The Variance of Gravity

This paper documents a fundamental problem in applied gravity estimation: the variance of gravity estimates becomes prohibitively large when policy regressors are parse. I define sparse regressors as dummy variables that equal one in fewer than 100 to 500 observations depending on the setting; a common characteristic of trade policies such as free trade agreements. Through Monte Carlo simulations calibrated to match previously established data generating processes in the literature, I demonstrate that the variance of coefficient estimates is approximately inversely proportional to the number of treated observations, making reliable statistical inference impossible when policy variables are infrequent. This variance problem is distinct from well-known issues related to high-dimensional fixed effects and affects both OLS and PPML estimators regardless of specification complexity. The severity of this variance problem depends on the magnitude of the true underlying coefficient: the variance problem is severe and practically prohibitive for moderate coefficients (such as those typically found for many trade policy effects), but becomes negligible for large effects. To address this issue, I propose Ridge regularization as a practical solution that reduces estimate variance while introducing minimal bias. The main contribution however is not advocating for Ridge regularization, but rather highlighting that variance is often the dominant source of uncertainty in gravity estimation when dealing with sparse policy variables, underscoring fundamental limitations of gravity models for evaluating infrequent policies with moderate effect sizes. These findings have implications not only for the international trade literature but also for other fields that employ gravitytype specifications, including migration and macroeconomics.