17  Shift-Share Instrumental Variables

using DataFrames
using LinearAlgebra
using Statistics
using Random
using CairoMakie
using Panelest
using StatsModels
using CSV

include("../../software/ShiftShareIV.jl/src/ShiftShareIV.jl")
using .ShiftShareIV

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.

17.1 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. 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.

17.2 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.

17.2.1 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). 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?

17.2.2 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.

17.3 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 the bhj_collapse function.
  • AKM SE (Adão, Kolesár, Morales 2019): explicit formula accounting for shock-level correlation; requires custom implementation or a dedicated package.

17.4 Simulation: build intuition

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

n_region   = 500
n_industry = 20
beta_true  = 0.5

df     = CSV.read("data/shift_share_sim.csv", DataFrame)
shares = Matrix(CSV.read("data/shift_share_shares.csv", DataFrame))
shocks = CSV.read("data/shift_share_shocks.csv", DataFrame).shock
X = df.X; Y = df.Y; u = df.u
first(df[:, [:region, :X, :Y, :u]], 6)

A naive OLS suffers from the confounder u:

ols  = feols(df, @formula(Y ~ X))
ivss = feiv(df, @formula(Y ~ 1), endo = :X, inst = :B)

@printf("%-25s %8s %8s\n", "Estimator", "Coef", "SE")
@printf("%-25s %8.3f %8.3f\n", "OLS (biased)",
        coef(ols)[end], stderror(ols)[end])
@printf("%-25s %8.3f %8.3f\n", "Shift-share IV",
        coef(ivss)[1], stderror(ivss)[1])
@printf("True beta: %.3f\n", beta_true)

OLS is biased upward by the confounder; the shift-share IV recovers a value close to the true effect.

17.4.1 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:

rw = rotemberg_weights(shares, shocks, X, Y)

@printf("%-10s %8s %8s %8s\n", "Industry", "Shock", "α_k", "β_k")
for row in eachrow(rw)
    @printf("%-10d %8.3f %8.3f %8.3f\n",
            row.industry, row.shock, row.alpha, row.beta_k)
end
@printf("\nGPSS identity: Σ α_k β_k = %.4f\n", sum(rw.alpha_beta))
@printf("Shift-share IV estimate: %.4f\n", coef(ivss)[1])

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 are 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.
fig = Figure(size = (640, 380))
ax  = Axis(fig[1, 1],
           xlabel = "Industry",
           ylabel = "Just-identified IV estimate (β_k)",
           title  = "Industry-level IV estimates and Rotemberg weights")

# Bubble plot: position = beta_k, size = |alpha_k|
scatter!(ax, rw.industry, rw.beta_k,
         markersize = 30 .* sqrt.(abs.(rw.alpha)),
         color = (:steelblue, 0.7))
hlines!(ax, [beta_true], color = :firebrick, linestyle = :dash,
        linewidth = 2, label = "True β = $(beta_true)")
axislegend(ax, position = :rb, framevisible = false)
fig

17.4.2 What goes wrong when shares are endogenous

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

v = CSV.read("data/shift_share_bad_v.csv", DataFrame).v
shares_bad = copy(shares)
shares_bad[:, 1] .= max.(0.01, shares[:, 1] .+ 0.15 .* v)
shares_bad = shares_bad ./ sum(shares_bad, dims=2)   # renormalise

bad_noise = CSV.read("data/shift_share_bad_noise.csv", DataFrame)
B_bad = bartik_iv(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  = DataFrame(X = X_bad, Y = Y_bad, B = B_bad)
iv_bad  = feiv(df_bad, @formula(Y ~ 1), endo = :X, inst = :B)

@printf("True beta: %.3f\n", beta_true)
@printf("Shift-share IV with bad share 1: %.3f\n", coef(iv_bad)[1])

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 shock-level inference) would detect the problem.

17.5 BHJ shock-level inference

Collapse the location-level data to the shock level and run a weighted IV at the shock (industry) level. This reframes the identifying assumption: shocks must be uncorrelated with the shock-level aggregated outcome residual.

collapsed = bhj_collapse(shares, shocks, Y, X)

# BHJ shock-level 2SLS: regress Y_agg ~ X_agg | shock, weighted by weight
df_collapsed = collapsed
iv_bhj = feiv(df_collapsed, @formula(Y_agg ~ 1),
              endo = :X_agg, inst = :shock, weights = :weight)

@printf("Location-level shift-share IV: %.3f\n", coef(ivss)[1])
@printf("BHJ shock-level IV:             %.3f\n", coef(iv_bhj)[1])
@printf("(Should be numerically equal under standard conditions)\n")

The BHJ shock-level regression is equivalent to the location-level IV when there are no controls. Its value is interpretive: validity is now a statement about shocks (20 observations, one per industry) rather than locations (500 observations), and testing shock-level exogeneity is more transparent.

17.6 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

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

where the shocks \(\Delta M^{\text{other}}_k\) are growth in Chinese imports into other rich countries (a “leave-one-out” construction that strips US-specific demand from the shocks).

# David Dorn distributes the replication data at ddorn.net
# Files needed:
#   workfile_china.csv  (commuting zone data, 1990–2007)
#   industry_shares.csv (czone × industry shares)

using CSV
cz     = DataFrame(CSV.File("data/workfile_china.csv"))
shares = Matrix(select(cz_wide, r"share_"))   # L × K share matrix
shocks = cz_wide.import_shock_other            # K-vector of "other-countries" shocks

B   = bartik_iv(shares, shocks)               # Bartik instrument
df_adh = hcat(cz, DataFrame(B = B))

# First-stage + second-stage
iv_adh = feiv(df_adh, @formula(d_sh_empl_mfg ~ 1),
              endo = :d_tradeusch_pw, inst = :B,
              vcov_type = Vcov.cluster(:statefip))

# Rotemberg decomposition
rw_adh = rotemberg_weights(shares, shocks, df_adh.d_tradeusch_pw,
                           df_adh.d_sh_empl_mfg)
# Top 5 industries by |alpha|:
sort(rw_adh, :alpha, rev=true)[1:5, :]

# BHJ shock-level regression
collapsed_adh = bhj_collapse(shares, shocks, df_adh.d_sh_empl_mfg,
                             df_adh.d_tradeusch_pw)
iv_bhj_adh = feiv(collapsed_adh, @formula(Y_agg ~ 1),
                  endo = :X_agg, inst = :shock, weights = :weight)

Modern best-practice extensions of the basic ADH regression:

  1. Rotemberg weights (via rotemberg_weights): identify which industries drive the estimate; flag weight concentration and heterogeneous \(\hat\beta_k\).
  2. BHJ shock-level regression (via bhj_collapse): reframe validity as a statement about 20 industry shocks rather than 500 commuting zones.
  3. AKM standard errors (Adão, Kolesár, Morales 2019): account for shock-level correlation in inference. Requires custom implementation; see the accompanying R chapter for an R-based version.

17.7 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 2SLS estimate into industry-specific just-identified IVs. Concentration warnings and heterogeneity warnings come for free.
  • Inference: cluster-on-region underestimates uncertainty. Use BHJ shock-level regressions, and if available, AKM standard errors.
  • ShiftShareIV.jl: provides bartik_iv, rotemberg_weights, and bhj_collapse. Panelest.jl provides feols/feiv for the regressions.