Many prominent studies in macroeconomics, labor, and trade use panel data on regions to identify the local effects of aggregate shocks. These studies construct regional-exposure instruments as an observed aggregate shock times an observed regional exposure to that shock. We argue that the most economically plausible source of identification in these settings is uncorrelatedness of observed and unobserved aggregate shocks. Even when the regression estimator is consistent, we show that inference is complicated by cross-regional residual correlations induced by unobserved aggregate shocks. We suggest two-way clustering, two-way heteroskedasticity- and autocorrelation-consistent standard errors, and randomization inference as options to solve this inference problem. We also develop a feasible optimal instrument to improve efficiency. In an application to the estimation of regional fiscal multipliers, we show that the standard practice of clustering by region generates confidence intervals that are too small. When we construct confidence intervals with robust methods, we can no longer reject multipliers close to zero at the 95% level. The feasible optimal instrument more than doubles statistical power; however, we still cannot reject low multipliers. Our results underscore that the precision promised by regional data may disappear with correct inference.