# Shift-Share Instrumental Variables
```{r}
#| include: false
library(tidyverse)
library(fixest)
library(ggplot2)
library(knitr)
```
Shift-share (or "Bartik") instruments combine local industry shares with national
industry growth rates to construct an instrument for local employment shocks.
They are workhorses in labour economics, trade, and urban economics — but the
identifying assumptions and inference procedures have been reworked over the
last five years. This chapter covers the modern view.
## 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*. The classic application is Bartik (1991), who used local industry mix
to predict employment growth. Autor, Dorn, and Hanson (2013) — "the China
shock" — used local industry shares interacted with industry-level Chinese
import growth in other rich 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.
## Two identification views
A 2020 paper by Goldsmith-Pinkham, Sorkin, and Swift (GPSS) and a 2022 paper
by Borusyak, Hull, and Jaravel (BHJ) showed that shift-share IV has two
distinct identification stories, with different validity conditions.
### 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). 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 is the key diagnostic:
which shares are driving the estimate?
### Shock view (BHJ 2022)
Treat the shocks $g_k$ as the source of identifying variation — they should
behave as-if randomly assigned across industries. Shares are exposure weights.
Under this view, validity requires that shocks are uncorrelated with
unobservables in the second stage, conditional on industry-level controls.
BHJ derive an equivalent IV regression at the shock level (one observation
per industry rather than per region), which is easier to validate.
The two views are not mutually exclusive but place the burden of
exogeneity on different objects. In an application, you should defend at
least one of them and report diagnostics for both.
## 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, Morales 2019)**: explicit formula for SE accounting
for shock-level correlation. Available in the `ShiftShareSE` R package.
## Simulation: build intuition
Set up a small DGP with regions, industries, and a known causal effect.
```{r}
#| label: sim-data
#| cache: true
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))
shocks <- read_csv("data/shift_share_shocks.csv", show_col_types = FALSE)$shock
u <- df$u
head(df)
```
A naive OLS suffers from the confounder `u`:
```{r}
#| label: naive-ols
#| cache: true
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)
```
OLS is biased upward by the confounder; the shift-share IV recovers a
value close to the true effect of `r beta_true`. With concentrated industry
shares and 20 industries, the first-stage F is well above 100.
### 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:
```{r}
#| label: rotemberg
#| cache: true
# 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), ")."))
```
Two diagnostics jump out:
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.
```{r}
#| label: rotemberg-plot
#| cache: true
#| fig-height: 4
#| fig-width: 7
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()
```
### What goes wrong when shares are endogenous
Now break the share-exogeneity assumption: make industry 1's share correlated
with the second-stage error.
```{r}
#| label: bad-shares
#| cache: true
# 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")
cat("Shift-share IV with bad share 1:", round(coef(ivss_bad)["fit_X"], 3), "\n")
```
The estimate is biased — but the bias comes from a *single industry's share*
covarying with unobservables. Rotemberg weights would have flagged industry 1
as carrying disproportionate identifying weight; a robustness check that
drops industry 1 (or applies BHJ's shock-level inference) would have
detected the problem.
## Empirical: the China shock (Autor, Dorn, Hanson 2013)
The canonical shift-share application is ADH (2013). Local labour markets
(commuting zones) are differently exposed to rising Chinese import competition
because they had different baseline industry mixes. 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 rich countries* (a "leave-one-out" construction that strips US-specific
demand from the shocks).
```{r}
#| label: adh-skeleton
#| eval: false
#| echo: true
# 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:
```{r}
#| label: adh-modern
#| eval: false
#| echo: true
# 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)
```
These three diagnostics — Rotemberg weights, AKM standard errors, and BHJ
shock-level regressions — should appear in any new shift-share paper.
## Summary
- **Shift-share IV** = local shares × national shocks. Identifies effects of
local economic shocks by leveraging differences in industry mix.
- **Two identification views**: shares exogenous (GPSS) or shocks as-if-random
(BHJ). Defend at least one; report diagnostics for both.
- **Rotemberg weights** decompose the OLS-IV estimate into industry-specific
just-identified IVs. Concentration warnings + heterogeneity warnings come
for free.
- **Inference**: cluster-on-region underestimates uncertainty. Use AKM SE
(Adão, Kolesár, Morales 2019) or BHJ shock-level regressions.
- **R packages**: `fixest` for the regression, `bartik.weight` for Rotemberg
decomposition, `ShiftShareSE` for AKM standard errors, `ssaggregate` for BHJ
shock-level regressions.
