Share Exogeneity in Shift-Share Designs

Updated 4 July 2026

The Share Exogeneity Framework is a method in shift-share designs that assumes baseline exposure shares are exogenous conditional on unit-level controls.
It establishes identification by ensuring that the regression error is orthogonal to share weights while allowing shocks to be non-random or correlated with aggregate outcomes.
It underpins IV/GMM estimation in both cross-sectional and panel settings and employs diagnostics like Rotemberg decomposition to assess share balance.

The share exogeneity framework is an identifying strategy for shift-share, or Bartik-style, designs in which shocks are treated as fixed and identification is attributed to exogeneity of the exposure shares. In canonical notation, a shift-share regressor or instrument takes the form $X_i = \sum_{k=1}^K s_{ik} g_k$ or $Z_i = \sum_{s=1}^S w_{is} g_s$ . Under this framework, the econometric content is that shares are orthogonal to the regression error, possibly conditional on unit-level controls, while shocks may be non-random and even correlated with the aggregate outcome process. The framework is therefore conceptually distinct from shock or shifter exogeneity, which allows endogenous shares and instead requires exogenous shocks (Borusyak et al., 2018, Park, 24 Feb 2026).

1. Canonical setup and formal definition

In the cross-sectional formulation emphasized in political science and in the Goldsmith-Pinkham, Sorkin, and Swift tradition summarized by Park, the exposure variable is

$X_i = \sum_j w_{ij} D_j,$

and the outcome equation is

$Y_i = \alpha + \beta X_i + \epsilon_i.$

The shifts $D_j$ are treated as fixed, and unit-level objects $(w_{ij}, \epsilon_i, W_i)$ are i.i.d. across $i$ . The defining orthogonality condition is, for all $j$ with $D_j \neq 0$ ,

$\operatorname{cov}(w_{ij}, \epsilon_i)=0,$

or, with controls $Z_i = \sum_{s=1}^S w_{is} g_s$ 0,

$Z_i = \sum_{s=1}^S w_{is} g_s$ 1

In panel designs, the canonical exposure uses baseline shares,

$Z_i = \sum_{s=1}^S w_{is} g_s$ 2

with outcome equation

$Z_i = \sum_{s=1}^S w_{is} g_s$ 3

or, in IV implementations,

$Z_i = \sum_{s=1}^S w_{is} g_s$ 4

The panel exogeneity condition is

$Z_i = \sum_{s=1}^S w_{is} g_s$ 5

for all $Z_i = \sum_{s=1}^S w_{is} g_s$ 6 with $Z_i = \sum_{s=1}^S w_{is} g_s$ 7. The baseline-share requirement is substantive: shares are fixed at $Z_i = \sum_{s=1}^S w_{is} g_s$ 8 to avoid post-treatment bias (Park, 24 Feb 2026).

This formulation implies that identification comes from exogenous cross-sectional heterogeneity in exposure intensities, not from quasi-random assignment of shocks. Park states the interpretation directly: under share exogeneity, the shares are “as-good-as-random” conditional on controls, while shocks can be non-random and even correlated with the outcome process at the aggregate level. In this sense, the framework treats the shift-share object as an exposure design built on comparability among units rather than comparability among shocks (Park, 24 Feb 2026).

2. Identification logic and its IV/GMM interpretation

The simplest way to see identification is through the bias decomposition in the two-shift model. If

$Z_i = \sum_{s=1}^S w_{is} g_s$ 9

then

$X_i = \sum_j w_{ij} D_j,$ 0

with

$X_i = \sum_j w_{ij} D_j,$ 1

If the share exogeneity condition holds, OLS is consistent for $X_i = \sum_j w_{ij} D_j,$ 2. The same logic generalizes to many shifts. With a single common shock and varying but exogenous exposure shares, Park describes the design as akin to difference-in-differences with continuous treatment intensity; with multiple shifts, one pools many such contrasts. Identification does not require “many shocks”; the shocks may be few and even strongly correlated, because the exogenous objects are the shares (Park, 24 Feb 2026).

A central result is the IV/GMM equivalence. Consider the first stage

$X_i = \sum_j w_{ij} D_j,$ 3

and second stage

$X_i = \sum_j w_{ij} D_j,$ 4

If the GMM weights are set proportional to the realized shocks, $X_i = \sum_j w_{ij} D_j,$ 5, then the fitted $X_i = \sum_j w_{ij} D_j,$ 6 equals the original shift-share $X_i = \sum_j w_{ij} D_j,$ 7. Park therefore states that shift-share OLS under share exogeneity can be seen as a special overidentified GMM/IV estimator using the vector of shares as instruments. This observation places the framework within standard IV logic: the identifying variation is the component of exposure explained by valid share instruments (Park, 24 Feb 2026).

This differs sharply from the shock-exogeneity approach. Borusyak, Hull, and Jaravel show that shift-share orthogonality can instead be represented as

$X_i = \sum_j w_{ij} D_j,$ 8

with identification delivered by

$X_i = \sum_j w_{ij} D_j,$ 9

even when shares are endogenous. In that framework, the operative moment condition is at the shock level rather than the unit level, and the identifying assumptions concern shock orthogonality, many-shock asymptotics, and cross-shock independence or weak dependence (Borusyak et al., 2018).

Because the share exogeneity framework is algebraically equivalent to an IV problem with shares as instruments, estimation follows familiar procedures. If the shift-share variable itself is the regressor and shares are exogenous, OLS is consistent. If the shift-share variable is used as an instrument, the second stage is estimated by 2SLS. Park emphasizes that standard heteroskedasticity-robust IV standard errors are valid under the maintained assumptions, since the framework treats shocks as fixed and places the exogeneity restriction on the shares (Park, 24 Feb 2026).

This inference logic is distinct from the “AKM-type” corrections associated with shifter exogeneity. Under shock exogeneity, special inferential issues arise because the identifying variation is the shock variation and inference must be done at the shock level. Borusyak, Hull, and Jaravel recommend estimating the equivalent shock-level IV regression and using conventional heteroskedasticity-robust, cluster-robust, or HAC standard errors there. Under share exogeneity, by contrast, the key requirement is that shares are valid instruments conditional on unit-level controls, so the special shock-level corrections are not required (Borusyak et al., 2018).

In panel settings, Park notes that the canonical asymptotic environment allows time-series dependence in $Y_i = \alpha + \beta X_i + \epsilon_i.$ 0 within $Y_i = \alpha + \beta X_i + \epsilon_i.$ 1 but assumes i.i.d. across $Y_i = \alpha + \beta X_i + \epsilon_i.$ 2. There is not yet a fully general theory for “geographic” clustering with many overidentifying share instruments. The practical guidance is therefore conventional clustering, for example by unit, by time, or both when appropriate. The econometric content remains unchanged: shares should be baseline, controls should be unit-level and pre-treatment where possible, and the inferential justification is still the standard IV/GMM interpretation of shares as instruments (Park, 24 Feb 2026).

4. Diagnostics, robustness checks, and recurring pitfalls

The most widely used diagnostic under share exogeneity is the Rotemberg decomposition. Park summarizes the core result as

$Y_i = \alpha + \beta X_i + \epsilon_i.$ 3

where $Y_i = \alpha + \beta X_i + \epsilon_i.$ 4 is the coefficient obtained when using only $Y_i = \alpha + \beta X_i + \epsilon_i.$ 5 as the instrument and the $Y_i = \alpha + \beta X_i + \epsilon_i.$ 6 depend only on the covariates. The practical implication is that shares with large absolute Rotemberg weights drive the estimate. Researchers should therefore compute and report Rotemberg weights, substantively justify the exogeneity of the high-weight shares, and probe robustness to excluding high-weight components (Park, 24 Feb 2026).

Share balance tests are the natural analogue of covariate balance in conventional quasi-experimental work. The recommended procedure is to regress key shares $Y_i = \alpha + \beta X_i + \epsilon_i.$ 7, especially those with large Rotemberg weights, on covariates $Y_i = \alpha + \beta X_i + \epsilon_i.$ 8 and on plausible proxies $Y_i = \alpha + \beta X_i + \epsilon_i.$ 9 for the regression error $D_j$ 0, checking whether

$D_j$ 1

In panels with a clear onset of shocks, pre-trend or placebo-period tests are also central. If the exposure measure predicts outcomes in pre-shock periods, the share exogeneity claim is weakened. Overidentification logic provides a further robustness check: one can compare the shift-share estimate to IV estimates that use alternative combinations of the shares and assess sensitivity to particular share components (Park, 24 Feb 2026).

Several failure modes recur in applications. One is the use of time-varying shares in panels, which induces post-treatment bias; the recommended practice is baseline shares $D_j$ 2. A second is mischaracterizing the instrument when additional scaling changes the actual share object. Park’s example is an instrument of the form $D_j$ 3, where the effective share is $D_j$ 4, not $D_j$ 5. A third is ignoring effect heterogeneity: with heterogeneous $D_j$ 6, the shift-share IV estimand is a weighted average of heterogeneous effects, but the weights need not be positive. Park therefore recommends documenting exposure distributions, such as Herfindahl indices, and being explicit about which shares carry identifying leverage (Park, 24 Feb 2026).

5. Nonlinear treatment effects and exogenous shares

Recent work extends the share exogeneity framework beyond linear IV. In the nonlinear treatment-effects formulation of shift-share designs with exogenous shares, the structural system is

$D_j$ 7

The share exogeneity assumption is

$D_j$ 8

The accompanying monotonicity condition is that $D_j$ 9 is continuously distributed given $(w_{ij}, \epsilon_i, W_i)$ 0 and $(w_{ij}, \epsilon_i, W_i)$ 1 is strictly monotonic in $(w_{ij}, \epsilon_i, W_i)$ 2, while the “common shocks” assumption states that one observes a single draw $(w_{ij}, \epsilon_i, W_i)$ 3 of $(w_{ij}, \epsilon_i, W_i)$ 4 within a period (Garzon et al., 29 Jul 2025).

The core device is a control function,

$(w_{ij}, \epsilon_i, W_i)$ 5

Under exogenous shares and monotonicity,

$(w_{ij}, \epsilon_i, W_i)$ 6

This yields the identified conditional mean

$(w_{ij}, \epsilon_i, W_i)$ 7

The framework then identifies four parameters: the Average Structural Function

$(w_{ij}, \epsilon_i, W_i)$ 8

the Local Average Response

$(w_{ij}, \epsilon_i, W_i)$ 9

the Average Derivative

$i$ 0

and the Policy Effect

$i$ 1

The estimation strategy is a three-stage procedure: first-stage quantile regression to estimate $i$ 2, second-stage OLS of $i$ 3 on $i$ 4 to obtain $i$ 5 and its derivative, and third-stage plug-in estimation of ASF, PE, AD, and LAR. Inference uses a weighted bootstrap with exponential weights and uniform confidence bands via maximal $i$ 6-statistics (Garzon et al., 29 Jul 2025).

This nonlinear extension alters the interpretation of linear shift-share IV. The paper shows that the 2SLS estimand can be written as a weighted average of marginal effects with weights $i$ 7, and when $i$ 8 changes sign, the weights can be negative. The resulting estimand is then “not weakly causal.” In the China-shock application, this explains why pooled 2SLS estimates are negative and significant while the Average Derivative is small or near zero and not statistically significant (Garzon et al., 29 Jul 2025).

6. Relation to shifter exogeneity, applications, and scope

The principal alternative to share exogeneity is the shifter, or shock, exogeneity framework. In Borusyak, Hull, and Jaravel, identification follows from quasi-random assignment of shocks while exposure shares are allowed to be endogenous. Their core moment condition is the shock-level orthogonality

$i$ 9

possibly conditional on shock-level controls $j$ 0, together with many-shock asymptotics such as

$j$ 1

and weak cross-shock dependence. Estimation is operationally equivalent to an $j$ 2-weighted shock-level IV regression, and inference is correspondingly shock-level rather than unit-level (Borusyak et al., 2018).

Park’s synthesis for political science stresses that the two frameworks correspond to different substantive defenses. Share exogeneity is appropriate when baseline shares are plausibly exogenous conditional on unit-level controls and the design emphasizes comparability across units. Shifter exogeneity is appropriate when shares are plausibly endogenous but shocks are many, small, and exogenous after demeaning or conditioning on shift-level covariates. Park observes that political science articles tend to rely on share exogeneity, suggesting that the shifter exogeneity framework is underutilized despite its comparable prevalence in economics (Park, 24 Feb 2026).

Empirically, the share exogeneity framework underlies many canonical applications. Park discusses immigration and other Bartik-style instruments in political science, where baseline shares are defended as predetermined and shocks are common across units. Borusyak, Hull, and Jaravel use the China-shock setting to illustrate the contrasting shock-exogeneity route. Borusyak and coauthors’ nonlinear treatment-effects paper returns to the same empirical domain but imposes exogenous shares, $j$ 3, to recover heterogeneous and nonlinear responses (Borusyak et al., 2018, Garzon et al., 29 Jul 2025, Park, 24 Feb 2026).

Taken together, these results define the modern meaning of the share exogeneity framework. It is not a generic synonym for exogeneity, but a specific identifying strategy for shift-share designs: shifts are treated as fixed, shares are the exogenous objects, estimation is interpretable as IV/GMM with shares as instruments, diagnostics focus on Rotemberg weights, balance, and pre-trends, and extensions to nonlinear treatment effects require control-function structure rather than linear IV alone. Its credibility therefore depends less on the randomness of shocks than on the substantive and statistical case that exposure shares are orthogonal to latent outcome determinants, conditional on the controls carried by the design (Garzon et al., 29 Jul 2025, Park, 24 Feb 2026).