Accounting for Unobservable Heterogeneity in Cross Section Using Spatial First Differences
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We propose a simple cross-sectional research design to identify causal effects that is robust to unobservable heterogeneity. When many observational units are adjacent, it may be sufficient to regress the "spatial first differences" (SFD) of the outcome on the treatment and omit all covariates. This approach is conceptually similar to first differencing approaches in time-series or panel models, except the index for time is replaced with an index for locations in space. The SFD approach identifies plausibly causal effects so long as local changes in the treatment and unobservable confounders are not systematically correlated between immediately adjacent neighbors. We illustrate how this approach can mitigate omitted variables bias through simulation and by estimating returns to schooling along 10th Avenue in New York and I-90 in Chicago. We then more fully explore the benefits of this approach by estimating effects of climate and soil on maize yields across US counties. In each case, we demonstrate the performance of the research design by withholding important covariates during estimation. SFD has multiple appealing features, such as internal robustness checks that exploit rotation of the coordinate system or double-differencing across space, it is immediately applicable to spatially-gridded data sets, and it can be easily implemented in statical packages by replacing a single index in pre-existing time-series functions.
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