Shifter Exogeneity in Shift-Share Designs

Updated 4 July 2026

The paper’s central contribution is an equivalence result that transforms unit-level orthogonality into shock-level exogeneity in shift-share designs.
It imposes quasi-random shock assignment, many dispersed shocks, and limited cross-shock dependence to ensure reliable causal identification.
The framework guides empirical applications by shifting diagnostics, inference, and balance tests to the shock level, improving methodological clarity.

The Shifter Exogeneity Framework is a quasi-experimental interpretation of shift-share, or Bartik, research designs in which identification comes from exogenous shocks rather than from exogenous exposure shares. In its canonical form, a unit-level instrument or treatment is written as a weighted average of common shocks, and causal identification is obtained by imposing orthogonality conditions on the shocks at the shock level, while allowing the shares to be endogenous. The framework’s central contribution is an equivalence result: observation-level orthogonality of a shift-share instrument with an unobserved residual can be rewritten as orthogonality between the underlying shocks and a shock-level unobservable. This reparameterization changes how identification, diagnostics, inference, and practical implementation are understood in shift-share designs (Borusyak et al., 2018).

1. Formal setup and object of identification

In the basic shift-share setup, the instrument is

$z_{\ell}=\sum_{n}s_{\ell n}g_{n},$

where $\ell$ indexes observational units, $n$ indexes shocks, $s_{\ell n}$ are exposure shares, and $g_n$ are shocks. The associated structural equation is

$y_{\ell}=\beta x_{\ell}+w_{\ell}^{\prime}\gamma+\varepsilon_{\ell},$

with controls $w_{\ell}$ , treatment $x_{\ell}$ , and residual $\varepsilon_\ell$ defined as orthogonal to $w_\ell$ in population. The SSIV estimator can be written, after Frisch-Waugh-Lovell residualization, as

$\ell$ 0

where $\ell$ 1 are weights normalized so that $\ell$ 2 (Borusyak et al., 2018).

The object of identification is the orthogonality moment

$\ell$ 3

The key algebraic step is to exchange sums over $\ell$ 4 and $\ell$ 5, yielding

$\ell$ 6

where

$\ell$ 7

This shows that the exclusion restriction can be analyzed at the shock level rather than at the unit level. In substantive terms, $\ell$ 8 must be orthogonal to the exposure-weighted average residual among units most exposed to shock $\ell$ 9 (Borusyak et al., 2018).

A more general shift-share variable is often written as

$n$ 0

with nonnegative shares satisfying $n$ 1 and $n$ 2. This notation is common in political-science treatments of shift-share designs and emphasizes that the same structure can be interpreted either through exogenous shares or through exogenous shifts (Park, 24 Feb 2026).

2. Identification logic and assumptions

The baseline identifying restriction is quasi-random shock assignment: $n$ 3 Under this condition, the shocks are mean-independent of the shock-level unobservable $n$ 4 and the exposure-weight vector $n$ 5. This is the framework’s defining exogeneity condition: the shocks, not the shares, are the object whose exogeneity matters (Borusyak et al., 2018).

Consistency additionally requires many dispersed shocks and limited cross-shock dependence: $n$ 6 The first condition is a dispersed-exposure or large-effective-sample requirement; the second is mutual shock independence or uncorrelatedness conditional on the unobservables. Under these conditions, together with relevance and regularity conditions, $n$ 7 (Borusyak et al., 2018).

The framework also admits conditional shock exogeneity. With observed shock-level controls $n$ 8,

$n$ 9

and, defining $s_{\ell n}$ 0,

$s_{\ell n}$ 1

In that case, exposure-weighted shock controls,

$s_{\ell n}$ 2

must be included in the unit-level regression. A central practical implication is that controls with shift-share structure are the observation-level way of partialling out shock-level confounders (Borusyak et al., 2018).

Political-science expositions of the framework restate these requirements as mean-independence of shifts, shift centering or demeaning, many small negligible shares, independence or weak dependence across shifts, and control variables at the shift level. If shift means vary systematically with observed shift-level covariates $s_{\ell n}$ 3, the framework allows

$s_{\ell n}$ 4

so one can residualize shifts using $s_{\ell n}$ 5 (Park, 24 Feb 2026).

A further complication arises with incomplete shares. If $s_{\ell n}$ 6 varies across observations, the instrument may mix in variation from the sum of shares itself. In that case, one must control for the sum of shares, or more generally for the exposure-weighted sum of shock-level controls. This is important in applications where the shocks apply only to a subset of sectors, such as manufacturing (Borusyak et al., 2018).

3. Equivalent shock-level regression, diagnostics, and inference

A second equivalence result shows that the SSIV estimator is numerically identical to the coefficient from a weighted shock-level IV regression. Define

$s_{\ell n}$ 7

the exposure-weighted average of $s_{\ell n}$ 8 among units exposed to shock $s_{\ell n}$ 9. Then

$g_n$ 0

which is the coefficient from an $g_n$ 1-weighted shock-level IV regression using $g_n$ 2 as the instrument (Borusyak et al., 2018).

This equivalence is interpretive as well as computational. The economic meaning of the estimand remains at the observation level, but estimation, balance testing, and inference can be moved to the shock level. In this representation, conventional heteroskedasticity-robust standard errors from the shock-level regression are asymptotically valid under additional assumptions, and the approach extends naturally to clustered or HAC shocks. It can also be used for first-stage diagnostics and falsification tests (Borusyak et al., 2018).

The framework therefore recommends shock-level balance tests. One can regress placebo or proxy variables on the shift-share instrument with exposure-robust inference, or directly regress shock-level confounders on shocks. This is a substantial methodological shift: when shock exogeneity is the identifying assumption, balance tests should be conducted at the shock level rather than only at the unit level (Borusyak et al., 2018).

Political-science adaptations emphasize related implementation tools: shift-level standard errors and clustering, an effective $g_n$ 3-statistic in the inverted regression framework, and a unit-of-analysis clarification according to which, under shift exogeneity, whether one works at the district or region level should not matter asymptotically so long as the weighting is handled correctly (Park, 24 Feb 2026).

The effective sample size of shocks is also central. In empirical work it is often summarized by the inverse Herfindahl,

$g_n$ 4

which indicates whether the many-shocks approximation is plausible. A design with many observational units but few effective shocks may be weak or poorly approximated by the asymptotic theory (Borusyak et al., 2018).

Shift-share designs can be identified in two different ways: by exogenous shares or by exogenous shifts. The distinction is methodological rather than terminological. Under share exogeneity, the shares are exogenous conditional on covariates and the shifts are treated as fixed. Under shift exogeneity, the shares may be endogenous, but the shifts are exogenous conditional on shares and covariates (Park, 24 Feb 2026).

Dimension	Share exogeneity	Shift exogeneity
Exogenous component	Shares	Shifts
Potentially endogenous component	Shifts	Shares
Main identifying logic	Comparability across units	Many small, comparable shocks

Under share exogeneity, the design has a difference-in-differences-like logic: all units face the same shocks, but they are exposed in different degrees, and those degrees are exogenous. Under shift exogeneity, that analogy breaks down because relative differences across units are driven by endogenous shares. Identification instead comes from the idea that if there are many small, comparable shifts, the bias from any one endogenous share “washes out” in aggregate, much like a law-of-large-numbers argument (Park, 24 Feb 2026).

This difference changes diagnostics and empirical emphasis. Share exogeneity motivates balance tests for shares, pre-trend tests, overidentification tests, and examination of key Rotemberg-weighted shares. Shift exogeneity instead motivates shift-level balance, residualization of shift means, attention to effective shock sample size, and clustered or weak-dependence-aware inference at the shock level (Park, 24 Feb 2026).

Several recurrent mistakes follow from collapsing the two frameworks into a single “Bartik” template. A shift-share design is not automatically valid; the source of exogeneity must be justified. Researchers may mischaracterize the source of exogeneity by stating that shares are exogenous when the actual identifying story depends on shocks, or vice versa. Ordinary clustered standard errors may be incorrect when dependence runs through the shifts. Initial shares are not automatically exogenous, because they can be equilibrium outcomes related to unobservables (Park, 24 Feb 2026).

5. Extensions and conceptual boundaries

Recent work on nonlinear treatment effects extends shift-share analysis in a different direction. In a triangular system,

$g_n$ 5

the control function

$g_n$ 6

is used to correct for endogenous treatment under an exogenous-shares condition,

$g_n$ 7

together with monotonicity of the treatment assignment equation. This framework identifies the Local Average Response, Average Derivative, Average Structural Function, and Policy Effect, and it shows that standard 2SLS can be a weighted average of marginal treatment effects with potentially negative weights when the first-stage derivative changes sign (Garzon et al., 29 Jul 2025).

For the history of the Shifter Exogeneity Framework, the most important point is that this nonlinear extension does not provide a general nonlinear version of shock exogeneity. The paper explicitly notes that an alternative strategy based on exogenous shifts is not pursued because, in a nonlinear setting, the usual equivalence arguments are not available. This marks a boundary of the linear shock-level equivalence: the standard region-industry inversion that underlies shifter exogeneity is a linear result, whereas the nonlinear paper preserves identification from exogenous shares given common shifters (Garzon et al., 29 Jul 2025).

The nonlinear paper also emphasizes that shift-share applications are typically interpreted in first differences, not levels, so the causal objects are effects on changes in outcomes rather than levels. This suggests that even rich nonlinear extensions inherit a basic limitation of the broader shift-share design (Garzon et al., 29 Jul 2025).

A separate use of the term “exogeneity” appears in causal-direction research. There, exogeneity is defined through a factorization

$g_n$ 8

with $g_n$ 9, so that the parameters governing the marginal and conditional distributions are variation free. That framework uses bootstrap-based dependence testing on nonparametric estimates of $y_{\ell}=\beta x_{\ell}+w_{\ell}^{\prime}\gamma+\varepsilon_{\ell},$ 0 and $y_{\ell}=\beta x_{\ell}+w_{\ell}^{\prime}\gamma+\varepsilon_{\ell},$ 1 to infer causal direction. The shared vocabulary should not obscure that this is a different criterion from shifter exogeneity in shift-share designs (Zhang et al., 2015).

6. Empirical applications and substantive significance

The framework is closely associated with the China-shock literature. In the Autor, Dorn, and Hanson setting, observational units are U.S. commuting zones, shocks are manufacturing-industry import growth shocks measured using eight non-U.S. developed economies, and the shift-share instrument is constructed from lagged regional industry shares. The framework interprets this design as relying on quasi-random variation in industry-specific import shocks rather than on exogenous regional shares. This is consequential because the shares are plausibly endogenous, while the shocks are more plausibly random conditional on observables (Borusyak et al., 2018).

Empirically, the framework changes what is inspected. In the ADH application, including the non-manufacturing “shock” at zero produces a concentrated shock distribution and a small effective sample size; excluding the missing service sector yields a more regular shock distribution. The paper finds moderate clustering of shocks at some industry-group levels, recommends appropriate shock clustering for inference, and performs shock-level balance tests for predetermined industry and regional covariates. The main SSIV estimates of the effect of import competition on manufacturing employment remain negative and statistically meaningful across specifications, while controls for lagged manufacturing share and its period interactions matter for the incomplete-shares problem (Borusyak et al., 2018).

In political science, the framework is described as underutilized relative to share exogeneity, even though it is often the more natural identification story in applications involving many comparable shocks. The literature reviewed in that field increasingly uses shift-share designs for trade shocks, immigration, technology shocks, capital movement, and foreign aid, but many papers do not fully articulate the relevant identification assumptions (Park, 24 Feb 2026).

An illustrative replication revisits a European China-shock study on nationalism and far-right voting. There, the design is argued to be naturally a shift-exogeneity setting because the identifying story concerns exogenous variation in Chinese supply conditions rather than endogenous regional shares. The replication introduces shift transformation or residualization using country-industry and year fixed effects, and bases inference on the shift-level structure rather than conventional district-level clustering. Once the design is implemented with these tools, the results are less robust, and some originally significant findings weaken or disappear (Park, 24 Feb 2026).

These applications clarify the framework’s substantive significance. It does not merely provide an alternative proof of consistency. It reassigns the evidentiary burden: the crucial questions become whether shocks are quasi-randomly assigned, whether they are centered or properly residualized, whether there are many sufficiently dispersed shocks, whether dependence across shocks is adequately modeled, and whether diagnostics and inference are conducted at the level where identification actually resides (Borusyak et al., 2018).