Shift-Share Designs in Econometrics

• Shift-share designs are empirical strategies that combine unit-specific exposure weights with common shocks to identify causal effects in various research settings.
• They enable estimation through both reduced-form measures and instrumental-variable strategies, linking unit-level outcomes with shock-level variation.
• Inference in shift-share designs must account for exposure-induced residual dependence, prompting the use of shock-level variance estimators and overidentification tests.

Shift-share designs are empirical strategies built around variables that combine unit-specific exposure weights with common shocks. In canonical notation, the shift-share regressor is $X_i=\sum_{s=1}^S w_{is}X_s$, while in instrumental-variables form it is often written as $z_\ell=\sum_n s_{\ell n}g_n$; in Bartik applications with lagged shares, a generic instrument is $s_{lt}=\sum_{j=1}^J z_{jlt_0}g_{jt}$. These constructions are used both to summarize heterogeneous exposure to many shocks and to instrument endogenous local treatments with aggregate shifters. Modern econometric work treats shift-share designs as a distinct class of causal designs whose interpretation depends on where exogeneity is located—shares or shocks—and whose inference must respect exposure-induced dependence rather than relying on conventional cross-sectional intuition (Adão et al., 2018, Borusyak et al., 2018, Apfel, 2019).

1. Formal structure and basic objects

A shift-share design links an outcome unit to shocks defined at another level of aggregation. In the baseline setup of regional applications, one observes

$$Y_i=\beta X_i+Z_i'\delta+\epsilon_i,\qquad X_i=\sum_{s=1}^S w_{is}X_s,\qquad w_{is}\ge 0,\quad \sum_{s=1}^S w_{is}\le 1.$$

The indices need not denote regions and sectors specifically: units may be commuting zones, countries, districts, firms, or individuals, while shocks may be industries, occupations, origin countries, donors, or time-specific disturbances. The shift-share variable is therefore an inner product between an exposure vector and a shock vector, and the same algebra covers reduced-form exposure measures and Bartik-style instruments (Adão et al., 2018, Park, 24 Feb 2026).

A central formal insight is that the unit-level design admits a shock-level representation. Defining

$$s_n=\sum_\ell e_\ell s_{\ell n},\qquad \bar v_n=\frac{\sum_\ell e_\ell s_{\ell n}v_\ell}{\sum_\ell e_\ell s_{\ell n}},$$

the SSIV estimator

$$\hat\beta=\frac{\sum_\ell e_\ell z_\ell y_\ell^\perp}{\sum_\ell e_\ell z_\ell x_\ell^\perp}$$

is numerically equal to the coefficient from a shock-level IV regression that uses $g_n$ as the instrument for $\bar x_n^\perp$, weighted by $s_n$. This equivalence re-expresses the unit-level orthogonality condition as a shock-level orthogonality condition and makes explicit that the relevant sampling variation can come from shocks rather than units (Borusyak et al., 2018).

The potential-outcomes formulation used in the modern literature makes this structure explicit. One common representation is

$$Y_i(x_1,\dots,x_S)=Y_i(0)+\sum_{s=1}^S w_{is}x_s\beta_{is},$$

so that a unit increase in shock $z_\ell=\sum_n s_{\ell n}g_n$0 changes region $z_\ell=\sum_n s_{\ell n}g_n$1’s outcome by $z_\ell=\sum_n s_{\ell n}g_n$2. Under this formulation, the shift-share coefficient is generally not a simple average treatment effect. In the shock-exogeneity framework of Adão, Kolesár, and Morales, the OLS estimand converges to

$z_\ell=\sum_n s_{\ell n}g_n$3

a variance-weighted average of heterogeneous region-sector effects (Adão et al., 2018).

2. Competing identification frameworks

The modern literature distinguishes two major identification strategies. Under share exogeneity, the identifying assumption is that the exposure shares are exogenous conditional on controls. In the simplest formulation,

$z_\ell=\sum_n s_{\ell n}g_n$4

or, in panel form with initial shares,

$z_\ell=\sum_n s_{\ell n}g_n$5

In this view, the aggregate shift-share regressor can be reinterpreted as IV/GMM with the individual shares serving as instruments. The design is therefore naturally tied to Rotemberg decompositions: the shift-share estimate can be written as a weighted average of just-identified estimates using one share at a time, so empirical credibility depends disproportionately on the shares with the largest absolute Rotemberg weights (Park, 24 Feb 2026).

Under shock exogeneity, by contrast, shares may be endogenous equilibrium objects, and identification comes from quasi-random assignment of the shifters. Borusyak, Hull, and Jaravel formalize the key equivalence: $z_\ell=\sum_n s_{\ell n}g_n$6 with

$z_\ell=\sum_n s_{\ell n}g_n$7

A baseline assumption is quasi-random shock assignment,

$z_\ell=\sum_n s_{\ell n}g_n$8

possibly conditional on shock-level covariates $z_\ell=\sum_n s_{\ell n}g_n$9. In the conditional case, valid unit-level controls are not arbitrary local covariates but exposure-weighted shock-level controls of the form

$s_{lt}=\sum_{j=1}^J z_{jlt_0}g_{jt}$0

When shares are incomplete, variation in total exposure can itself be confounding, so one should control for the sum of shares as well (Borusyak et al., 2018).

These frameworks imply different asymptotics and different empirical burdens. Under shock exogeneity, consistency requires many shocks and dispersed average exposure, captured by a Herfindahl-style condition such as

$s_{lt}=\sum_{j=1}^J z_{jlt_0}g_{jt}$1

At the same time, first-stage relevance is strengthened when units are locally specialized, so successful shift-share designs often combine shock-level dispersion with unit-level concentration. This logic differs sharply from the share-exogeneity interpretation, where the number of shocks does not itself rescue endogenous shares (Borusyak et al., 2018).

3. Inference, exposure dependence, and the effective sample size

A defining econometric feature of shift-share designs is that residual dependence is induced by similar exposure structures. Adão, Kolesár, and Morales show this directly in a placebo exercise using actual labor market outcomes across 722 U.S. commuting zones and actual 1990 industry employment shares across 396 manufacturing industries, but placebo shift-share regressors constructed with randomly generated sectoral shocks. With a nominal $s_{lt}=\sum_{j=1}^J z_{jlt_0}g_{jt}$2 level, rejection rates are as high as $s_{lt}=\sum_{j=1}^J z_{jlt_0}g_{jt}$3 with heteroskedasticity-robust standard errors, up to $s_{lt}=\sum_{j=1}^J z_{jlt_0}g_{jt}$4 with state-clustered standard errors, and never below $s_{lt}=\sum_{j=1}^J z_{jlt_0}g_{jt}$5. The problem is not geography per se: residuals are correlated across regions with similar share vectors even when those regions are geographically distant (Adão et al., 2018).

This motivates shock-level variance estimation. Let

$s_{lt}=\sum_{j=1}^J z_{jlt_0}g_{jt}$6

denote the shock-level residual aggregate. With controls, AKM propose

$s_{lt}=\sum_{j=1}^J z_{jlt_0}g_{jt}$7

and the standard error

$s_{lt}=\sum_{j=1}^J z_{jlt_0}g_{jt}$8

AKM0 imposes the null in residual construction and often yields better finite-sample size control; its confidence intervals are often asymmetric around $s_{lt}=\sum_{j=1}^J z_{jlt_0}g_{jt}$9. Borusyak, Hull, and Jaravel reach the same practical conclusion from the shock-level regression equivalence: standard heteroskedasticity-robust or clustered errors should be computed at the shock level, with clustering over shock groups when shocks are correlated within broader categories (Adão et al., 2018, Borusyak et al., 2018).

The literature also emphasizes that the relevant effective sample size is governed by shocks, not by the number of observational units. Borusyak, Hull, and Jaravel recommend reporting concentration measures such as $$Y_i=\beta X_i+Z_i'\delta+\epsilon_i,\qquad X_i=\sum_{s=1}^S w_{is}X_s,\qquad w_{is}\ge 0,\quad \sum_{s=1}^S w_{is}\le 1.$$0 or its inverse $$Y_i=\beta X_i+Z_i'\delta+\epsilon_i,\qquad X_i=\sum_{s=1}^S w_{is}X_s,\qquad w_{is}\ge 0,\quad \sum_{s=1}^S w_{is}\le 1.$$1, interpreted as an effective sample size. If a few shocks dominate the design, the many-shock approximation becomes weak even in very large cross sections (Borusyak et al., 2018).

When shocks are few or concentrated, studentized randomization inference provides an alternative. The proposed statistic

$$Y_i=\beta X_i+Z_i'\delta+\epsilon_i,\qquad X_i=\sum_{s=1}^S w_{is}X_s,\qquad w_{is}\ge 0,\quad \sum_{s=1}^S w_{is}\le 1.$$2

combines finite-sample validity under stronger assumptions on the shock assignment mechanism with asymptotic validity under weaker assumptions, including some forms of treatment-effect heterogeneity and misspecification of the simulated shock distribution. This is designed precisely for shift-share settings where asymptotic shock-robust methods may have poor finite-sample behavior (Alvarez et al., 2022).

A separate methodological warning concerns design-based simulations. Simulations that hold realized outcomes fixed and resample shocks can be misleading when the true treatment effect is nonzero, because the realized treatment component is mechanically absorbed into the simulated residuals. In shift-share applications, this can create residual dependence that mimics spatial correlation even if the true errors are not spatially correlated. Alternative designs based on placebo or pre-treatment outcomes, or on residualized outcomes $$Y_i=\beta X_i+Z_i'\delta+\epsilon_i,\qquad X_i=\sum_{s=1}^S w_{is}X_s,\qquad w_{is}\ge 0,\quad \sum_{s=1}^S w_{is}\le 1.$$3, are proposed to align the simulation DGP more closely with the true one (Ferman, 11 Mar 2026).

4. Overidentification, invalid components, and robustness to exclusion failures

In the shares-exogeneity interpretation, a Bartik instrument is not a single primitive instrument but a weighted combination of many share-specific instruments. Apfel makes this explicit by writing the structural equation as

$$Y_i=\beta X_i+Z_i'\delta+\epsilon_i,\qquad X_i=\sum_{s=1}^S w_{is}X_s,\qquad w_{is}\ge 0,\quad \sum_{s=1}^S w_{is}\le 1.$$4

with first-stage representations

$$Y_i=\beta X_i+Z_i'\delta+\epsilon_i,\qquad X_i=\sum_{s=1}^S w_{is}X_s,\qquad w_{is}\ge 0,\quad \sum_{s=1}^S w_{is}\le 1.$$5

The conventional exclusion restriction becomes “all shares are valid.” Apfel relaxes this through invalid-IV selection. Under the majority condition

$$Y_i=\beta X_i+Z_i'\delta+\epsilon_i,\qquad X_i=\sum_{s=1}^S w_{is}X_s,\qquad w_{is}\ge 0,\quad \sum_{s=1}^S w_{is}\le 1.$$6

Adaptive Lasso can consistently select invalid shares; under the weaker plurality exclusion restriction

$$Y_i=\beta X_i+Z_i'\delta+\epsilon_i,\qquad X_i=\sum_{s=1}^S w_{is}X_s,\qquad w_{is}\ge 0,\quad \sum_{s=1}^S w_{is}\le 1.$$7

the Confidence Interval Method can identify $$Y_i=\beta X_i+Z_i'\delta+\epsilon_i,\qquad X_i=\sum_{s=1}^S w_{is}X_s,\qquad w_{is}\ge 0,\quad \sum_{s=1}^S w_{is}\le 1.$$8 even when most instruments are invalid, provided the valid shares form the largest cluster of single-share IV estimands. Post-selection, selected-invalid shares are not dropped; they are included as controls, and the corrected shift-share instrument is reconstructed as

$$Y_i=\beta X_i+Z_i'\delta+\epsilon_i,\qquad X_i=\sum_{s=1}^S w_{is}X_s,\qquad w_{is}\ge 0,\quad \sum_{s=1}^S w_{is}\le 1.$$9

This is a point-identification strategy under weaker assumptions than universal share validity (Apfel, 2019).

A related development concerns the testability of identifying assumptions. “Overidentification in Shift-Share Designs” shows that Bartik designs are often structurally overidentified even when the researcher uses only one scalar instrument. Under share exogeneity, the whole vector of shares implies moment conditions such as

$$s_n=\sum_\ell e_\ell s_{\ell n},\qquad \bar v_n=\frac{\sum_\ell e_\ell s_{\ell n}v_\ell}{\sum_\ell e_\ell s_{\ell n}},$$0

under shock exogeneity, the residualized Bartik instrument must satisfy

$$s_n=\sum_\ell e_\ell s_{\ell n},\qquad \bar v_n=\frac{\sum_\ell e_\ell s_{\ell n}v_\ell}{\sum_\ell e_\ell s_{\ell n}},$$1

which implies orthogonality with many functions of shares, controls, and residuals. The paper proposes max-type overidentification tests that remain valid in high-dimensional regimes and are robust to heteroskedasticity and clustering. In the Autor-Dorn-Hanson application, the homogeneous-effects model is rejected under both the share-exogeneity and shock-exogeneity strategies once these implications are tested. The same paper argues that positive-weight heterogeneous-effects interpretations of TSLS can require implausible conditions on the covariance structure of shares or shocks (Hahn et al., 2024).

These contributions shift the practical question from whether a Bartik instrument is valid in the abstract to whether the underlying shares or shocks admit a sufficiently credible valid subset, and whether the implied extra moment conditions survive empirical scrutiny. This suggests a robustness workflow centered on decomposition, testing, and selective reconstruction rather than on treating the shift-share variable as a primitive black box.

5. Extensions beyond linear many-shock models

Recent work extends shift-share econometrics beyond linear constant-effect environments. “Nonlinear Treatment Effects in Shift-Share Designs” studies a nonseparable triangular model with exogenous shares,

$$s_n=\sum_\ell e_\ell s_{\ell n},\qquad \bar v_n=\frac{\sum_\ell e_\ell s_{\ell n}v_\ell}{\sum_\ell e_\ell s_{\ell n}},$$2

under the key restriction

$$s_n=\sum_\ell e_\ell s_{\ell n},\qquad \bar v_n=\frac{\sum_\ell e_\ell s_{\ell n}v_\ell}{\sum_\ell e_\ell s_{\ell n}},$$3

The control function

$$s_n=\sum_\ell e_\ell s_{\ell n},\qquad \bar v_n=\frac{\sum_\ell e_\ell s_{\ell n}v_\ell}{\sum_\ell e_\ell s_{\ell n}},$$4

recovers the rank of the treatment unobservable, delivering

$$s_n=\sum_\ell e_\ell s_{\ell n},\qquad \bar v_n=\frac{\sum_\ell e_\ell s_{\ell n}v_\ell}{\sum_\ell e_\ell s_{\ell n}},$$5

This identifies four target objects: the Local Average Response, the Average Derivative, the Average Structural Function, and a Policy Effect based on a counterfactual treatment-assignment rule $$s_n=\sum_\ell e_\ell s_{\ell n},\qquad \bar v_n=\frac{\sum_\ell e_\ell s_{\ell n}v_\ell}{\sum_\ell e_\ell s_{\ell n}},$$6. A central implication is that linear 2SLS is not generally estimating the Average Derivative, the Local Average Response, or the slope of the Average Structural Function. Its weights depend on the first-stage derivative, and if $$s_n=\sum_\ell e_\ell s_{\ell n},\qquad \bar v_n=\frac{\sum_\ell e_\ell s_{\ell n}v_\ell}{\sum_\ell e_\ell s_{\ell n}},$$7 changes sign, 2SLS can involve negative weighting and lose a weak causal interpretation (Garzon et al., 29 Jul 2025).

Another extension studies a limiting case in which the many-shock logic collapses. “Microfoundations and the Causal Interpretation of Price-Exposure Designs” analyzes single-shock exposure designs of the form

$$s_n=\sum_\ell e_\ell s_{\ell n},\qquad \bar v_n=\frac{\sum_\ell e_\ell s_{\ell n}v_\ell}{\sum_\ell e_\ell s_{\ell n}},$$8

or $$s_n=\sum_\ell e_\ell s_{\ell n},\qquad \bar v_n=\frac{\sum_\ell e_\ell s_{\ell n}v_\ell}{\sum_\ell e_\ell s_{\ell n}},$$9. These designs are shift-share-like but differ from canonical many-shock Bartik settings because there is effectively one focal shock process rather than many. The paper shows that 2SLS and TWFE estimands can be contaminated by covariance with other prices and by general-equilibrium output responses in other sectors. Inference also changes fundamentally: the effective number of independent shock realizations is governed by the time dimension $$\hat\beta=\frac{\sum_\ell e_\ell z_\ell y_\ell^\perp}{\sum_\ell e_\ell z_\ell x_\ell^\perp}$$0, not by the number of exposed regions $$\hat\beta=\frac{\sum_\ell e_\ell z_\ell y_\ell^\perp}{\sum_\ell e_\ell z_\ell x_\ell^\perp}$$1. This motivates the price-exposure variance estimator

$$\hat\beta=\frac{\sum_\ell e_\ell z_\ell y_\ell^\perp}{\sum_\ell e_\ell z_\ell x_\ell^\perp}$$2

a time-series analogue of AKM’s shock-level aggregation (Moreno-Louzada et al., 10 Dec 2025).

Taken together, these extensions show that the standard linear many-shock framework is not exhaustive. Nonlinearity, heterogeneous marginal effects, counterfactual assignment rules, and single-shock exposure designs all require modifications to the canonical shift-share toolkit.

6. Empirical domains and disciplinary diffusion

Shift-share designs are pervasive in regional and labor economics. Canonical applications include regional labor market outcomes regressed on exposure to sectoral demand shocks, local labor supply elasticity estimated with Bartik-style instruments built from national industry growth, and import-competition designs in the style of Autor, Dorn, and Hanson (Adão et al., 2018). The same logic appears in immigration applications, where local exposure is constructed from historical settlement shares and national migrant inflows, and in commodity-price settings where export composition interacts with world prices (Apfel, 2019, Boulat, 2024).

Empirical reanalyses have shown that shift-share conclusions can be highly design-sensitive. In Apfel’s immigration application, conventional estimates for high-skilled wages are roughly $$\hat\beta=\frac{\sum_\ell e_\ell z_\ell y_\ell^\perp}{\sum_\ell e_\ell z_\ell x_\ell^\perp}$$3 (SSIV), $$\hat\beta=\frac{\sum_\ell e_\ell z_\ell y_\ell^\perp}{\sum_\ell e_\ell z_\ell x_\ell^\perp}$$4 (2SLS), and $$\hat\beta=\frac{\sum_\ell e_\ell z_\ell y_\ell^\perp}{\sum_\ell e_\ell z_\ell x_\ell^\perp}$$5 (LIML), but invalid-share selection—especially CIM with an Anderson-Rubin stopping rule—moves the high-skilled wage coefficient from $$\hat\beta=\frac{\sum_\ell e_\ell z_\ell y_\ell^\perp}{\sum_\ell e_\ell z_\ell x_\ell^\perp}$$6 to $$\hat\beta=\frac{\sum_\ell e_\ell z_\ell y_\ell^\perp}{\sum_\ell e_\ell z_\ell x_\ell^\perp}$$7 while remaining statistically significant. In the China-shock application, baseline SSIV reproduces $$\hat\beta=\frac{\sum_\ell e_\ell z_\ell y_\ell^\perp}{\sum_\ell e_\ell z_\ell x_\ell^\perp}$$8; using all shares separately in overidentified 2SLS gives $$\hat\beta=\frac{\sum_\ell e_\ell z_\ell y_\ell^\perp}{\sum_\ell e_\ell z_\ell x_\ell^\perp}$$9; and invalid-share procedures can make the adjusted SSIV more negative, around $g_n$0, although confidence intervals still include the original estimate (Apfel, 2019).

Commodity-price Bartik applications illustrate both the portability and fragility of the design. In “Conflicts and the New Scramble for African Resources,” the main instrument is

$g_n$1

with an incomplete-share control

$g_n$2

Using industry-clustered inference, the paper reports that a \$1 million increase in mineral trade raises the number of conflicts by $g_n$3; oil and fuels drive the results on conflict counts, while the HS26 group of ores, slag, ash, and rare minerals is linked more strongly to fatalities (Boulat, 2024).

The design has also diffused rapidly into political science. A review of 35 articles reports no such studies before 2015 and rapid growth after 2020. The same review argues that the vast majority rely on share exogeneity, whereas shifter exogeneity is underutilized despite its comparable prevalence in economics. It recommends making the final variable explicit in shift-share form, stating clearly whether identification comes from shares or shifters, tailoring controls to that choice, and using the corresponding diagnostics: Rotemberg-weight inspection and share balance tests under share exogeneity, or shock-level controls, shock concentration metrics, and shock-level inference under shifter exogeneity (Park, 24 Feb 2026).

Across these literatures, the central lesson is stable. A shift-share design is not identified merely because it is Bartik-like. Its causal interpretation depends on a defensible account of the exogenous component, on the granularity and dependence structure of the shocks, on whether exclusion can survive decomposition into underlying shares or shocks, and on inference procedures that respect the level at which the quasi-random variation actually enters the design.