Shift-Share Designs: Theory and Inference*

Abstract

We study inference in shift-share regression designs, such as when a regional outcome is regressed on a weighted average of sectoral shocks, using regional sector shares as weights. We conduct a placebo exercise in which we estimate the effect of a shift-share regressor constructed with randomly generated sectoral shocks on actual labor market outcomes across U.S. commuting zones. Tests based on commonly used standard errors with 5% nominal significance level reject the null of no effect in up to 55% of the placebo samples. We use a stylized economic model to show that this overrejection problem arises because regression residuals are correlated across regions with similar sectoral shares, independent of their geographic location. We derive novel inference methods that are valid under arbitrary cross-regional correlation in the regression residuals. We show using popular applications of shift-share designs that our methods may lead to substantially wider confidence intervals in practice.

The superstar and the followers: Intra-firm product complementarity in international trade

“…Borusyak, Hull, and Jaravel (2018) show that there is a numerical equivalence between a standard shift share IV regression coefficients and one that has been transformed and estimated at the level of the "shock". In this transformed regression they show that the standard errors are valid ("exposure robust") and asymptotically equivalent to those suggested by Adão, Kolesár, and Morales (2019).…”

“…Borusyak, Hull, and Jaravel (2018) show how valid standard errors can be found by transforming a standard shift share IV regression to the level of the "shock" and derive valid standard errors that are "exposure robust". The standard errors obtained this way are asymptotically equivalent to those suggested by Adão, Kolesár, and Morales (2019). Using the single-core product definition as a special case, I follow the framework proposed by Borusyak, Hull, and Jaravel (2018) and…”

“…Third, there is a recent literature that has specifically focused on inference and identification in settings with shift-share instruments as employed in this paper (see Adão, Kolesár, and Morales, 2019, Borusyak, Hull, and Jaravel, 2018and Goldsmith-Pinkham, Sorkin, and Swift, 2019. Borusyak, Hull, and Jaravel (2018) develop a framework that rests on an exact numerical equivalence between a standard IV shift share regression and one that is estimated at the level of the identifying variation ("shock").…”

“…In columns 5-6 products within the same one or two digit HS product category as any of the core products are dropped. Adão, Kolesár, and Morales (2019) find that in shift share regressions standard errors are often underestimated and the problem stems from the regression residuals being correlated across observations that have similar exposure shares. Borusyak, Hull, and Jaravel (2018) develop a novel framework that addresses this issue and provides valid standard errors.…”

“…In this appendix I focus on the specification that uses a single-core product per firmdestination. Using the single core specification for this exercise is applicable since, if anything, the problem discussed by Adão, Kolesár, and Morales (2019) should be exacerbated by this definition of the core. This stems from the fact that all firms that are exposed to the same shock will have identical exposure shares (since s ipd = 1 for all observations) while the shares vary in the baseline multi-product core definition.…”