14  Shift-Share Instrumental Variables

Shift-share, or Bartik, instruments combine local industry shares with industry-level shocks. The idea is simple: a location with a large baseline share in an industry is more exposed to shocks in that industry. The difficult part is not constructing the instrument. The difficult part is stating what is assumed exogenous: the shares, the shocks, or both.

14.1 The construction

For each location \(\ell\) and industry \(k\), let \(s_{\ell k}\) be the share of local employment in industry \(k\) at baseline, and let \(g_k\) be a national growth rate (or trade shock) for that industry. The shift-share instrument is

\[ B_\ell \;=\; \sum_{k=1}^{K} s_{\ell k}\, g_k. \]

This is “shift-share” because it combines a local share with an industry-level shift. Bartik (1991) used local industry mix to predict local employment growth. Autor, Dorn, and Hanson (2013) used local industry shares interacted with industry-level Chinese import growth in other high-income countries.

The first-stage regression is then

\[ \Delta L_\ell \;=\; \pi_0 + \pi_1 B_\ell + X_\ell'\gamma + \epsilon_\ell, \]

with \(B_\ell\) instrumenting for the observed local shock in a 2SLS for the outcome of interest.

14.2 Two identification views

There are two ways to justify a shift-share instrument. They put the exogeneity assumption on different objects.

14.2.1 Share view (GPSS 2020)

Treat the shares \(s_{\ell k}\) as the source of identifying variation, with the shocks \(g_k\) acting as weights. The instrument is valid if shares are exogenous to the outcome (conditional on controls). Goldsmith-Pinkham, Sorkin, and Swift (2020) (henceforth GPSS) prove that the shift-share IV is numerically equivalent to a GMM combination of \(K\) just-identified IVs, one per industry share:

\[ \hat\beta^{SS} \;=\; \sum_{k=1}^{K} \hat\alpha_k\, \hat\beta_k, \]

where \(\hat\beta_k\) is the just-identified IV using share \(s_{\cdot k}\) alone and \(\hat\alpha_k\) is the Rotemberg weight. This gives an important diagnostic: which industry shares are driving the estimate?

14.2.2 Shock view (BHJ 2022)

Treat the shocks \(g_k\) as the source of identifying variation. Shares are exposure weights. Under this view, shocks should be uncorrelated with the second-stage unobservables, conditional on industry-level controls. Borusyak, Hull, and Jaravel (2022) (henceforth BHJ) show how to rewrite the regression at the shock level.

The two views are not mutually exclusive. In an application, we should be clear which one is being defended and report diagnostics for both.

14.3 Inference

Standard OLS or cluster-robust standard errors on the regional regression underestimate uncertainty because the same industry shocks appear in many regions, inducing cross-region correlation that region-level clustering does not capture. Two corrections are now standard:

• Cluster on shocks (BHJ): equivalent to running the regression at the industry-shock level. Implemented via shock-level reweighting.
• AKM SE (Adão, Kolesár, and Morales 2019): explicit formula for SE accounting for shock-level correlation. Available in the ShiftShareSE R package.

14.4 Simulation: build intuition

Use a small DGP with regions, industries, and a known causal effect.

n_region   <- 500
n_industry <- 20
beta_true  <- 0.5

df     <- read_csv("data/shift_share_sim.csv", show_col_types = FALSE)
shares <- as.matrix(read_csv("data/shift_share_shares.csv", show_col_types = FALSE))
# This simulation is constructed to satisfy the Borusyak et al. (2022)
# shock-exogeneity assumption: the industry-level shocks are drawn
# independently of the region-level confounder `u` and of the shares
# themselves. The shares are allowed to be endogenous (correlated with
# region characteristics) -- it is specifically the shocks that must be
# "as good as randomly assigned" for this identification argument, which is
# a different (and in this simulation, the maintained) assumption from the
# Goldsmith-Pinkham et al. (2020) shares-exogeneity view discussed above.
shocks <- read_csv("data/shift_share_shocks.csv", show_col_types = FALSE)$shock
u      <- df$u
head(df)

# A tibble: 6 × 5
  region      X      Y       B      u
   <dbl>  <dbl>  <dbl>   <dbl>  <dbl>
1      1 -0.518  0.117 -0.138  -0.308
2      2  0.213 -1.07   0.547  -0.539
3      3  0.652  0.892  0.351   1.37 
4      4  0.286  0.861  0.0385  1.10 
5      5 -0.334 -0.330  0.421  -0.112
6      6  0.376 -1.73   1.17   -1.75 

A naive OLS suffers from the confounder u:

ols  <- feols(Y ~ X, data = df)
ivss <- feols(Y ~ 1 | X ~ B, data = df)
etable(ols, ivss, headers = c("OLS", "Shift-share IV"),
       digits = 3, digits.stats = 3, fitstat = ~ . + ivf)

                                   ols             ivss
                                   OLS   Shift-share IV
Dependent Var.:                      Y                Y
                                                       
Constant              -0.154** (0.047)    0.077 (0.059)
X                      1.45*** (0.075) 0.531*** (0.131)
_____________________ ________________ ________________
S.E. type                          IID              IID
Observations                       500              500
R2                               0.428            0.025
Adj. R2                          0.427            0.022
F-test (1st stage), X               --            370.5
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

OLS is biased upward by the confounder; the shift-share IV recovers a value close to the true effect of 0.5. With concentrated industry shares and 20 industries, the first-stage F is well above 100.

14.4.1 Rotemberg weights

The GPSS decomposition says the shift-share IV is a weighted average of \(K\) just-identified IVs (one per industry share). Compute the weights:

# Rotemberg weight for industry k (using centered moments to match the
# demeaned 2SLS regression):
#   alpha_k = g_k * Cov(s_{.k}, X) / sum_j g_j * Cov(s_{.j}, X)
# Each industry's just-identified IV estimate uses s_{.k} as instrument:
#   beta_k = Cov(s_{.k}, Y) / Cov(s_{.k}, X)
# GPSS prove the shift-share IV estimate equals sum_k alpha_k * beta_k.

cov_sk_X <- sapply(1:n_industry, function(k) cov(shares[, k], df$X))
cov_sk_Y <- sapply(1:n_industry, function(k) cov(shares[, k], df$Y))
denom    <- sum(shocks * cov_sk_X)

alpha  <- shocks * cov_sk_X / denom
beta_k <- cov_sk_Y / cov_sk_X

rotemberg <- tibble(industry = 1:n_industry,
                    shock    = shocks,
                    weight   = alpha,
                    beta_k   = beta_k)

kable(rotemberg, digits = 3,
      caption = paste0("Rotemberg decomposition. Sum of alpha*beta_k = ",
                       round(sum(alpha * beta_k), 3),
                       " (= shift-share IV estimate of ",
                       round(coef(ivss)["fit_X"], 3), ")."))

Rotemberg decomposition. Sum of alpha*beta_k = 0.531 (= shift-share IV estimate of 0.531).

industry	shock	weight	beta_k
1	1.250	0.054	0.604
2	-0.959	0.048	0.244
3	-0.212	0.005	2.576
4	0.745	0.002	-10.225
5	2.663	0.387	0.469
6	-1.068	0.082	1.025
7	0.100	-0.001	-1.627
8	-0.718	0.025	-0.689
9	1.181	0.058	0.853
10	-0.197	0.004	1.188
11	0.741	0.023	-0.112
12	0.051	-0.001	-0.055
13	-0.442	0.014	0.611
14	0.270	0.003	-0.013
15	0.631	0.003	-3.240
16	2.271	0.204	0.746
17	-0.212	0.001	0.174
18	-0.115	0.000	-39.532
19	0.020	0.000	1.234
20	-1.183	0.089	0.554

Two diagnostics matter:

1. Weight concentration: if one or two industries carry most of the weight, the identifying variation is essentially coming from those industries’ shares. Their exogeneity is the assumption you’re really leaning on.
2. Heterogeneous \(\hat\beta_k\): in this benign simulation, the industry-specific IVs are all near the true effect (with sampling noise). In real data, if a few industries give wildly different estimates, the “average” shift-share IV is averaging over heterogeneous treatment effects you should report explicitly.

ggplot(rotemberg, aes(x = factor(industry), y = beta_k, size = abs(weight))) +
  geom_hline(yintercept = beta_true, linetype = "dashed", color = "red") +
  geom_point(alpha = 0.7) +
  scale_size_continuous(name = "|Rotemberg weight|") +
  labs(x = "Industry", y = "Just-identified IV estimate",
       title = "Industry-level IV estimates and weights",
       subtitle = paste("Red line = true effect =", beta_true)) +
  theme_minimal()

14.4.2 What goes wrong when shares are endogenous

Now break the share-exogeneity assumption: make industry 1’s share correlated with the second-stage error.

# Re-do shares so industry 1's share covaries with a new confounder v
v <- read_csv("data/shift_share_bad_v.csv", show_col_types = FALSE)$v
shares_bad <- shares
shares_bad[, 1] <- pmax(0.01, shares[, 1] + 0.15 * v)
shares_bad <- shares_bad / rowSums(shares_bad)   # renormalise to sum to 1

bad_noise <- read_csv("data/shift_share_bad_noise.csv", show_col_types = FALSE)
B_bad <- as.numeric(shares_bad %*% shocks)
X_bad <- B_bad + 0.3 * u + bad_noise$noise_x
Y_bad <- beta_true * X_bad + u + 0.6 * v + bad_noise$noise_y

df_bad <- tibble(X = X_bad, Y = Y_bad, B = B_bad)
ivss_bad <- feols(Y ~ 1 | X ~ B, data = df_bad)
cat("True beta:", beta_true, "\n")

True beta: 0.5 

cat("Shift-share IV with bad share 1:", round(coef(ivss_bad)["fit_X"], 3), "\n")

Shift-share IV with bad share 1: 0.798 

# Rotemberg weights recomputed under the bad shares
cov_sk_X_bad <- sapply(1:n_industry, function(k) cov(shares_bad[, k], X_bad))
alpha_bad    <- shocks * cov_sk_X_bad / sum(shocks * cov_sk_X_bad)
cat("Rotemberg weight of industry 1:", round(alpha_bad[1], 3),
    "-- rank", rank(-abs(alpha_bad))[1], "of", n_industry, "\n")

Rotemberg weight of industry 1: 0.11 -- rank 3 of 20 

# Leave-one-industry-out: rebuild the instrument without industry 1
B_loo    <- as.numeric(shares_bad[, -1] %*% shocks[-1])
ivss_loo <- feols(Y ~ 1 | X ~ B, data = tibble(X = X_bad, Y = Y_bad, B = B_loo))
cat("Shift-share IV without industry 1:", round(coef(ivss_loo)["fit_X"], 3), "\n")

Shift-share IV without industry 1: 0.48 

The estimate is biased, and the bias comes from one industry’s share being related to the error. Note which diagnostic catches it. Rotemberg weights measure influence — whose exogeneity the estimate leans on — not endogeneity: the offending industry carries a weight of only 0.11, third largest of twenty, yet it alone moves the estimate from 0.5 to 0.8. The leave-one-industry-out check is what isolates it: rebuilding the instrument without industry 1 restores the estimate to 0.48.

14.5 Empirical: the China shock (Autor, Dorn, Hanson 2013)

The standard shift-share application is the China-shock study of Autor, Dorn, and Hanson (2013) (henceforth ADH). Commuting zones are exposed differently to Chinese import competition because their baseline industry mixes differ. The instrument is

\[ \text{IV}_\ell \;=\; \sum_k \frac{L_{\ell k,\,1990}}{L_{\ell,\,1990}}\, \frac{\Delta M^{other}_{k}}{L_{k,\,1990}}, \]

where the shocks \(\Delta M^{other}_k\) are growth in Chinese imports into other high-income countries. This leave-one-out construction removes US-specific demand from the shock. In the actual ADH design the employment shares are lagged one census behind the outcome period (e.g. 1980 employment for the 1990–2000 stack), precisely to mitigate share endogeneity; we date everything to 1990 here to keep the notation light.

# David Dorn distributes the replication data at
# https://www.ddorn.net/data.htm — files needed:
#   workfile_china.dta  (commuting zone data, 1990-2007)
#   industry_shares.csv (czone-industry shares)
#   industry_imports.csv (industry-level imports)

library(haven)
library(fixest)

cz   <- read_dta("workfile_china.dta")
fit  <- feols(d_sh_empl_mfg ~ 1 | d_tradeusch_pw ~ d_tradeotch_pw_lag,
              data = cz,
              cluster = ~ statefip)
summary(fit)

Modern best-practice extensions of the basic ADH regression. These blocks are schematic — they do not run here, so check argument names against the current package documentation before adapting them:

# 1. Rotemberg weights via the bartik.weight package (Goldsmith-Pinkham)
# devtools::install_github("paulgp/bartik-weight")
library(bartik.weight)
rw <- bw(cz, master = master_spec, y = "d_sh_empl_mfg",
         x = "d_tradeusch_pw", weight = "timepwt48",
         G = G_growth, Z = Z_shares)

# Plot the Rotemberg weights to see which industries drive the estimate
plot(rw)

# 2. Adão-Kolesár-Morales standard errors
# devtools::install_github("kolesarm/ShiftShareSE")
library(ShiftShareSE)
ivreg_ss(d_sh_empl_mfg ~ d_tradeusch_pw + controls,
         X = "d_tradeotch_pw_lag",
         data = cz, W = shock_weights, region_cvar = "czone",
         method = "akm0")

# 3. BHJ shock-level inference: collapse to shock (industry-period) level
# devtools::install_github("borusyak/shift-share")
library(ssaggregate)
shocks_data <- ssaggregate(data = cz, vars = c("d_sh_empl_mfg",
                                                "d_tradeusch_pw"),
                            shock = "d_tradeotch_pw_lag", weights = "timepwt48",
                            l = "czone", n = "industry", t = "year",
                            s = "share")
# Now regress at shock level — interpretation: shock-level IV
feols(d_sh_empl_mfg ~ 1 | d_tradeusch_pw ~ shock, data = shocks_data)

For a new shift-share paper, I would expect Rotemberg weights, AKM standard errors, and a BHJ shock-level version or an explanation for why it is not appropriate.

14.6 Summary

• Shift-share IV combines local shares and industry shocks.
• The GPSS view puts exogeneity on shares; the BHJ view puts it on shocks.
• Rotemberg weights show which industry shares drive the 2SLS estimate.
• Region-level clustering is usually too optimistic. Use AKM standard errors or shock-level inference when possible.
• In R, fixest, bartik.weight, ShiftShareSE, and ssaggregate cover most of the workflow.

Adão, Rodrigo, Michal Kolesár, and Eduardo Morales. 2019. “Shift-Share Designs: Theory and Inference.” Quarterly Journal of Economics 134 (4): 1949–2010.

Autor, David H., David Dorn, and Gordon H. Hanson. 2013. “The China Syndrome: Local Labor Market Effects of Import Competition in the United States.” American Economic Review 103 (6): 2121–68.

Bartik, Timothy J. 1991. Who Benefits from State and Local Economic Development Policies? W.E. Upjohn Institute.

Borusyak, Kirill, Peter Hull, and Xavier Jaravel. 2022. “Quasi-Experimental Shift-Share Research Designs.” Review of Economic Studies 89 (1): 181–213.

Goldsmith-Pinkham, Paul, Isaac Sorkin, and Henry Swift. 2020. “Bartik Instruments: What, When, Why, and How.” American Economic Review 110 (8): 2586–2624.