thompson estimator
• Proposition 3: Asymptotic Distribution for the Hajek estimator
• Proposition 4
• Lemma A.1
• proof
• proof
• Lemma A.2
• proof
• proof