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Shift-Share Designs in Political Science
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A comprehensive introduction to shift-share methods lacks despite a growing interest within political science. The need arises partly from the complexity of the methods themselves, and partly from that of the literature. The historical literature can be confusing without backgrounds as the term shift-share has appeared in various contexts. Moreover, recent methodological developments have yet to be organically synthesized for those unfamiliar with the methods. \footnote{BorusyakHullJaravel2025b synthesize the literature from a methodological viewpoint but assume some familiarity with shit-share methods as they are prevalent in modern economics. This article can be read together with theirs.BorusyakHullJaravel2025a a technical review.} This article aims to fill both gaps by introducing the usage of the methods in the literature and the new identification strategies, while maintaining their relevance for political scientists.
The first half of the article reviews the economics literature. Shift-share designs refer to research de- signs with shift-share measures. Suppose $n$ units are subject to $m$ common treatments by varying degrees. Formally, unit $i$ is exposed to the $j$th treatment $D_j$ (“shift”) by a non-negative weight $w_{ij}$ (“share”), where weights sum to less than $1$ in each unit: $w_{ij} \ge 0 \quad \text{and} \quad \sum_{j=1}^{m} w_{ij} \le 1$. The shift-share measure for unit $i$ is defined as
We use the China shock for the working example. Other leading examples are in Section (ref). AutorDornHansonMajlesi2020 study whether imports from China caused political polarization in the United States. Regions are exposed to common national trade shocks, but by varying degrees depending on the presence of each industry in the regions. Under the above notation, units $i$ are regions, treatment units $j$ are industries, and weights $w_{ij}$ are regional employment shares of each industry. Shifts $D_j$ compute the decadal national change in Chinese imports divided by the national demand for each industry, referred to as import penetration.
The shift-share measure performs two tasks. First, it provides a summary measure for comparable yet potentially heterogeneous shifts. While different industries may have different effects on regional electoral outcomes, a single regional import shock measure may reasonably summarize these varied effects. Second, it imputes an unobservable unit-level treatment with proxy shifts. A direct measure of the import shock would exploit the regional import change divided by the regional demand for each industry. Since trade data is often only available at the national level, the shift-share measure proxies the direct one by combining national import data with regionally available data.
However, employing a measure that combines two sets of variables complicates the statistical analysis. Establishing the exogeneity of even a single source is already a demanding task that warrants a dedicated section in most empirical articles. For example, shifts might not be exogenous to the outcome variable in our working example. Consider two regions that have similar industry portfolios. They are likely to share similar demographic composition and economic interests, and hence prone to similar unobserved electoral shocks. These confounders may lead to the omitted variable bias if unaccounted for. The concerns about identification have emerged only more recently despite the long history of the use of the methods.
The literature thus proceeded with the exogeneity of either shifts or shares instead of both. The share exogeneity framework assumes that shares are exogeneous conditional on covariates GoldsmithPinkhamSorkinSwift2020. This framework allows shifts to be determined endogenously to unit-level outcomes,\footnote{Since shifts are common across units, such endogeneity may exist with respect to the joint distribution of the unit-level outcomes --- e.g. when shifts covary with the average of the stochastic terms in the outcome variable.} yet the relative differences in shifts effectively applied across units are exogenously determined through the shares. This setting is a classical difference-in-differences design if the multiple shifts were considered separately.
The shift exogeneity framework assumes that shifts are exogeneous conditional on covariates AdaoKolesarMorales2019,BorusyakHullJaravel2022. The difference-in-differences analogy breaks down since the relative differences are now determined by endogenous shares. One might instead hope that if no single share is too large, the biases due to the endogeneity in the relative differences cancel out much like the law of large numbers. Such cancellation requires not only small shares but also a large number of shifts. BorusyakHull2023 propose an alternative method using randomized inference under a stronger assumption that shifts follow the same distribution and are interchangeable.
The second half of this article reviews the use of shift-share designs in political science. Shift-share designs are increasingly common, especially in areas close to economics such as technology shock, capital movement and foreign aid. Many studies directly import shift-share variables developed in economics into political science applications. While these studies have generated important findings in their own right, the methods appear to hold even greater potential within political science. It is especially striking that American politics has seen the fewest applications of shift-share methods given the field’s unparalleled access to rich data on complex interactions among different sets of actors, including elections and donation networks.
However, identification is not often justified thoroughly even in recently published articles. Only about half of the articles published since 2021 cite one of the above two exogeneity frameworks. Moreover,Many of those that do merely state the assumptions without substantively arguing for them or conducting falsification tests to assess their plausibility. The fact that articles mostly rely on share exogeneity may suggest either that the framework is being misapplied or that the shift exogeneity framework remains underutilized. I propose three dimensions of shift-share designs to consider when designing shift-share variables and their identification strategies.
I replicate ColantoneStanig2018b to illustrate how to discuss identifying assumptions. The authors study the impact of Chinese import on electoral outcomes, similarly to AutorDornHansonMajlesi2020 but in European countries instead of the United States. BorusyakHullJaravel2022 argue that the original China shock measures by AutorDornHanson2013 that use the US regional employment shares align with the shift exogeneity framework. This article finds that their European counterpart needs an additional shift transformation to justify the framework, and that correct procedures overturn the statistical significance of the findings. This exercise demonstrates that researchers must take extra care to tailor shift-share designs to their specific contexts. The transformation and residualization schemes developed here can be applied in other shift-share designs. talk more about residualization.
The paper is organized as follows. Section (ref) introduces three leading examples that further motivate shift-share designs. Section (ref) surveys the historical use of shift-share designs in economics. Section (ref) synthesizes two main identification strategies under the share exogeneity and the shift exogeneity. Section (ref) reviews the use of shift-share designs in political science and makes comments. Section (ref) illustrates the shift exogeneity framework through replicating a paper. Section (ref) concludes the paper.
AutorDornHansonMajlesi2020 Continuing from Section (ref), both shares and shifts can be endogenous to electoral outcomes if unobserved domestic confounders affect employment, import and politics at the same time. \footnote{AdaoKolesarMorales2019 a structural model that induces non-trivial correlations between employment, import and wage in the context of AutorDornHanson2013} The authors construct a shift-share instrument that interacts local US employment with Chinese import to other advanced economies comparable to the US: \[ Z_i = \sum_j w_{ij} D_j^{*}, \] where $D_j^{*}$ denotes the counterfactual shifts. This instrument isolates the export shock originating from China, which is plausibly exogenous to US electoral outcomes. The implied identification strategy is shift exogeneity.
FoukaTabellini2022 The authors ask whether Mexican immigrants to the United States have changed white Americans’ perception of Black Americans. Since immigrants can endogenously choose where to move in upon observing local characteristics, they construct a “Bartik” instrument that interacts the initial regional share of Mexicans $w_i$ with the national inflow of Mexicans in subsequent decades $D_t$: $Z_{it} = w_i D_t$.\footnote{$Z_{it}$ is then scaled by the predicted regional population. The methodological implications will be discussed in Section (ref).} The instrument redistributes the national immigrant inflow accordingly to the initial distribution of immigrants. The national inflow may correlate with the joint distribution of regional perceptions (see footnote 2), but the initial regional share $w_i$ is free from any time-varying confounders. The identification strategy is share exogeneity.
NunnQian2014 The authors asks how US food aid affects the incidence of conflict in recipient countries. Since food aid may be triggered by conflict, they instrument aid with US wheat production, noting that wheat aid has been used as a way to institutionally handle surplus food production. To maximize the instrument power, they further interact wheat production with the fraction of years a recipient country received food aid during the study period: $Z_{it} = w_i D_t$ where $w_i$ is the fraction and $D_t$ is wheat production. The instrument captures the prospective amount of food aid determined by a potentially endogenous history $w_i$ (referred to as “propensity score” in some articles) and an exogenous food production $D_t$. The implied identification strategy is shift exogeneity.
This section traces the historical development of shift-share methods and shows their wide application in the economics literature. The term “shift-share” was first coined in the context of shift-share analysis or shift-share decomposition, a technique that descriptively decomposes a single variable into several components Dunn1960,EstebanMarquillas1972,Lemieux2002. As one of the first examples, Perloff1957 calculates the “expected” income per capita of each state and the deviance from the expected income using the national average income and the local industry shares. Denote $D_{ij}$ for the income per capita of state $i$ and industry $j$, and $w_{ij}$ for the employment share of industry $j$ in state $i$. The income per capita $X_i$ of state $i$ can be decomposed into \[ X_i = \sum_j w_{ij} D_{ij} = \underbrace{\sum_j w_{ij} \bar D_{\cdot j}}_{\text{expected income}} + \underbrace{\sum_j w_{ij}\left(D_{ij}-\bar D_{\cdot j}\right)}_{\text{regional income shock}} , \]
where $\bar D_{\cdot j}$ is the national average income of industry $j$.
$Z_{it}$ is then scaled by the predicted regional population. The methodological implications will be discussed in Section (ref).
If the share $w_{ij}$ varies over time so $w_{ijt}$ denotes the share $w_{ij}$ at time $t=0,1,\ldots$, we can express $w_{ijt}$ as the sum of the initial share and the change over time $w_{ij0} + (w_{ijt}-w_{ij0})$ and break down $X_i$ into three components:
Equation (ref) has two deviance terms each due to the change in shares and to idiosyncratic income shocks. This technique is often used to assess the relative explanatory power of multiple possible causes of phenomena BoundJohnson1992,OlivettiPetrongolo2016,BursteinMoralesVogel2019,FreemanGanguliHandel2020. For example, Equation (ref) can shed light on whether industry-level income shocks or changes in shares contributed more to the average change in income. While modern literature has advanced into non-linear or nonparametric decompositions, this linear method remains popular for its simplicity.
Shift-share methods later appeared in structural models as Bartik instruments. Bartik (1991) regresses local wage on local labor supply to estimate the inverse elasticity of labor supply. Since price and quantity are simultaneously determined in the market, the author introduces an instrument inspired by equation (2). Redefining $D_{ijt}$ as labor supply of region $i$, industry $j$ and time $t$, the idea is that the expected labor supply $\sum_j w_{ij0}\bar D_{\cdot jt}$ is correlated with the actual labor supply $X_{it}$ through the initial share $w_{ij0}$ but not with time-varying confounders. The argument relies on the assumption that national labor supply $\bar D_{\cdot jt}$ reflects labor demand shocks rather than labor supply shocks, leaving open the critique noted in footnote 2.
This instrumentation differs from shift-share decomposition in that the dimension $j$ is not predefined. Researchers are free to choose any dimension $j$ to construct Bartik instruments as long as the independent variable $X_i$ can be decomposed along the dimension $j$ and a plausible “average” $\bar D_{\cdot j}$ exists. Therefore, multiple instruments may exist for the same independent variable depending on how it is decomposed (the dimension of $j$) and how individual shifts are approximated (the nature of $\bar D_{\cdot j}$).
Bartik instruments subsequently gained popularity in the wider economics literature. As region and industry are two natural orthogonal levels of analysis, economists have constructed Bartik instruments by interacting the same industry shares with various national outcomes such as earnings Luttmer2005, diamond2016location, hours worked BoundJohnson1992, BoundHolzer2000, productivity and technology shocks GouldWeinbergMustard2002, AcemogluRestrepo2020, and immigration inflow Card2001. Regional variables can be decomposed along other dimensions such as immigration inflow by skill groups Card2009, income growth by income groups BoustanFerreiraWinklerZolt2013, credit shifts by banks GreenstoneMasNguyen2020, and price changes by housing characteristics GrahamMakridis2023.
Sometimes the term Bartik is used to emphasize the approximation of regional variables with national shifts when no decomposition is involved. construct a counterfactual number of local Mexican immigrants using national immigrant inflows rather than decomposing the number of local immigrants. GabrielKleinPessoaForthcoming study the effect of austerity on political extremism and instrument regional austerity measures with national austerity measures, multiplied by the ratio of regional to national per capita government spending. The multiplier represents each region’s sensitivity to government spending.
Shift-share designs or variables focus on the special inner product structure of the variable as in Equation (ref) with Bartik instruments as a special case. A prominent example is AutorDornHanson2013 who first introduced the China shock measure and instrument. The authors use shift-share variables not only for instruments but also for noisy estimates of local exposure to Chinese imports. The “China shock” measure has drawn wide attention and has been applied to various macro outcomes such as employment AcemogluAutorDornHansonPrice2016, mortality PierceSchott2020, marriage rate AutorDornHanson2019, public goods provision FelerSenses2017, and polarization AutorDornHansonMajlesi2020; as well as to contexts outside the US DauthFindeisenSuedekum2017,BaroneKreuter2021.
Another difference between the China shock approach and Bartik instruments is that the former consider horizontal counterfactual shifts (Chinese export to the US versus to comparable economies) instead of hierarchical counterfactual shifts (regional shifts versus national average) to construct the instrument. This is to isolate the exogenous component of Chinese import attributable to Chinese factors rather than domestic ones, thereby making the shift exogeneity framework more appropriate. StuenMobarakMaskus2012 study the effect of foreign doctoral student supply at US universities on scientific publications, and similarly build an instrument by interacting shares of students from each source country at a given university-field with the number of doctoral students choosing other host countries.
Other examples highlight the versatility of shift-share designs through creative choices of units and shares. Kovak2013 measures regional exposure to trade liberalization by interacting price differentials per industry with composite shares derived from a structural model, which consist of regional industry labor shares, the elasticity of substitution between production factors, and their cost shares. HummelsJorgensenMunchXiang2014 define firms as units and measure their exposure to transportation costs by interacting changes in national transportation cost with shares of input source countries for each firm. AcemogluLinn2004 define new drug categories as units and measure their exposure to demographic changes by interacting demographic changes per age group and age profiles of users for each drug category. XuChenzi2022 defines ports as units and measures their exposure to bank failure by interacting bank failure rates and their credit exposure to each bank.
The methodological definition of shift-share distills the common identification challenges that arise in any study using shift-share variables. While it does not prescribe how to construct shift-share measures or instruments compared to Bartik instruments, it conversely broadens applicability of the methods. Researchers are free to use any shift-share designs that satisfy the identifying assumptions.
This section summarizes identification and inference strategies in shift-share designs. BorusyakHullJaravel2025b provide high-level intuition for the two frameworks and practical guidance to commonly asked questions. This section produces a mathematical summary of the frameworks using consistent notations. This is to help potential users of the methods engage with the original articles that proposed these strategies from different motivations and perspectives. Appendix B contains technical details.
The biggest challenge in shift-share designs is that shares and shifts are rarely both exogenous. One plausibly exogenous variation is already hard to find in observational studies; finding two is much harder. Two distinct statistical problems may arise. First, the identification problem: OLS or 2SLS estimates involving shift-share variables might be biased if units systematically select into different treatments due to non-randomness in shifts or shares. We will see that this does not happen under the identifying assumptions of either framework.
Second, the inferential problem: the typical cluster-robust standard error estimator may underestimate the true standard error even when the point estimate is valid. AdaoKolesarMorales2019 show that this can occur when shares exhibit a non-trivial correlation structure that does not align with the chosen clustering scheme. Consider the China shock example. Errors are typically clustered at the state level in geographical studies. However, units with similar industry shares may have correlated errors regardless of geographical proximity since employment and wages---the outcome variable in AutorDornHanson2013---are functions of the same supply and demand shifters, potentially generating complex correlation structures. The authors therefore propose an inference procedure that accommodates arbitrary correlations in the error term. This inferential problem does not arise under share exogeneity.
This section summarizes findings in GoldsmithPinkhamSorkinSwift2020Goldsmith-Pinkham. Share exogeneity holds when shares in the shift-share measure are orthogonal to the error term. We treat shifts as fixed to allow for an arbitrary correlation structure, and everything else as i.i.d.\ across units.
\paragraph{Identification} Consider a simple model with two shifts: $Y_i = \alpha + \beta X_i + \epsilon_i \quad \text{and} \quad X_i = w_{i1}D_1 + w_{i2}D_2$. The regression of $Y_i$ on $X_i$ yields $\hat\beta = \frac{\widehat{\operatorname{cov}}(Y_i,X_i)}{\widehat{\operatorname{var}}(X_i)} = \beta + \frac{\widehat{\operatorname{cov}}(X_i,\epsilon_i)}{\widehat{\operatorname{var}}(X_i)}$, and the covariance between $X_i$ and $\epsilon_i$ is $\operatorname{cov}(w_{i1}D_1,\epsilon_i) + \operatorname{cov}(w_{i2}D_2,\epsilon_i) = D_1\cdot\operatorname{cov}(w_{i1},\epsilon_i) + D_2\cdot\operatorname{cov}(w_{i2},\epsilon_i)$.
The last operation uses that $D_1$ and $D_2$ are fixed.\footnote{Fixing shifts means that if we allow shifts to be stochastic, consistency holds as long as the share exogeneity is satisfied for each realized value of shifts $(D_1,D_2)$. This of course would hold if shares were randomized.} Share exogeneity assumes that shares are uncorrelated with the error term: $\operatorname{cov}(w_{ij},\epsilon_i)=0$ for all $j$ such that $D_j\neq0$. Therefore, the OLS estimate $\hat\beta$ consistently estimates the true $\beta$ under the identifying assumptions.
Identification under share exogeneity is analogous to difference-in-differences designs. In a shift-share design with one shift, units are exposed to a single common shock but to varying degrees exogenously determined by shares. Share exogeneity thus ensures comparability between units. Designs with multiple shifts merely pool multiple difference-in-differences designs. This implies that neither the nature nor the number of shifts helps identification.
\paragraph{Inference} We can view the OLS regression as an IV regression where $X_i$ instruments itself. Consider an IV model where exogenous shares $w_{i1}$ and $w_{i2}$ instrument the shift-share measure $X_i$ separately: \[ X_i = \gamma + \delta_1 w_{i1} + \delta_2 w_{i2} + \eta_i, \] \[ Y_i = \alpha + \beta X_i + \epsilon_i . \] Note that the model is identified as we use two instruments for one independent variable. The Generalized Method of Moments (GMM) estimator can impose arbitrary weights on the first-stage coefficients $\delta_1$ and $\delta_2$, but inference (and consistency) is valid regardless of the specific values at which weights are fixed. Weights $\delta_1/\delta_2=D_1/D_2$ immediately yield $\delta_1=D_1$, $\delta_2=D_2$ and the fitted first-stage value $\hat X_i=X_i$. This shows that shift-share regressions are a special case of GMM under share exogeneity.
\paragraph{Control variables} Zero covariance conditions may be weaker than mean-independence $\mathbb{E}[\epsilon_i \mid w_{ij}]=0$ or full independence $\epsilon_i \perp w_{ij}$, but they remains strong as they must hold for every shift $j$. We consider two relaxations. First, we allow shares to be exogenous conditionally on control variables: $\operatorname{cov}(w_{ij},\epsilon_i \mid \Pi_i)=0$ where $\Pi_i$ denotes controls. These controls can be included directly in the OLS regression. To see why this works, observe that $\Pi_i$ can be included as exogenous variables in the IV regression.
\paragraph{Rotemberg decomposition} Second, we may instead require zero covariance conditions to hold only for the shifts that matter most for the final estimate. Let $\hat\beta_j$ denote the estimate instrumented with the $j$th share alone. The shift-share regression coefficient can then be expressed as a weighted average of $\hat\beta_j$: \[ \hat\beta_{\text{shift-share}}=\sum_{j=1}^{m}\hat\alpha_j\hat\beta_j \] where the constants $\hat\alpha_j$, known as Rotemberg weights, depend only on the covariates and sum to $1$. $\hat\beta_j$ is not consistent if the $j$th share violates exogeneity. This implies that the shift-share estimate will be reasonably accurate as long as the shares with the largest absolute Rotemberg weights are exogenous. Researchers must be ready to defend the exogeneity of these shares more than others. R package bartik.weight and Stata package bartik-weight are available for weight calculation.
\paragraph{Panel settings} We write the full panel model as follows: \[ Y_{it}=\beta X_{it}+\gamma^\top\Pi_{it}+\epsilon_{it}, \] \[ Z_{it}=\sum_{j=1}^{m} w_{ij0}D_{jt} \] where $t=0,\ldots,T$ denotes time, $Z_{it}$ instruments $X_{it}$, and $\{\epsilon_{i0},\ldots,\epsilon_{iT}\}$ are independent across $i$.
Two points are worth noting. First, the model allows for temporal correlations in the error term but not spatial correlations across units, meaning it cannot accommodate geographically clustered errors. This is due to a lack of a theory of inference under clustering and overidentification (footnote 14 in GoldsmithPinkhamSorkinSwift2020), although BorusyakHullJaravel2025b recommend conventional clustering for practical purposes. Second, the instrument $Z_{it}$ fixes shares at their initial values. This is to avoid post-treatment bias that can arise if current shares were partly shaped by past shocks. FoukaTabellini2022 use the initial spatial distribution of Mexican immigrants for this reason.
\paragraph{Diagnostic tests}Share exogeneity assumptions can be indirectly tested. First, the conditional covariance $\operatorname{cov}(w_{ij},T_i \mid \Pi_i)$ must be zero for shares $j$, especially with the largest Rotemberg weights, if $T_i$ proxies the error term $\epsilon_i$. $T_i$ can be any variables that affect the dependent variable not through the share instruments such as any labor supply shocks in Bartik1991. This can be practically done by regressing shares on covariates and interpret the regression results GoldsmithPinkhamSorkinSwift2020. Second, the estimate $\hat\beta$ should be zero in pre-treated periods if exist, analogous to pre-trend tests in difference-in-differences designs. The immigration regime change of $1965$ in FoukaTabellini2022 is an example of the onset period. Third, since the shift-share instrument is one specific way to combine individual share instruments, the estimate should be robust to alternative ways to exploit multiple instruments. Overidentification tests formalize this idea GoldsmithPinkhamSorkinSwift2020.
\paragraph{Effect heterogeneity} The discussion so far has assumed units are homogeneous. How does the method fare under effect heterogeneity? Suppose the second-stage regression is replaced by $Y_i=\alpha+\beta_i X_i+\epsilon_i$ where $\beta_i$ denotes heterogeneous treatment effects. It turns out that the shift-share estimate converges in probability to a weighted average of individual treatment effects $\beta_i$, but the weights are positive only if all unit-level effects are either uniformly positive or negative and all Rotemberg weights are positive.\footnote{If heterogeneity is also introduced in the first stage by writing $X_i=\sum_j w_{ij} t_{ij}D_j$ where $t_{ij}$ captures the true marginal effect of the $j$-th shift on the $i$-th unit’s outcome, the weights are convex only when shares are uncorrelated with one another. This condition is not attainable when shares are complete HahnKuersteinerSantosWilligrod2024.} Since achieving both conditions is challenging, the shift-share regression may not provide a reliable estimate under effect heterogeneity.
Shift exogeneity holds when the shift distribution is mean-independent of the error term and shares. This framework applies when there are no viable selection-on-observables strategies for shares. It instead identifies conditions where comparable shifts can be combined into a single “proper” treatment. We treat shifts as random and independent, and everything else as fixed. A non-random sequence of shares and errors (a triangular array) will be considered in asymptotics. Formal discussions are relegated to Appendix.
\paragraph{Invalid instruments} Under shift exogeneity, the shift-share regression is the same IV regression as in share exogeneity but with invalid instruments KolesarChettyFriedmanGlaeserImbens2015. Since shares are endogenous, the Rotemberg decomposition implies that the shift-share estimate is a weighted average of biased estimates. AdaoKolesarMorales2019 conditions under which these biases cancel out analogously to the law of large numbers.
Two conditions are needed. First, shifts must be demeaned. This is to purge out the systematic variation in the shift-share variable. Consider the two-shift model from the previous section.
As we are treating shares as fixed, units select into different aggregate treatments by the potentially endogenous share distribution unless $\mathbb{E}[D_1]=\mathbb{E}[D_2]=0$.\footnote{As noted in footnote 5, these expectations have already been conditioned on shares and errors by considering them fixed.}
Second, no shares can asymptotically dominate the others. This is for each endogenous share to contribute only a limited bias to the shift-share estimate so that the biases collectively cancel out. Shifts cannot also be strongly correlated as otherwise cancellation would fail from correlations among biases. Note that identification requires a large number of shifts as well as units unlike the share exogeneity framework.
\paragraph{Regression inversion} BorusyakHullJaravel2022, Hull and Jaravel (2022) propose an alternative way to understand identification. Consider the two-shift model again, with complete shares: $\sum_j w_{ij}=1$ for all $i$. Incomplete shares are discussed below. We first invert the regression turning shifts into “observations.” Taking a weighted average of the original second-stages $Y_i=\alpha+\beta X_i+\epsilon_i$ where each unit $i$ is weighted by its $j$-th share $\frac{w_{ij}}{\sum_i w_{ij}}$ yields
\[ \left\{
\right. \]
This is a regression model with two “observations.” Note that the inversion does not affect the coefficients $\alpha$ and $\beta$. It turns out that the shift-share estimate is mechanically equivalent to the IV estimate in this inverted regression if shifts $D_1$ and $D_2$ instrument “endogenous variables” $\frac{\sum_i w_{i1}X_i}{\sum_i w_{i1}}$ and $\frac{\sum_i w_{i2}X_i}{\sum_i w_{i2}}$ with particular weights $\sum_i w_{i1}$ and $\sum_i w_{i2}$.\footnote{Weights here apply to units similarly to some geographical regressions that adjust for regional population size. Compare these with the GMM weights applied to overidentified moment conditions in the share exogeneity framework.}
Identification is achieved as long as shifts $D_j$ are orthogonal to the inverted errors $\epsilon'_j$ that consist of errors $\epsilon_i$ and shares $w_{ij}$. Since errors and shares are fixed, shifts must be not only mean-independent but also mean-zero to achieve the zero-correlation condition $\operatorname{cov}(D_j,\epsilon'_j)=0$. Consistency of the IV estimate further requires many uncorrelated shifts and restrictions on shares so that each regression weight $\sum_i w_{ij}$ becomes asymptotically negligible.
\paragraph{Inference} The equivalence result between the shift-share regression and the inverted regression applies only to the point estimate and not to the standard error. The typical heteroskedasticity-robust estimator in the inverted regression produces valid standard errors by the Central Limit Theorem if shifts are independent, or the cluster-robust estimator if shifts are clustered. Note that the inference remains valid under any dependence structure among shares and errors, and only the dependence structure of shifts matters in the inference procedure provided that shares satisfy negligibility. Available in R/Stata package ssaggregate and R package ShiftShareSE.
\paragraph{Control variables} If shifts are exogenous but centered around different means, one may directly model the means. Assume shifts are linear in controls such as shift-level covariates or dummies that represent ex-ante known shift groups: \[ \mathbb{E}[D_j]=\gamma^\top p_j . \] The demeaned shift-share measure \[ \sum_j w_{ij}(D_j-\gamma^\top p_j)=X_i-\gamma^\top\Bigl(\sum_j w_{ij}p_j\Bigr) \] would achieve identification if other assumptions are satisfied. $\gamma$ will be consistently estimated in the regression of shifts $D_j$ on their covariates $p_j$ as the number of shifts increases. Finally, estimating $\hat\gamma$ and demeaning $D_j$ is equivalent to controlling for an additional vector $\sum_j w_{ij}p_j$ in the shift-share regression by the Frisch--Waugh--Lovell theorem.
The inverted regression method also requires controls $\sum_j w_{ij}p_j$ before inversion, which are inverted using the same weights $\frac{w_{ij}}{\sum_j w_{ij}}$ as the shift-share measure. As a special case, this control scheme accommodates incomplete shares: $\sum_j w_{ij}<1$. Introduce a hypothetical shift $D_{m+1}$ that is identically zero with a complementary share $w_{i(m+1)}=1-\sum_j w_{ij}$. Define a shift-level dummy by setting $p_j=1$ for $1\le j\le m$ and $p_{m+1}=0$. Shares are now complete with the new shift, and the regression can be inverted with an additional control $\sum_{j=1}^{m+1} w_{ij}p_j=\sum_{j=1}^{m} w_{ij}$. This control is also needed in the invalid instrument approach.
\paragraph{Panel settings} We write the full panel model as follows: \[ Y_{it}=\beta X_{it}+\gamma^\top\Pi_{it}+\epsilon_{it}, \] \[ Z_{it}=\sum_{j=1}^{m} w_{ijt}D_{jt}, \] where $t=1,\ldots,T$ denotes time and $Z_{it}$ instruments $X_{it}$. Rewrite the shift-share measure $Z_{it}$ in long form by including all $T\times m$ shifts across periods and assigning zero weights to shifts outside period $t$. Both approaches immediately apply without having to fix shares at their initial values. Clustering may be required if shifts are temporally correlated.
\paragraph{Diagnostic tests} Share conditions and shift correlation can be directly tested, while shift exogeneity assumptions can be indirectly tested similarly to share exogeneity. First, \[ \operatorname{cov}\!\left(D_j,\frac{\sum_i w_{ij}T_i}{\sum_i w_{ij}}\right)=\operatorname{cov}(D_j,T'_j)=0 \] for unit-level covariates $T_i$ and shift-level covariates $T'_j$ that proxy error terms $\epsilon_i$ and $\epsilon'_j$, after partialling out shift-level covariates $p_j$. Second, pre-trend tests if there exists the onset period. Third, HahnKuersteinerSantosWilligrod2024 propose an overidentification test. The intuition is that if shifts are mean-independent of shares and errors, any functions of them cannot be correlated with the demeaned shifts, hence coefficients $\gamma$ to the shift-level controls $p_j$ being overidentified. Section (ref) illustrates diagnostic tests with an example.
\paragraph{Effect heterogeneity} The shift-share estimate $\hat\beta$ always converges in probability to a convex average of individual treatment effects when both the second-stage effects and the shifts are allowed to be heterogeneous. However, if randomness is introduced into shares as well, negative weights may occur unless every pair of shifts is positively correlated HahnKuersteinerSantosWilligrod2024.
The previous identification strategy had two major challenges. First, individual biases had to cancel out, which required a large number of shifts and a complex asymptotic analysis to justify consistency under arbitrary share and error distributions. Second, if shifts were not mean-zero, they had to be demeaned by modeling and estimating their means.
Both challenges can be bypassed by instead assuming that shifts are independent of errors conditional on shares and can be grouped so that they are exchangeable within each group (Borusyak and Hull 2023).\footnote{Exchangeability means that the joint distribution of shifts is invariant under permutation: $(D_1,\ldots,D_n)\overset{d}{=}(D_{\sigma(1)},\ldots,D_{\sigma(n)})$ for any permutation $\sigma$. This condition implies identical distributions but relaxes independence. For example, jointly normal shifts with a common correlation are not i.i.d.\ but exchangeable.}
The testing procedure consists of two steps. First, fix the effect size $\beta$ at some value. Second, randomize shifts and calculate the cross-moment between the shift-share instruments $Z_i$ and the residuals $Y_i-\beta X_i$ for the test statistic. Repeat the process for many values of $\beta$. The point estimate is the $\beta$ such that the test statistic under real shifts is the mean of the test statistics under randomization, and the confidence interval collects all $\beta$ such that the test statistic under real shifts is not too extreme in the test statistic distribution under randomization.\footnote{This randomization test is valid in more general settings where both instrument formula and the design are known, meaning that instruments are known functions of exogenous and endogenous components and the shift assignment process is prespecified.}
Asymptotics can be dispensed with since randomization inference is exact. Demeaning can be also dispensed with since it merely shifts each test statistic by a constant: \[ \mathbb{E}\big[(\tilde Z_i-\mu_i)(Y_i-\beta X_i)\big] =\mathbb{E}\big[\tilde Z_i(Y_i-\beta X_i)\big] -\mathbb{E}\big[\mu_i(Y_i-\beta X_i)\big], \] where $\tilde Z_i$ is the recalculated instrument with randomized shifts and $\mu_i=\mathbb{E}[Z_i]$. Since the second term does not vary under shift randomization, demeaning does not affect the point estimation nor the interval estimation.
We conclude the section with three remarks. First, if the second-stage includes covariates, residualize the outcome variable $Y_i$ and the independent variable $X_i$ over the covariates before performing the permutation test (Appendix C.6). Second, although the test is exact under finite shifts, it imposes a stronger requirement that shifts be identically distributed, implying that all moments must coincide instead of the first moment in the previous section. Finally, as randomization inference tests the sharp null, results are harder to interpret under potential effect heterogeneity.
Share exogeneity and shift exogeneity rely on different sets of identifying assumptions. The former exploits comparability among units, while the latter leverages comparability among shifts. Empirical designs that researchers have in mind may suggest which framework is more suitable. Share exogeneity is implied when they focus on similarity among units (through share exogeneity), or shocks to specific industries that are key to identification (those with potentially large Rotemberg weights). Shift exogeneity is relevant when they emphasize many comparable shocks or finite identical shocks unrelated to unit characteristics.
Share exogeneity might not be applicable if shares are codetermined with the outcome variable in a sort of equilibrium as in AutorDornHanson2013 where both shares and errors have similar shift-share structures. The theory also does not yet accommodate geographically clustered errors. Shift exogeneity requires covariates that fully capture variation existing in shift means, as well as stronger assumptions on shifts such as mean-independence or identical distribution, in contrast to the zero correlation condition under share exogeneity. Neither framework provides fully satisfactory causal interpretation under effect heterogeneity.
What does one do if both shifts and shares are endogenous? One may consider exogenize one of them by fixing shares at their initial values FoukaTabellini2022 or finding counterfactual exogenous shifts AutorDornHansonMajlesi2020. HahnKuersteinerSantosWilligrod2024 explore the possibility that identification comes partly from shifts and partly from shares. In rare examples where both are exogenous, share exogeneity typically yields smaller standard errors than shift exogeneity as noted by AdaoKolesarMorales2019, and produces the same standard error as the shift-share regression by the equivalence result in Section (ref).
This section surveys the application of shift-share designs in political science, identifies the strengths and weaknesses of current studies, proposes a framework for understanding and developing shift-share designs, and discusses opportunities for future research. The review is based on Table A.1 that compiles thirty-five articles that use shift-share designs published in political science journals. These shift-share articles were identified through keyword searches for “shift-share” and “Bartik” on Google Scholar, as well as by tracking papers that cited milestone shift-share articles such as Card2001 and AutorDornHanson2013. I note that creating an exhaustive list is challenging as some articles might have used shift-share designs as a measurement strategy without explicitly stating it.
Table (ref) presents a breakdown by year and journal. The first shift-share article was published in 2015, followed by five more before 2020. Twenty-nine articles have been published since then in the last five years including six forthcomings, indicating growing interest in shift-share designs within political science. The table also reveals another pattern: while shift-share designs were initially found in general-interest top journals, a growing number of articles are found in field-specific journals. This means that the shift-share methods are becoming a standard tool for studying specific topics.
The use of shift-share designs in political science largely mirrors their application in economics. Trade shocks are the most popular topic, with sixteen in total. Most adapt the research design of AutorDornHanson2013 to alternative outcome variables or contexts; the notable exception is BisbeeRosendorff2024 who create new trade shock measures using occupation-level data. Studies on technology shocks apply shift-share designs both for measurement and as instruments similar to trade shocks, but replacing industry-level import exposures with technological innovations AutorDornHanson2015. Immigration and migration follow with six studies, often adopting the Bartik instrument of Card2001. Capital movement, foreign aid and natural resources articles exploit the “propensity score” design introduced by NunnQian2014.
Despite the growing popularity of the methods, the associated identification problems are not as widely recognized in the field. Only one-third of the articles published in or after 2021 acknowledged potential en- dogeneity in shifts or shares and cited relevant source papers from the previous section. Only a few among those explicitly discussed identifying assumptions and conducted diagnostic tests, while most stopped at merely noting the possible methodological concerns. Moreover, the vast majority of studies relied on share exogeneity, knowingly or not, including articles on trade shocks that have been more associated with the shift exogeneity framework in economics. While this does not necessarily invalidate the designs, but the heavy reliance on share exogeneity suggests either a misapplication of the framework or an underutiliza- tion of shift exogeneity. These observations underscore the need for a more principled way to understand shift-share designs, particularly those less familiar with them.
Table (ref) shows three basic dimensions to consider. The first dimension is the type of shift-share variables indicating why shift-share variables are needed in the first place. An example of measurement is the China shock where regional import exposure is approximated with regional employment shares and national import changes. This application of shift-share designs needs justification of the linear approximation to the unobserved true independent variable as well as the exogeneity of shifts or shares. Section (ref) provides examples of the other two applications. Although the type does not directly affect the validity of research designs, it may help identify a shift-share variable relevant to the research as a first step.
The second dimension concerns the definition of the shift-share variable. While units are determined by the research question, researchers can be creative in selecting shifts and shares based on data availability. Section 3 lists examples that illustrate diverse designs at the end. Note that when the shift-share variable is used as an instrument, shifts and shares do not necessarily have a natural shift and share interpretation. The propensity score approach in NunnQian2014 is an example, with more examples in Appendix A of BorusyakHullJaravel2025b. This formal definition of shift-share designs broadens their potential use compared to the decomposition-based substantive definition.
When one has decided on an overall measurement or instrumenting approach, it is good practice to write down the final variable in shift-share form. FoukaTabellini2022 construct a Bartik instrument using the initial regional immigrant share $w_i$ with the national inflow $D_t$, but scaled by the predicted populatioñ $\tilde P_{it}$ of region $i$ at time $t$. Therefore, the actual instrument is \[ Z_{it}=\frac{w_i D_t}{\tilde P_{it}} \] instead of $Z_{it}=w_iD_t$ (footnote 4). Since the predicted populatioñ $\tilde P_{it}$ varies across units, the 2SLS approach is valid only if $\frac{w_i}{\tilde P_{it}}$ can be viewed as exogenous shares conditional on shifts $D_t$. Express $w_i$ as the ratio of the initial numbers of regional to national immigrants $\frac{I_{i0}}{I_0}$ and note that the initial number of national immigrants $I_0$ is common across all instruments. It then suffices to establish that the ratios of the initial number of immigrants to the predicted population size $\frac{I_{i0}}{\tilde P_{it}}$, instead of the initial immigrant share $w_i$, are exogenous “shares” conditional on shifts $D_t$. Diagnostic tests and regression specification may be misled in general if shifts and shares are mischaracterized.
The third dimension is the source of exogeneity and the corresponding identification strategies. Section 4 and checklists in BorusyakHullJaravel2025b apply. Note that although certain topics tend to favor one identification strategy over the other in the literature, the link is not deterministic. ScheveSerlin2023 identify the effect of their trade shock measure using share exogeneity while acknowledging most studies on trade shocks rely on shift exogeneity. If exogeneity can be argued for both shifts and shares as in CarreriDube2017, the analogy to difference-in-differences designs, and thereby reliance on the share exogeneity framework, suffices as standard errors remain unchanged without shift exogeneity. The empirical case where exogeneity comes from the combination of shifts and shares, as explored theoretically in HahnKuersteinerSantosWilligrod2024, remains to be seen.
Formalism can help clarify identification with non-standard shift-share variables. Consider Ziaja (2020) who extends the propensity score approach to multiple donors in the context of foreign aid: \[ Z_{it}=\sum_j p_{ij}D_{jt} \] for $p_{ij}$ the fraction of years in which recipient country $i$ was aided by donor $j$ and $D_{jt}$ the total for- eign aid by donor country $j$ in year $t$. Rewrite $p_{ij}$ as “fixed” shares $p_{ij0}$ to rewrite the instrument into \[ Z_{it}=\sum_j p_{ij0}D_{jt}. \] This is the typical panel shift-share structure under share exogeneity, and identification follows from standard assumptions that $\operatorname{cov}(p_{ij0},\epsilon_{it})=0$ for every $j$. Now extend shares $p_{ij}$ to $p_{ij\,t,t'}$ with additional year indices $t,t'$, setting $p_{ij\,t,t'}=p_{ij}$ if $t=t'$ and $0$ otherwise. The instrument then turns into \[ Z_{it}=\sum_{j,t'} p_{ij\,t,t'}D_{jt'}, \] which is the typical long-form structure in panel settings under shift exogeneity. Identification follows from suitable identifying assumptions.
I conclude the review with two suggestions for future research. First, shift-share designs can be a useful tool when treatments or instruments are correlated. Network studies have sometimes used the designs unknowingly BorusyakHullJaravel2025b. Imagine an experimental design where random nodes are seeded on a network and their neighbors select into the treatment depending on the fraction of seeders among their neighbors. Instruments are not independently determined as they arise from the interaction between the common seeder status and the underlying network structure: \[ Z_i=\sum_j w_{ij}D_j \] where the share $w_{ij}$ is the inverse of the number of neighbors of node $i$, and the shift $D_j=1$ if node $j$ is seeded and zero otherwise. 2SLS may underestimate the standard error if the network was endogenously formed. Although not listed in Table A.1, similar examples may exist in political science that went unnoticed when the correlation structure needed to be accounted for. The standard error can be computed either analytically or via simulation under a given correlation structure among $Z_i$. One may alternatively use the shift exogeneity framework provided that the network has a sufficiently diffuse structure.
Second, shift-share designs may hold greater potential in the American politics literature. Only five of thirty-five articles in Table A.1 study American politics. This may reflect that topics such as trade and immigration where shift-share designs are common are less frequently studied in the American context, or that richer data reduces the need for such proxy variables. Nevertheless, I argue that the granularity of data makes shift-share designs have wider applicability in the field. One could measure legislators’ exposure to donor-level shocks via the donor network or industry-level shocks via the lobbying network -- the sheer number of donors and industries may constitute many independent shifts required in the asymptotic shift exogeneity framework. Endogenous policymaking across regions could be instrumented using fractional exposure to nationwide shocks. Behavioral outcomes such as racial agenda or sentiments often refract through racial composition in the region documented in the census. These examples show potential utility of the methods in core topics of American politics.
Given the low awareness of shift exogeneity, this section illustrates the framework by replicating ColantoneStanig2018b. See FoukaTabellini2022 or ScheveSerlin2023 for examples of share exogeneity. The authors ask how imports from China led to the rise of nationalism and far-right parties in European countries, using an identical shift-share design to AutorDornHanson2013 apart from outcome variables and the geographical context. In light of the replication exercise of BorusyakHullJaravel2022, I highlight two additional procedures required due to the different geographical context: shift transformation and shift residualization.
The authors regress electoral outcomes of electoral districts on the shift-share trade measure using data from 15 European countries spanning from 1988 to 2007. The electoral outcomes of interest are (1) the vote share-weighted mean and median nationalism score and the nationalist autarchy score of parties, and (2) vote shares of far-right parties. The independent variable is the interaction between local manufacturing industry employment shares and two-year differences in Chinese imports scaled by total national employment of the industry:
where $c$ indexes countries, $r$ regions, $j$ industries, $t$ years. The last expression rewrites the shift-share variable in long form where $j'$ indexes country-industries and $t'$ years (see Section (ref)). Local employment $L_{rjt}$ is measured at the NUTS-2 level, and industries $j$ are classified at the two-letter NACE level, 14 in total, compared to 20 two-digit manufacturing SIC codes.\footnote{NUTS is the administrative geographical unit and NACE is the industry code system developed by Eurostat. AutorDornHanson2013 use the 4-digit Standard Industrial Classification (SIC), which BorusyakHullJaravel2022 find are clustered at the 3-digit level.} The instrument replaces $\Delta \mathrm{Import}_{cjt}$ in Equation (ref) with two-year differences in Chinese imports into the United States, $\Delta \mathrm{Import}_{US,jt}$. The authors aim to isolate the effect of “exogenous changes in supply conditions in China, rather than \dots\ domestic factors that could be correlated with electoral outcomes.”\footnote{The authors list two possible sources of shift endogeneity. First, politicians may strategically protect certain districts from Chinese imports, in which case electoral outcomes determine the local exposure to trade with China. Second, both the shifts and the dependent variable may be affected by idiosyncratic local shocks such as economic fluctuation or political performance of incumbents that are not attributable to China.} This clearly implies identification based on shift exogeneity, provided that correlations between the two import changes are primarily driven by supply shocks within China AutorDornHanson2013 .
\paragraph{Endogeneity concern.} Regions with different industry structures are likely to have different political orientation due to unique demographic composition, economic interests and historical backgrounds. These confounders may cause regions with similar industry structures to electorally respond to import shocks differently from regions with dissimilar industry structure. This observation has two implications. First, shares are likely correlated not only with levels of the error term but also with its changes in ways not easily addressed by conditioning on observables, making less feasible the application of share exogeneity as advocated in GoldsmithPinkhamSorkinSwift2020. Second, for valid inference, errors should be clustered as implied by the share structure as well as at the NUTS-2 region-year level, as in the original article. The shift exogeneity framework takes shares and errors as fixed and exploits the randomness in shifts only, allowing for arbitrary correlations among shares and errors.
\paragraph{Unique challenges.} This shift-share design has three structural differences from AutorDornHanson2013. First, both shift-share variables are measured at a higher level than the outcome variable as NUTS-2 regions contain multiple electoral districts: if $Y_i$ denotes the electoral outcome of district $i$, region $r$ corresponds to multiple regions $i$. The authors use district as the unit of analysis and cluster the errors by region-year. I show in Appendix B.2 that the choice of the unit of analysis does not matter under shift exogeneity since the inverted regression is unaffected when districts and regions are properly weighted by their population. I pick region as the unit of analysis to better illustrate the effective sample size.
Second, the instrument suffers from a divide-by-zero problem in its shifts as Chinese imports to the US are divided by the national industry employment of much smaller European countries, some of which even lack the industries under analysis. Compare this with AutorDornHanson2013 who divide the Chinese import to a group of comparably sized European economies by US industry employments. This motivates the shift transformation in the next section that neglects shifts with small denominators.
Third, electoral outcomes are not as frequently measured as independent variables and instruments, virtually creating a missing data problem. Note that identification and inference under shift exogeneity partly comes from the ability to isolate the stochastic part in realized shift values, or equivalently, to center shifts by properly modeling their means. I propose a residualization scheme that uses annual trade data and then estimates the effect size by regressing outcomes on the residuals.
\paragraph{Data reconstruction.} The replication is based on the reconstructed data covering the shorter period from 2001 to 2007. The replication files of the authors only contain the aggregate shift-share variables and not the individual shift and share components. I reconstruct the individual shifts and shares using unlicensed, publicly available data following the protocol in Appendix C. The reconstructed independent and instrumental variables have the correlation of 0.87 and 0.90 with the aggregate shift-share variables used in the original article.\footnote{The differences mainly stem from share estimates. While the original variables use national employment statistics sourced from each country, the reconstructed ones use Eurostat employment statistics that are noisier and limited in coverage.} Table F.1 replicates their main results using the original variables, the original variables censored from 2001 to 2007, and the imputed and predicted reconstructed variables.\footnote{Imputed variables predict missing employments using linear regression, and predicted variables replace all observed values with values predicted by linear regression.} Censored and reconstructed estimates were close enough, especially the effects on average nationalism scores. The following analyses use the nationalism score for the main dependent variable. Consequently, the main lesson of the replication exercise will be the importance of using the correct statistical procedures rather than a reversal of the authors' substantive findings.
Shifts must be processed first if they are not mean-zero and independent. BorusyakHullJaravel2022 propose residualizing the scaled import changes over period fixed effects and clustering shifts at a level lower than NUTS-2.
I assume the following structural model for shifts:
where $u_{cj}(\equiv u_{j'})$ is the country-industry fixed effect, $v_t$ is the time fixed effect, and $\eta_{cjt}$ is the mean-zero incidental deviation in the shift to be properly clustered. The residualization here is two-way to account for the baseline differences in the denominator across country-industry pairs. Since changes in the employment are slower than changes in the trade flow, I argue that country-industry fixed effects can reasonably control for the baseline differences. The incidental term $\eta_{cjt}$ is expected to be independent across industries according to the findings of BorusyakHullJaravel2022, but not across countries due to the common term $\Delta \mathrm{Import}_{US,jt}$.
Both AdaoKolesarMorales2019 and BorusyakHullJaravel2022 implicitly estimate the incidental term $\hat{\eta}_{cjt}$ by controlling for aggregate fixed-effects $\sum_{j',t'} w_{rj't,t'} u_{j'}$ and $\sum_{j',t'} w_{rj't,t'} v_t$ in the shift-share regression. This approach estimates the incidental component of the shifts only using those from the country-years when elections were held, and therefore yields less precise estimates than directly obtaining $\hat{\eta}_{cjt}$ from the annual trade data. Appendix D extends estimation and inference in the shift-share regression to with residualized shifts $\hat{\eta}_{cjt}$. I note that this estimation may come at the cost of the easy implementation of the inverted regression, but the cost might be offset by dispensing with the need to introduce aggregate fixed-effects as in our example.
In addition to the residualization, I propose shift replacement to ensure the identical mean assumption. Since individual European economies are much smaller than the U.S., the shifts in this study tend to be more volatile than those in AutorDornHanson2013. Moreover, some countries barely have certain industries, causing the divide-by-zero problem in the shifts. This problem did not arise in the other paper as their sizes of their numerator and the denominator were comparable. I replace shifts with zero if the sum of the regional shares of a certain industry in the country is less than 0.03, or if $w_{cjt}=\sum_{r\in c} w_{rjt}$ is less than 0.03 with zero. 24% of the shifts were replaced, most of which is in the petroleum and nuclear fuel industry. Appendix E provides theoretical justifications. Statistically, the replacement scheme better protects the identical expectation assumption from possible misspecification of shift-level covariates and the measurement error from data reconstruction. Substantively, the replacement scheme limits the analysis to industries that align more closely with the structural model underlying the shift-share measure. Note that the scheme is not to replace all industry shocks with a national employment share of less than three percent. This scheme can be applied to the share in the independent variable as well.
The estimation procedure is as follows. First, MISSING FROM THE ORIGINAL DRAFT
Asymptotic inference under shift exogeneity requires the mean-independence and the linear expectation of shifts, independence across shift clusters, and asymptotically negligible cluster shares, where shifts and shares refer to those in the instrumental variable. Mean-independence holds if the factors driving the trade shock between China and the U.S.\ do not affect the domestic politics of European countries. The other three assumptions are not discussed in ColantoneStanig2018b. The identical expectation and cross-cluster independence assumptions will depend on the way shifts are manipulated.
\paragraph{Shift independence.} To test the shift independence conditions, I include all shifts from the countries and time period of interest regardless of whether elections occurred in a given country and year.\footnote{This is to secure test power, but might be problematic if election timing is correlated with the shifts. However, even if election timing is a function of trade shocks in parliamentary countries, the assumptions would not be violated as long as the function is not time-varying and does not depend on the past shifts.} Table (ref) reports shift summary statistics. The odd columns use replaced shifts and the even columns use residualized replaced shifts.\footnote{Shifts were weighted by their aggregate share in the process of residualization.} Shifts are centered around zero before residualization, and they show a high variance even after residualization. The standard deviation of residualized shifts is around three times of that in AutorDornHanson2013, as expected from the relative size of economies of the U.S.\ and the European countries. Residualization retains around 24 percent of the variation.
The lower panel tests the dependence among shifts. Autocorrelations measure the correlation between shifts at year $t$ and $t-1$, or year $t$ and $t-2$. Intra-class correlation coefficients (ICC) estimate the following unweighted random-effect models:
where $\mathbb{I}$ are indicators and all random terms are normal.\footnote{Only one of $\beta_{ct}$ and $\gamma_{jt}$ is treated random and the other is fixed in the unresidualized model due to the insufficient number of years. The residualized model includes only one of $\beta_{ct}$ and $\gamma_{jt}$ due to collinearity, and $\delta_{cj}$ and $\nu_t$ are fixed.} Large coefficients imply dependence within the groups and suggest the need for clustering. The results align with our specification. Residualized shifts are uncorrelated at least two years apart, and exhibit little ICCs across country-year but high ICCs across industry-year.\footnote{The apparent statistical significance of country-year ICCs is due to the asymmetric confidence intervals.} One-year autocorrelations are significant because $\Delta US\ \mathrm{Import}_{jt}$ and $\Delta US\ \mathrm{Import}_{j(t-1)}$ both contain the import change from year $t-2$ to year $t-1$. Since only one pair of elections in the sample was held one year apart in the same country, 2002 and 2003 in the Netherlands, I drop the 2002 election from the data.
Table (ref): to show exogeneity wrt the outcome variable, political proxies require additional data collection at subnational level\dots
\paragraph{Shift exogeneity.} Table (ref) tests the shift mean-independence. The upper panel tests if shifts predict pre-shock shift-level variations, and the lower panel tests if shifts corresponding to the elections predict pre-shock unit-level variations. Economic placebos being tested are initial national employment shares by industry and initial worker composition by region. Political placebos could also be considered, but they must come from subnational variation due to the residualization at the country-industry level.\footnote{Pre-shock political leaning or nationalism score would be some examples.} These variables can affect the election outcomes independently of the trade shock and proxy the error term in the inverted regression that the shifts are intended to instrument. All specifications use residualized shifts, or equivalently control for industry-country and year fixed effects. Results do not reject the null hypotheses that instruments are uncorrelated with pre-shock variables, except in the case of the shift-level test. The strong significance in the shift-level test might be an artefact of data availability. The true pre-shock period should be before 1988 when China had hardly entered international trade, but the placebo variable relies on 1999 employment shares.\footnote{The mean and standard deviation of the initial national industry share are 1.09% and 0.84%, so the effect size can be substantially significant as well.} Unit-level placebo tests use pre-1988 variables. The shift-level test is presented for illustrative purposes and should not be substantively interpreted much. Table F.2 reports similar results with unreplaced shifts.
\paragraph{Share negligibility.} Given the industry-year shift clusters, asymptotic negligibility holds if the aggregate European economy is not dominated by certain industries or if elections have occurred frequently enough during the given period. To assess the share negligibility assumption, I focus on regional employment shares that match with the election data and calculate cluster shares. The total number of clusters is 98. Define cluster shares $w_{jt}=\sum_{c} w_{cjt}$. Table (ref) presents share summary statistics. The first and second columns measure the ratio of the largest $w_{jt}$ (or $w_{jt}^2$) and the sum of $w_{jt}$ (or $w_{jt}^2$). These two metrics show as how negligible the largest cluster is and should be as close to zero as possible AdaoKolesarMorales2019. The third column measures the inverse of the sum of each cluster's relative size squared, or the inverse of their Herfindahl Index (HHI). This shows the effect sample size in the inverted regression, so it should be as large as possible BorusyakHullJaravel2022. All metrics indicate that resulting estimates will be consistent and asymptotically valid.
\paragraph{Pre-trend test.} Not available, especially due to the data limitation.
Given the absence of strong evidence failing falsification tests, I replicate the main results using the new shift-share estimator. Regression specifications differ from the original paper in two ways. First, election outcomes are averaged by NUTS-2 region for congruence between the unit of analysis and the unit of shifts. Second, shift-level controls are added to satisfy the identical mean assumption.
Table (ref) summarizes the main results. The dependent variables are the median and weighted average of nationalism scores in each electoral district. Estimates measure how much an extra unit of Chinese import drove parties towards nationalism. The cluster row uses the same 2SLS estimators as in the original paper where standard errors are clustered by NUTS-2 region-year pair. The BHJ row uses inverted regression estimators following BorusyakHullJaravel2022, with standard errors clustered by industry-year pair.
A comparison between columns (1) and (2) and between columns (4) and (5) suggests that the results are not much affected by aggregation to the district level. However, BHJ estimator reports much smaller effects and larger standard errors. The difference in points estimates are due to the effect heterogeneity. While the other estimator by AdaoKolesarMorales2019 would preserve the point estimate, it is not applicable in this example since the number of shifts is larger than the number of units. Larger standard errors reflect the endogeneity problem between shares and electoral outcomes. This suggests that regions with the similar industrial composition have correlated electoral outcomes unaccounted by trade shocks, and NUTS-2 region-year clusters fail to capture this correlation by treating them as independent units.
Columns (3) and (6) control for shift-level fixed effects aggregated to the unit level, in addition to the unit-level region-year fixed effects in the original specification. Our residualization strategy specifies that shifts are only comparable when partialled out by country-industry and time fixed effects. These controls reflect the initial share of each industry and the total manufacturing share in the region when aggregated.\footnote{The controls are not collinear by design. Each aggregated country-industry fixed effect contains initial shares of the industry in the country across all years, while each aggregated time fixed effect contains all total manufacturing shares in the year across all countries and industries.} A comparison between columns (2) and (3) and between columns (5) and (6) indicates that the results are not robust to the addition of these new control variables either.\footnote{Note that this comparison is not affected by measurement errors caused by share imputation.} In the shift exogeneity framework, the shift-share estimate balances out the error terms in the inverted regression (B.4) under the assumption that shifts are independent and have the identical mean. The inconsistent results with respect to residualization suggest that the original results might have been driven by a group of large, correlated shifts. The shift-share variable does not capture the correct impact of shifts if different means are not adjusted for. Also, there is no way to account for dependence among shifts in conventional regression methods. Raw shifts are expected to have different means across country due to varying economic sizes, and would be highly correlated within industry as they share the same US imports as a component. Table F.3 shows that the results are similar when predicted shares or raw shifts are used instead of imputed shares or transformed shifts.
Take care of your shifts!
Again, Consequently, the main lesson of the replication exercise will be the importance of using the correct statistical procedures rather than a reversal of the authors' substantive findings.
This section finds that the replication data meets assumptions required in the shift exogeneity framework but the results are not robust either to the asymptotically correct standard error estimator or shift residualization. The only methodologically valid estimates are those in columns (3) and (6) and BHJ row, and they suggest that the original findings are not well supported from the causal perspective. This replication exercise makes two original contributions. First, the residualization and clustering strategy proposed here is more sophisticated compared to AutorDornHanson2013 that has an almost identical shift-share design. This is due to the complexity of the data structure that has three-way correlations compared to two-way in the original China shock paper. Second, I propose a shift transformation scheme that trims noisy outlier shifts that do not much affect the shift-share estimate. Shifts in this example involve division by small numbers that are not precisely observed. The scheme protects shifts from misspecifications, hence from erroneously rejecting the null hypothesis regarding the identical mean and independence assumption.
Having started as an imputation scheme, shift-share designs are now understood more generally as designs that involve inner-product variables between endogenous and exogenous variables. This article reviews how shift-share designs have been used in economics and political science from a methodological standpoint. Shift-share variables can measure factual unobservables using shares and observable shifts, or counterfactual unobservables using initial shares and counterfactual shifts such as the average value of the sample units or values from external units. Shift-share variables based on the propensity score are a statistical construct to enhance the relevance of instruments to the independent variable, but share the same statistical properties as other shift-share variables. This article synthesizes those statistical properties with unified notation and framework. Share exogeneity assumes units are comparable and independent similarly, but not entirely identical, to units in difference-in-differences. Shift exogeneity assumes shifts are comparable and independent so that errors balance out as in the law of large numbers.
Political science articles are increasingly adopting shift-share designs, but they rarely discuss or are fully aware of the required identifying assumptions. This article exemplifies how to assess those assumptions and how the new methods can affect results with a political science example. In the replication exercise, this article further proposes new methodological techniques that help to verify strict identifying assumptions that are otherwise not possible. Findings were compromised by either newly suggested standard error estimators or shift residualization, each indicating different identification issues underlying the original design. This article finds that despite complexities in shift-share designs, they still have significant potential in political science, particularly in American Politics where they are underutilized.
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