Shift-Share / Bartik Instruments

Motivating Example

Between 1990 and 2007, China's manufacturing exports exploded. Some US labor markets were devastated. Others barely noticed. Why? Because local economies have different industry compositions. A city dominated by furniture manufacturing was hammered by Chinese competition. A city dominated by software development was not.

Autor et al. (2013) wanted to estimate the causal effect of Chinese import competition on US manufacturing employment. The challenge is obvious: local employment changes are driven by many factors beyond Chinese trade, and areas that are declining for other reasons might also be more exposed to import competition.

Their solution was a shift-share instrument, also known as a Bartik instrument. This approach extends the instrumental variables framework by constructing the instrument from two components. The idea: construct a predicted measure of local Chinese import exposure by interacting:

• National shocks ("shifts"): the growth in Chinese imports in each industry nationwide
• Local shares: the share of each industry in each local labor market's initial employment

The resulting instrument $Z_l = \sum_k s_{lk} \cdot g_k$ varies across locations because different places have different industry mixes, but the variation comes from national import growth, which is plausibly unrelated to local labor demand shocks.

This construction is widely used in applied economics — trade, immigration, fiscal policy, technology adoption. But for decades, researchers lacked clarity about exactly what makes the instrument valid. Recent work has clarified this question substantially.

A. Overview

A shift-share instrument takes the form:

where:

• $l$ indexes locations (or other cross-sectional units)
• $k$ indexes industries (or other categories)
• $s_{lk}$ is the share of industry $k$ in location $l$ (typically measured in a base period)
• $g_k$ is the shift (national growth rate or shock in industry $k$)

The instrument exploits the idea that national shocks ($g_k$) affect different locations differently because of their pre-existing industry compositions ($s_{lk}$).

Two Frameworks

The breakthrough in recent scholarship has been recognizing that shift-share instruments can be justified through two fundamentally different sets of assumptions:

1. Shares-based identification (Goldsmith-Pinkham et al., 2020): The shares $s_{lk}$ are the source of exogenous variation. The instrument is essentially a GMM estimator using the shares as individual instruments, with the shocks serving as weights. This framework requires the shares to be uncorrelated with unobserved determinants of the outcome — essentially, initial industry composition must be exogenous.

2. Shocks-based identification (Borusyak et al., 2022): The shocks $g_k$ are the source of exogenous variation. This identification requires the shocks to be as-good-as-randomly assigned, but allows the shares to be endogenous. This approach is the more common justification in trade and immigration settings, where national-level changes (like China's industrial policy) are plausibly exogenous to any individual local labor market.

Common Confusions

"Are these two frameworks in conflict?" No. They are complementary. They identify the same parameter under different assumptions. The question is: in your setting, is it more plausible that the shares are exogenous or that the shocks are exogenous? The answer determines which diagnostics to run.

"Can I just run 2SLS and not worry about this distinction?" Technically yes — the first-stage regression and 2SLS machinery are the same regardless of interpretation. But you need to know why your instrument is valid so you can test the right assumptions. Under the shares interpretation, it is important to check that shares are balanced (uncorrelated with local covariates). Under the shocks interpretation, it is important to check that shocks are as-if-random.

"What about the exclusion restriction?" The exclusion restriction requires that the shift-share instrument affects the outcome only through the endogenous variable. In the China shock example, the instrument should affect local employment only through its effect on local import competition, not through other channels. This restriction is violated if, for example, areas with high manufacturing shares also experience technology shocks that affect employment independently of trade.

"How many industries do I need?" Under the Borusyak et al. framework, you need many shocks (large $K$) so that the law of large numbers kicks in and the instrument's exogeneity holds on average. If $K$ is small, each individual shock has too much influence, and the as-if-random assumption is harder to justify.

B. Identification

The Estimating Equation

The typical setup is:

Second stage: $Y_l = \alpha + \beta X_l + \gamma' \mathbf{W}_l + \varepsilon_l$

First stage: $X_l = \pi_0 + \pi_1 Z_l + \pi_2' \mathbf{W}_l + u_l$

where $X_l$ is the endogenous variable (e.g., change in local import exposure), $Z_l$ is the shift-share instrument, and $\mathbf{W}_l$ are controls.

Under the Shares Interpretation

Goldsmith-Pinkham et al. show that the shift-share IV estimator is numerically equivalent to a GMM estimator using the $K$ individual shares $\{s_{l1}, \ldots, s_{lK}\}$ as instruments:

where $\alpha_k \propto \hat{g}_k$ are Rotemberg weights that reflect each industry's contribution to identification. Key implication: a few industries typically drive the result. It is important to check which industries have the largest Rotemberg weights and verify that their shares are plausibly exogenous.

Diagnostics:

• Report Rotemberg weights (which industries matter most)
• Check balance: regress shares on local covariates for the high-weight industries
• Over-identification test: if $K > 1$, you have multiple instruments and can test over-identifying restrictions

Under the Shocks Interpretation

Borusyak et al. show that the shift-share IV can be recast as an estimator at the industry level. The key assumption is that shocks are as-good-as-randomly assigned:

where $\bar{\varepsilon}_k = \sum_l s_{lk} \varepsilon_l / \sum_l s_{lk}$ is the exposure-weighted average of local residuals for industry $k$.

Diagnostics:

• Check balance: regress shocks on industry-level characteristics
• Examine whether shocks are correlated with pre-period trends
• Verify that $K$ is large enough for the asymptotic approximation

C. Visual Intuition

Imagine a map of the United States. Each local labor market has a pie chart showing its industry composition. Now imagine that China's rise disproportionately affects certain industries (shown in red). The areas with the largest red slices are the most "exposed" — they receive the largest values of the shift-share instrument.

The identifying variation comes from comparing outcomes in areas with large red slices (high exposure) to areas with small red slices (low exposure), where the redness of each industry is determined by national-level Chinese import growth.

D. Mathematical Derivation

E. Implementation

1library(fixest)
2# Install from GitHub: devtools::install_github("jjchern/bartik.weight")
3library(bartik.weight)
4
5# Step 1: Construct shift-share instrument
6# Z_l = sum_k (share_lk * shock_k) — inner product of shares and shocks
7df$shift_share <- as.matrix(df[, share_cols]) %*% shocks
8
9# Step 2: Two-stage least squares using fixest
10# outcome ~ controls | fixed_effects | endogenous ~ instrument
11est <- feols(outcome ~ controls | 0 | exposure ~ shift_share,
12           data = df, vcov = "HC1")
13summary(est)
14
15# Step 3: Rotemberg weights (Goldsmith-Pinkham et al. 2020)
16# Decomposes the IV estimate into industry-level contributions
17# alpha_k shows how much each industry drives the overall estimate
18bw_result <- bw(
19master = df, y = "outcome", x = "exposure",
20controls = control_cols, weight = "weight_col",
21local = df, Z = share_cols,
22global = shocks_df, G = "shock_growth"
23)
24summary(bw_result)
25
26# Step 4: Examine top 5 industries by Rotemberg weight
27# These industries must pass exogeneity / balance tests
28top_weights <- sort(bw_result$alpha, decreasing = TRUE)[1:5]
29print(top_weights)

F. Diagnostics

1. First-stage F-statistic. As with any IV, check for weak instruments. F > 10 is the traditional threshold, though more recent guidance (Lee et al., 2022) suggests stricter thresholds.

2. Rotemberg weights. Report the top 5-10 industries by Rotemberg weight. Check whether these industries are plausibly exogenous. If one industry dominates, your results depend heavily on that industry.

3. Balance checks. Under shares-based identification: regress shares on pre-period covariates. Under shocks-based identification: regress shocks on industry-level characteristics.

4. Over-identification test. Under the shares interpretation, you have $K$ instruments but estimate only one parameter. The Sargan-Hansen J-test checks whether the over-identifying restrictions hold. A rejection suggests that at least some shares are not valid instruments.

5. Leave-one-industry-out. Re-estimate removing the top Rotemberg weight industry. If results change dramatically, they are fragile.

6. Pre-trend tests. If you have multiple periods, check whether the shift-share instrument predicts outcome changes in the pre-period.

Interpreting Your Results

When reporting shift-share IV results, it is important to be clear about which identification framework you are invoking. State explicitly: "We interpret the shift-share instrument under the [shares/shocks]-based framework of [Goldsmith-Pinkham et al./Borusyak et al.]."

Report the Rotemberg weights and discuss which industries drive identification. If 80% of the identifying variation comes from three industries, the reader needs to evaluate whether those three industries satisfy the exogeneity conditions.

G. What Can Go Wrong

H. Practice

Top Rotemberg weights:
  Mexico:           0.41
  Philippines:      0.09
  China:            0.08
  India:            0.07
  Vietnam:          0.06

I. Swap-In: When to Use Something Else

• Standard IV: When a single instrument with a clear exclusion restriction is available — the shift-share structure adds complexity that is only warranted when the instrument is inherently composed of shares and shocks.
• Difference-in-differences: When the shock creates a clear before/after comparison for exposed versus unexposed regions, and parallel trends is directly defensible.
• Synthetic control: When few regions are heavily exposed and constructing a data-driven counterfactual from donor regions is feasible.
• OLS with controls: When the exposure variable is exogenous conditional on observables and the primary concern is confounding rather than endogeneity — no IV structure is needed.

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