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Quasi-Experimental Shift-Share Research Designs

Kirill Borusyak, Peter Hull, Xavier Jaravel

arXiv:1806.01221v9econ.EM

TL;DR

The paper develops a quasi-experimental framework for SSIV validity that allows endogenous exposure shares by shifting identification to shock variation. Its equivalence results motivate shock-level orthogonality conditions and support viewing the application as leveraging exogenous shock variation, although shock orthogonality may still fail.

  • Problem

    SSIV validity must be established through conditions on the underlying shocks rather than assuming exposure-share exogeneity.

  • Method

    The framework allows exposure shares to be endogenous and represents instrument-residual orthogonality through orthogonality between underlying shocks and a shock-level unobservable.

  • Results

    The sensitivity, falsification, and overidentification exercises suggest that the approach can be reasonably viewed as leveraging exogenous shock variation.

  • Takeaways & Limitations

    Equivalent shock-level conditions provide a basis for assessing SSIV validity and the consistency of IV estimates as the number of shocks increases.

  • Takeaways & Limitations

    The conditions for shock orthogonality could still fail.

Abstract

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Many studies use shift-share (or ``Bartik'') instruments, which average a set of shocks with exposure share weights. We provide a new econometric framework for shift-share instrumental variable (SSIV) regressions in which identification follows from the quasi-random assignment of shocks, while exposure shares are allowed to be endogenous. The framework is motivated by an equivalence result: the orthogonality between a shift-share instrument and an unobserved residual can be represented as the orthogonality between the underlying shocks and a shock-level unobservable. SSIV regression coefficients can similarly be obtained from an equivalent shock-level regression, motivating shock-level conditions for their consistency. We discuss and illustrate several practical insights of this framework in the setting of Autor et al. (2013), estimating the effect of Chinese import competition on manufacturing employment across U.S. commuting zones.

LSE and CEPR∗

The paper lists contact information for the authors and acknowledges colleagues, seminar participants, and referees for helpful comments.

  • The listed contacts are k.borusyak@ucl.ac.uk, hull@uchicago.edu, and x.jaravel@lse.ac.uk.
  • The authors acknowledge Rodrigo Adão, Joshua Angrist, David Autor, and other scholars for helpful comments.
  • The acknowledgments include various seminar participants and five anonymous referees.

1 Introduction

The introduction develops a shock-based quasi-experimental interpretation of SSIV that permits endogenous exposure shares, derives equivalent shock-level representations, and presents extensions, inference tools, simulations, and an application to Chinese import competition.

  • Orthogonality between a shift-share instrument and an unobserved residual is equivalent to orthogonality between shocks and a shock-level unobservable.The shock-level unobservable captures average unobserved outcome determinants among observations most exposed to each shock.
  • The equivalent shock-level IV regression averages outcomes and treatments using exposure shares, then uses shocks to instrument aggregated treatment.This equivalence supplies a shock-level identification condition and applies to the shift-share instrument's structure.
  • Consistency is supported when shocks are as-good-as-randomly assigned and many sufficiently independent shocks have sufficiently small average exposure.Instrument relevance additionally requires shocks to affect treatment, even when units are exposed to only a small number of shocks.
  • The framework extends to conditional shock assignment, incomplete shares, panel data, multiple endogenous variables, and multiple shock sets.Shift-share controls can isolate conditional quasi-experimental variation, while incomplete shares require controlling for the sum of exposure shares.
  • Shock-level estimation offers practical inference and testing tools, with simulations supporting asymptotic approximations and SSIV finite-sample similarity to conventional shock-level IV.The approach can produce asymptotically valid standard errors under an additional controls assumption and can be implemented with standard software.
  • In the Autor et al. application, the evidence supports interpreting the SSIV as leveraging quasi-random industry-specific Chinese import shocks.The application studies Chinese import penetration and manufacturing employment across U.S. commuting zones.
  • A limitation is that industry growth rates can depend on unobserved regional labor supply shocks.The framework shows that such settings can still be handled by treating industry employment growth rates as shocks, but the dependence remains a concern.
  • SSIV identification can rely on quasi-random assignment of shocks rather than exogenous exposure shares.The framework focuses on conditions under which exposure-share exogeneity is unnecessary.

2 Setting and Motivation

The paper recasts SSIV identification and estimation at the shock level, allowing endogenous shares while requiring orthogonality of shocks and a shock-level unobservable. It establishes an equivalent weighted shock-level IV procedure and states conditions supporting consistency.

  • The paper motivates identification with two equivalence results that recast SSIV orthogonality and estimation at the shock level.These results support identification by exogenous shocks when exposure shares are endogenous.
  • The SSIV setting combines outcomes, treatments, controls, regression weights, shocks, and exposure shares across observations and shocks.The instrument uses observed shocks and nonnegative shares, with shares initially normalized to sum to one across shocks.
  • The structural goal is to estimate a causal or structural parameter β using a shift-share instrument for treatment variation.The motivating labor-supply example relates local wage and employment growth and uses import tariffs across industries as shocks.
  • The SSIV moment condition requires orthogonality between the shift-share instrument and the second-stage residual, together with a first stage for identification.The setup does not impose iid observations and allows dependencies generated by common shocks.
  • The shift-share orthogonality condition can be rewritten as a shock-level condition involving shocks and average unobserved determinants among highly exposed observations.When shares are endogenous, this shock orthogonality condition provides the identification route.
  • The framework does not rely on iid observations and interprets shock-level estimates as effects for the original regional or observational unit.In the labor-supply example, industry-level computation estimates regional rather than industry labor-supply elasticity and may capture local spillovers.
  • The SSIV estimator equals the coefficient from a share-weighted shock-level IV regression using shocks as the instrument.This equivalence motivates studying SSIV consistency through a non-standard shock-level IV procedure.

3 A Quasi-Experimental SSIV Framework

The framework identifies SSIV effects through quasi-randomly assigned shocks while allowing exposure shares to be endogenous. Its baseline conditions require dispersed, mutually uncorrelated shocks and a relevant first stage, with extensions for conditional assignment and weak dependence.

  • Baseline framework: SSIV identification can hold when shocks are as-good-as-randomly assigned, mutually uncorrelated, numerous, and sufficiently dispersed in average exposure.Relevance generally arises when observations concentrate exposure in a small number of shocks that affect treatment.
  • Baseline framework: Exposure shares may be endogenous because the framework conditions on them without restricting their dependence with unobserved residuals.The key identification condition instead concerns shock assignment conditional on shock-level unobservables and exposure weights.
  • Identification: Under quasi-random shock assignment, each shock has the same mean regardless of relevant unobservables and average exposures, satisfying the SSIV moment condition.In the labor-supply example, this excludes tariffs being strategically chosen based on labor-supply trends.
  • Identification: SSIV identifies the coefficient when the instrument is relevant, and stronger assignment and monotonicity conditions identify a convex average of heterogeneous treatment effects.The latter result generalizes conventional IV and reduced-form shift-share identification results.
  • Consistency: Consistency follows from baseline shock-assignment and many-shock assumptions together with relevance and additional regularity conditions.The many-shock condition requires the effective shock-level sample to grow while individual importance weights vanish.
  • Extensions: The framework extends to shocks that are conditionally quasi-random within observed groups and to weak mutual dependence such as clustering or serial correlation.These extensions relax the baseline assumptions while preserving consistency under corresponding conditions.

4 Extensions

The extensions address estimated or equilibrium shocks, incomplete shares, panel data, multiple instruments, and measurement noise. They show when feasible SSIV remains valid and identify boundaries where corrections, controls, or stronger assumptions are needed.

  • Estimated shocks: The framework accommodates shocks estimated within the sample, including the canonical Bartik design, and examines the resulting estimation noise.It treats observed shocks as noisy estimates of latent shocks satisfying quasi-random assignment and derives conditions for feasible SSIV.
  • Estimated shocks: When supply shocks are spatially uncorrelated, leave-one-out correction can address mechanical bias from using estimated shocks that aggregate local supply shocks.With spatially correlated supply shocks, leave-one-out adjustment may not suffice; more restrictive split-sample methods may be needed.
  • Estimated shocks: In practice, the leave-one-out correction does not materially change the SSIV estimate, especially without regional employment weights.The framework explains this pattern through a heuristic statistic that is larger without importance weights.
  • Incomplete shares: Incomplete shares can make SSIV leverage non-experimental variation in total exposure in addition to quasi-experimental shock variation.The issue arises even when the baseline shock assumptions hold.
  • Incomplete shares: With incomplete shares, conditional quasi-random assignment and suitable controls can isolate quasi-experimental manufacturing-shock variation.The resulting estimator has an exposure-weighted-sum rather than average interpretation.
  • Panel data: Consistency can follow from L,N →∞ under cross-sectional conditions or from T →∞ with weak serial dependence, even when L and N are small.Short panels or repeated cross-sections with fixed T require separate conditions.
  • Multiple instruments: With multiple shift-share instruments, SSIV again has an equivalent shock-level IV representation, although the equivalence is more complex under overidentification.The framework also extends to multiple treatments and instruments.

5 Shock-Level Inference and Testing

The shock-level representation supplies exposure-robust inference, relevance diagnostics, and falsification tests for SSIV. It delivers valid standard errors under stated conditions but imposes restrictions on controls and remains subject to finite-sample and exposure-structure considerations.

  • Exposure-robust inference: Under additional control and regularity conditions, conventional shock-level standard errors yield asymptotically valid confidence intervals for the SSIV coefficient.This provides exposure-robust inference by estimating at the level of as-good-as-random variation.
  • Shock-level equivalence: SSIV coefficients are numerically equivalent to coefficients from an appropriately weighted shock-level IV regression.The shock-level regression instruments aggregated treatment with the underlying shocks.
  • Inference limitations: The method restricts controls: shock-level confounding must be captured by shift-share controls, while other controls must not be asymptotically correlated with the instrument.Valid shift-share inference with general control vectors remains an open problem, although weaker assumptions yield conservative standard errors.
  • Exposure-robust inference: The inference approach is practical because it works with standard software, extends to clustered or autoregressive shocks, and applies when N > L.The authors also provide a Stata package for implementation.
  • Testing: Shock-level regressions support falsification tests using residual proxies or pre-trends and relevance tests based on the equivalent first-stage regression.These tests target the assumed quasi-random shock assignment rather than comparing identification frameworks.

6 Shift-Share IV in Practice

The framework interprets SSIV designs as leveraging quasi-random shock variation while allowing exposure shares to be endogenous. In the Autor et al. application, diagnostics and sensitivity analyses assess whether this interpretation is plausible and whether estimates remain stable.

  • Taxonomy of SSIV Settings: The framework distinguishes three SSIV settings according to how shocks and exposure shares are defined and how shocks can be viewed as instruments.It is intended for applications where shocks may be tailored to observed outcomes and treatments, including geographic regions.
  • Autor et al. application: The Autor et al. design uses industry import-growth shocks combined with lagged local industry employment shares to study manufacturing employment.The analysis emphasizes variation in the shocks across periods and industries.
  • Shock variation: 191.6 is the inverse HHI of shock-level average exposure, indicating sizable industry-level exposure variation and motivating its routine reporting.The authors recommend the inverse HHI as a simple diagnostic for shock-level average exposure.
  • Falsification and controls: Locations exposed to larger ADH trade shocks tend to have higher immigrant fractions, and balance tests fail to reject imbalance for ten of twelve potential confounders.The authors discuss alternative shocks or shock-level controls when balance failures challenge the identifying assumption.
  • Discussion: Taken together, sensitivity, falsification, and overidentification exercises suggest that the ADH shocks are plausibly unconfounded in this application.The authors state that the framework is useful in an influential setting where the alternative SSIV framework is inapplicable.
  • Practical implications: The ADH application illustrates how shock-level identifying assumptions translate into SSIV controls, balance tests, and exposure-robust inference.The framework also provides a way to understand the identifying variation in the ADH instrument.

7 Conclusion

The paper develops a quasi-experimental SSIV framework that identifies effects from quasi-random shocks while allowing exposure shares to be endogenous. Its shock-level equivalence results provide practical guidance for estimation, inference, and credibility assessment, while highlighting effective-sample-size and exposure-share constraints.

  • The framework connects SSIV to conventional shock-level IV estimation and facilitates practical assessment of SSIV credibility across several economic fields.The paper illustrates these implications using Autor et al. (2013)'s application to Chinese import competition and U.S. manufacturing employment.
  • Shift-share instruments are valid when shocks are idiosyncratic with respect to exposure-weighted unobserved factors, allowing endogenous exposure shares.
  • The framework represents SSIV orthogonality through orthogonality between underlying shocks and a shock-level unobservable.
  • SSIV coefficients can be obtained from an equivalent shock-level IV regression using shocks directly as instruments.
  • The framework guides practice by controlling for exposure-weighted shock-level confounders and conducting shock-level estimation, placebo tests, first-stage F-statistics, and exposure-robust inference.These procedures account for non-standard clustering arising from common shock exposure.
  • SSIV designs with few or insufficiently dispersed shocks may have effectively small samples despite many observations.Instruments whose exposure shares do not sum to a constant require appropriate controls.

Figures and Tables

The tables characterize the shocks, assess their balance and dependence, and report shift-share IV estimates for Chinese import competition and manufacturing employment in the Autor et al. setting.

  • Shock summary statistics: Table 1 summarizes the distribution of China import shocks across industries and periods using exposure-share weights.The shocks are measured from import flows from China in eight developed economies outside the United States.
  • Shock dependence: Table 2 reports intra-class correlation coefficients for the manufacturing shocks from a hierarchical model.The estimates use maximum likelihood with exchangeable industry and sector random effects and period fixed effects.
  • Balance tests: Table 3 reports industry- and regional-level balance tests relating Autor et al. shocks to covariates and pre-trends.The tests include controls and weighted regressions, with standard errors clustered at the three-digit SIC level for industry tests.
  • Shift-share IV estimates: Table 4 reports shift-share IV estimates of the effect of Chinese import competition on regional manufacturing employment growth.The specifications instrument import competition with predicted China import growth and include period, geographic, and start-of-period controls.

A.1 Heterogeneous Treatment Effects

This appendix studies how SSIV behaves with nonlinear treatments and heterogeneous effects. Under shock-level random assignment and first-stage monotonicity, the estimator identifies a convex average of rescaled treatment effects, while endogenous shares generally threaten consistency without random shocks.

  • Model: The framework considers nonlinear structural outcomes and treatments generated by multiple shocks, allowing treatment effects and first-stage responses to vary across observations.The derivatives βℓr and πℓnr represent marginal treatment effects and marginal shock effects, respectively.
  • Identification with heterogeneous effects: Under first-stage monotonicity, the large-sample SSIV coefficient is a convex average of rescaled treatment effects.The weights depend on first-stage effects, exposure shares, regression weights, treatment aggregation weights, and the shock distribution.
  • Identification with heterogeneous effects: Without treatment aggregation, the rescaling disappears, connecting the result to convexly weighted averages of heterogeneous effects for continuous treatments.With aggregation, SSIV instead averages treatment effects per aggregated unit.
  • Identification with heterogeneous effects: The result generalizes identification to aggregated treatments and establishes a convex average of rescaled treatment effects in the leading shift-share example.The appendix relates this result to prior work on reduced-form shift-share regressions.
  • Share exogeneity: Share endogeneity generally makes SSIV inconsistent unless the shocks are as-good-as-randomly assigned, even when individual shock importance weights converge to zero.The appendix defines share endogeneity through non-vanishing variance of the shock-level residual component.
  • Share exogeneity: Under the stated shock conditions and sustained first-stage relevance, at least one shock-level residual component retains variance bounded away from zero asymptotically.Proposition A2 states that max_n Var[¯ε_n] exceeds a positive constant for sufficiently large samples.

A.3 Comparing SSIV and Native Shock-Level Regression Estimands

This appendix compares regional SSIV and native industry-level IV estimands when outcomes and treatments are aggregated across region-by-industry cells. They generally differ with within-region spillovers or heterogeneous treatment effects because the regressions weight economic units differently.

  • Setup: The comparison defines regional and native industry-level regressions using the same cell-level outcomes and treatments aggregated with exposure or employment weights.The setup conditions on cell weights and permits share endogeneity.
  • General comparison: Regional SSIV and native industry-level IV generally differ when there are within-region spillovers or heterogeneous treatment effects.The comparison maintains first-stage relevance, shock exogeneity, shock independence, and appropriate laws of large numbers.
  • Within-region spillovers: With within-region spillovers, SSIV captures the treatment effect net of spillovers, whereas native industry-level IV subtracts only part of the spillover effect.The difference arises because spillovers are fully contained within regions but not within industries.
  • Within-region spillovers: The two estimands coincide with nonzero spillovers only when the average region is asymptotically concentrated in one industry.The condition is expressed through the limiting behavior of the local concentration index H_L.
  • Treatment-effect heterogeneity: With heterogeneous treatment effects, SSIV assigns relatively greater weight to cells representing larger fractions of the regional economy.Exposure shares enter both the regional outcome and shift-share instrument, while they enter only the outcome in the industry regression.
  • Treatment-effect heterogeneity: Heterogeneity in first-stage effects and shock variances affects the weighting schemes of the regional and industry estimands equivalently.The comparison treats treatment effects as varying across region-by-industry cells.

A.4 Connection to Rotemberg Weights

The appendix interprets Rotemberg weights as leverage measures in an equivalent shock-level IV regression rather than measures of sensitivity to share-exogeneity misspecification. Skewed weights need not threaten consistency, but high leverage can complicate inference.

  • Shock-level decomposition: The SSIV coefficient admits a shock-level decomposition in which each shock-specific IV estimate is combined using a Rotemberg weight.The decomposition parallels the interpretation in Goldsmith-Pinkham et al. while arising from the equivalent shock-level regression.
  • Interpretation of weights: In this framework, skewed Rotemberg weights do not measure sensitivity to share-exogeneity misspecification and do not threaten SSIV consistency.This conclusion holds when skewness reflects a heavy-tailed, high-variance shock distribution satisfying the regularity conditions and dispersed exposure shares.
  • Interpretation of weights: Share endogeneity means the shock-level residual components can remain nonzero, so Rotemberg weights do not have the same sensitivity-to-misspecification interpretation.The distinction concerns the framework’s allowance for endogenous exposure shares.
  • Inference: High-leverage observations can bias residual variance estimates toward zero and produce standard errors that are too small.The appendix notes that null-imposed confidence intervals can address this inference problem.
  • Inference: Monte Carlo simulations find satisfactory coverage for conventional exposure-robust confidence intervals even with Rotemberg weights as skewed as those in the cited applications.The result is reported for the simulation exercises in Appendix A.11.
  • Additional convergence conditions: The appendix also develops convergence arguments for controls and residual components under dispersed weights, weak dependence, clustering, and multiple shift-share terms.These cases include conventional or clustered local shocks and additional shift-share structures with different exposure shares.

A.6 Estimated Shocks

This appendix derives conditions for consistency of SSIV estimators when shocks are estimated, emphasizing leave-one-out corrections and a heuristic for when conventional estimation has little mechanical bias.

  • The appendix establishes formal conditions for SSIV estimation with estimated shocks and leave-one-out instruments.
  • Heuristic for Importance of LOO Correction: The proposed heuristic predicts that non-LOO SSIV is relatively insensitive to mechanical bias when the heuristic H is large.
  • LOO Identification and Consistency: LOO consistency requires cross-observation orthogonality between residuals and shock-estimation errors, but allows own-observation covariance.This excludes mechanical bias from the residual directly entering its own shock estimate.
  • LOO Identification and Consistency: The cross-observation condition may fail when regions share shocks, so excluding the own observation alone may be insufficient.
  • Heuristic for Importance of LOO Correction: H is large when employment is more concentrated across industries within regions than across regions within industries, making LOO correction less necessary.
  • Application to Bartik (1991): In the Bartik application, LOO and conventional estimates range from 1.2 to 1.3, indicating little practical role for the correction.

A.9 SSIV Relevance with Panel Data

The panel-data analysis shows that holding exposure shares fixed can weaken the SSIV first stage as the number of periods grows. Updated shares preserve relevance under a non-vanishing Herfindahl condition.

  • Fixed pre-period exposure shares are likely to weaken the SSIV first stage in panel regressions.
  • With updated shares, panel SSIV relevance requires a non-vanishing Herfindahl index for an average observation-period.
  • With fixed shares, the overlap between initial and current shares may weaken or vanish as T →∞.

A.10 SSIV with Multiple Endogenous Variables or Instruments

The appendix extends SSIV equivalence and quasi-experimental analysis to multiple endogenous variables or instruments. It also develops corrected first-stage diagnostics and efficient shock-level procedures.

  • The equivalence result extends to multiple instruments when exposure shares are common across instruments, allowing SSIV estimates to be obtained from shock-level IV regressions.
  • The framework generalizes to multiple treatment channels and discusses corresponding extensions of the quasi-experimental conditions.
  • Effective First-Stage F-statistics: A corrected effective first-stage F-statistic is derived for SSIV with multiple instruments and implemented in weakssivtest.
  • Efficient Shift-Share GMM: Efficient SSIV-GMM estimation uses shock-level IV regressions with a weighting matrix chosen from the inverse asymptotic variance.
  • In one application, the omnibus overidentification test statistic is 10.92 with 7 degrees of freedom and a p-value of 0.142.
  • The corrected effective first-stage statistic is 15.10, remaining above the conventional heuristic threshold of 10.

A.11 Finite-Sample Performance of SSIV: Monte-Carlo Evidence

Monte Carlo simulations based on the Autor et al. (2013) China-shock setting compare SSIV with conventional industry-level IV. The results indicate similar finite-sample performance and support Herfindahl and first-stage diagnostics.

  • Simulation design: The simulations generate 10,000 samples by redrawing shocks while holding estimated residual components fixed, isolating shock randomness.
  • Main results: Across simulation variations, SSIV performance is similar to industry-level IV, while many weak instruments produce bias that grows with the number of instruments.
  • The simulation findings may depend on the China-shock data-generating process, motivating analogous simulations for researchers’ own data.
  • Simulation design: Simulated estimators are approximately unbiased, with median bias at most 1% of the estimator’s standard deviation.
  • Main results: Rejection rates are close to the nominal 5% level for SSIV and conventional IV: 7.6% and 6.8%, respectively.
  • Main results: Imposing the null brings rejection rates closer to nominal: 5.2% for SSIV and 5.0% for conventional IV.
  • Main results: The Herfindahl index indicates the effective number of industries, and properly constructed first-stage F-statistics inform weak-instrument bias.

B.2 Proposition 5 and Related Results

Proposition 5 establishes exposure-robust inference through an equivalent shock-level regression. The related results characterize equivalence with existing procedures, conservativeness under weaker control assumptions, and finite-sample differences.

  • Proposition 5: The shock-level regression yields exposure-robust standard errors for SSIV estimates under the paper’s stated assumptions.The result is established under additional assumptions that largely follow Adão et al. (2019).
  • Related results: When controls consist only of a constant, the proposed heteroskedasticity-robust standard error is numerically equivalent to Adão et al. (2019)’s baseline IV standard error.The equivalence follows because the relevant shock-level projection has exact fit.
  • Proposition 5: The SSIV coefficient equals the coefficient from the corresponding shock-level IV regression after controlling for the relevant shock-level variables.The coefficient vector on the controls is numerically zero, leaving the instrumented regression coefficient unchanged.
  • Related results: Both the paper’s and Adão et al. (2019)’s standard errors are asymptotically valid under the stated assumptions because they capture the conditional asymptotic variance of the SSIV estimator.The resulting confidence intervals are asymptotically valid unconditionally as well.
  • Related results: Under weaker control assumptions, the proposed standard errors are asymptotically conservative, while under the baseline assumptions they are likely smaller in finite samples.The finite-sample comparison is proved for homoskedastic formulas and is only suggestive under heteroskedasticity.
  • Proposition 5: The equivalent shock-level regression also provides a convenient route to the null-imposed inference procedure of Adão et al. (2019).The procedure is obtained from the paper’s equivalent regression rather than from a separate SSIV calculation.

C Appendix Figures and Tables

The appendix reports visual, robustness, inference, period-specific, overidentified, and simulation results for the Autor et al. (2013) and Bartik (1991) applications. These materials illustrate the shock-level equivalence and compare alternative specifications and standard errors.

  • Appendix figures: The Appendix Figure C1 binned scatterplots show the industry-level first-stage and reduced-form relationships underlying the column 3 specification.The plotted slopes are 5.71 × 10^-3 and −1.52 × 10^-3, whose ratio is −0.267, the column 3 SSIV coefficient.
  • Robustness outcomes: Table C1 extends the Autor et al. (2013) analysis to unemployment, labor-force non-participation, and log average weekly wage growth.The specifications otherwise match the corresponding Table 4 columns and use SIC3-clustered exposure-robust standard errors.
  • Alternative inference: Table C2 compares conventional state-clustered, Adão et al. (2019), and null-imposed shock-level confidence intervals with the exposure-robust results.The conventional standard errors are generally too low according to the accompanying discussion.
  • Period-specific effects: Table C3 allows the treatment effect to vary across periods by interacting treatment and instruments with period indicators.The specifications retain the controls from column 3 of Table 4 and use equivalent shock-level regressions for exposure-robust inference.
  • Overidentified estimates: Table C5 reports overidentified shift-share IV estimates using imports from eight non-U.S. countries as instruments for Chinese import competition.It compares two-stage least squares, limited information maximum likelihood, and two-step optimal generalized method of moments estimates with period fixed effects.
  • Bartik application and simulations: The appendix also applies the framework to the Bartik (1991) setting and uses Monte Carlo tables to examine rejection rates and first-stage F-statistics.The simulations compare SSIV designs with conventional industry-level IV designs and vary the number of shocks in weak-instrument exercises.
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