Problem Set 4.5 — Shift-Share Exposure Weights and the Identification Critique
Covers Ch 4.5, and leans on Week 3 IV. Methods allowed: everything through Ch 4.5 — the shift-share / Bartik construction B_r=\sum_k s_{rk}\,g_k from baseline local exposure shares s_{rk} (with \sum_k s_{rk}=1) and national shifts g_k; the canonical local-shock design (instrument the realized local shock with B_r); the two identification views — Goldsmith-Pinkham–Sorkin–Swift (2020) “identification from the shares” (exogenous initial shares; Rotemberg-weight decomposition \hat\beta_1^{\text{Bartik}}=\sum_k\hat\alpha_k\,\hat\beta_{1,k}, \sum_k\hat\alpha_k=1) and Borusyak–Hull–Jaravel (2022) “identification from the shifts” (quasi-random shocks across many dispersed industries; effective number of shocks); the Adão–Kolesár–Morales (2019) inference problem (cross-regional error correlation from shared exposure); and all of Ch 3.4 IV (relevance \operatorname{Cov}(Z,D)\neq0 testable; exclusion \operatorname{Cov}(Z,\varepsilon)=0 untestable; first-stage F; LATE-style weighted-average estimands). You may treat OLS, FWL (Ch 2.3), clustering (Ch 2.4), and the IV machinery (Ch 3.4–3.5) as known. Show your reasoning; a boxed number with no argument earns no credit, and any time you invoke a condition you must name it — “relevance,” “exclusion,” “exogenous shares,” or “quasi-random shifts” — never just “the assumption.”
Total: 100 points. Six problems, escalating. Problem 1 constructs Bartik instruments by hand from shares and shifts (arithmetic). Problem 2 is the relevance / first-stage problem (why B_r predicts the realized local shock). Problem 3 is the GPSS “identification from the shares” view (what must be exogenous; which industries drive the estimate, via Rotemberg weights). Problem 4 is the BHJ “identification from the shifts” view (as-good-as-random shocks; the effective number of shocks). Problem 5 is a design-critique problem (argue which assumption is credible in a described design and how you’d diagnose it). Problem 6 is the connection-to-IV problem (relevance and exclusion translated into shift-share terms, plus AKM inference).
One idea runs through every problem, so hold it from the start. A shift-share instrument B_r=\sum_k s_{rk}\,g_k is a product of two ingredients — local shares and national shifts — and a product is exogenous to the error if either factor is. So the single hardest, untestable assumption in all of IV (exclusion) gets two independent shots in a shift-share design: defend the shares (GPSS) or defend the shifts (BHJ). Which shot is honest depends on your data — few concentrated shares push you toward the share story, many dispersed shocks toward the shift story. Every problem below is, at bottom, about that fork: which factor can you credibly call exogenous, and what diagnostic backs it up.
Throughout, Maya anchors the finance setting: regional bank-lending exposure and its effect on household credit / delinquency. Outcome Y_r is the change in region r’s household-debt delinquency rate; the endogenous regressor is the realized local credit-supply growth \Delta C_r; the instrument is the bank-mix prediction B_r.
Problem 1 — Construct the Bartik instrument by hand (16 points)
Maya builds a bank-lending shift-share instrument. There are three banks. The shifts g_b are each bank’s national lending growth over the year (computed leaving out region r, so the region cannot have caused them). The shares s_{rb} are the fraction of each region’s baseline deposits held by each bank (rows sum to one). Two of her regions:
| Bank b | National shift g_b | Lakeside share s_{rb} | Mill Hollow share s_{rb} |
|---|---|---|---|
| BigCommerce Bank | +4\% | 0.20 | 0.70 |
| Heartland Savings | -10\% | 0.50 | 0.10 |
| Coastal Credit Union | +1\% | 0.30 | 0.20 |
(a) (3 pts) Confirm both regions’ shares are valid exposure weights (state the condition the rows must satisfy and check it).
(b) (5 pts) Compute the Bartik instrument B_r=\sum_b s_{rb}\,g_b for Lakeside and for Mill Hollow. Show each product term. Report each B_r as a percentage.
(c) (3 pts) In one or two sentences, interpret the two numbers in Maya’s setting: what predicted credit-supply shock does each region get, and why do the two regions differ even though they face the identical three national shifts?
(d) (3 pts) Suppose every region in Maya’s sample held exactly the same deposit shares (0.40, 0.30, 0.30). Compute B_r for such a region, and explain in one sentence why a shift-share instrument built this way would instrument nothing — tie your answer to which kind of variation (share dispersion or shift dispersion) has been destroyed.
(e) (2 pts) Separately, suppose instead that all three banks had the same national shift, g_b=+2\% for every b, but regions kept their different shares. Compute B_r for Lakeside under this scenario and explain in one sentence why B_r is again a constant across regions — naming the other kind of variation that has now been destroyed.
Problem 2 — Relevance / the first stage (14 points)
Maya now treats B_r as an instrument for the realized local credit-supply growth \Delta C_r, in the spec (state it in the CONVENTIONS §4 discipline): outcome = change in regional delinquency rate Y_r; key regressor = realized regional credit growth \Delta C_r; controls = baseline region demographics; fixed effects = region and period; clustering = (to be revisited in Problem 6); sample = U.S. commuting zones over the period; identifying assumption = B_r is as-good-as-randomly assigned, defended in Problems 3–4.
(a) (4 pts) Write the first-stage regression Maya runs (regress what on what?), and state the relevance condition both as a covariance statement involving B_r and \Delta C_r and in plain words. Why should we expect relevance to hold here — i.e., why would a region predicted to ride contracting banks actually experience slower realized credit growth?
(b) (4 pts) Maya’s first stage gives a coefficient on B_r of \hat\pi_1=0.85 with standard error 0.10. Compute the t-statistic and, since there is a single instrument, the first-stage F. Does this clear the “F>10” rule of thumb? State explicitly which of the two IV conditions this F-statistic speaks to.
(c) (3 pts) A colleague says, “The first stage is significant, so relevance is settled — and that means my instrument is valid.” Correct him in two sentences: distinguish which condition the first-stage F can confirm and which it can say nothing about, and connect this to the testable/untestable asymmetry from Ch 3.4.
(d) (3 pts) Recall from Ch 3.5 that “significance is not strength.” Maya’s data are clustered and heteroskedastic. Name the kind of first-stage statistic she should report instead of the classical F, and in one sentence say why a weak shift-share first stage is dangerous — what does 2SLS drift back toward as the first stage weakens?
Problem 4 — Identification from the shifts: BHJ and the effective number of shocks (18 points)
Borusyak, Hull and Jaravel (2022) locate identification not in the shares but in the shifts, viewed as a quasi-experiment over industries (or, here, banks).
(a) (4 pts) State the BHJ identifying assumption (“quasi-random shifts”) in words: what is treated as the random object, what must be true of it, and how many of them do you need? Critically, what does the BHJ view let the shares be — and why is this exactly the opposite license from the GPSS view?
(b) (4 pts) Explain the law-of-large-numbers logic in two or three sentences: why does having many dispersed shocks make the exposure-weighted average B_r=\sum_k s_{rk}\,g_k behave like a clean region-level experiment, and what goes wrong with that logic when there are only a handful of shocks?
(c) (5 pts) BHJ warn that the nominal number of shocks can overstate the effective number when a few shocks carry most of the exposure weight. A simple Herfindahl-style effective count is K_{\text{eff}}=\frac{1}{\sum_k \omega_k^2}, where \omega_k\ge0 are the (normalized, \sum_k\omega_k=1) aggregate exposure weights on each shock. Consider two of Maya’s sub-samples, each with five banks:
- Sample A (dispersed): weights \omega=(0.20,\,0.20,\,0.20,\,0.20,\,0.20).
- Sample B (concentrated): weights \omega=(0.80,\,0.10,\,0.05,\,0.03,\,0.02).
Compute K_{\text{eff}} for each sample. (Show the sum of squares.) Even though both have five banks nominally, what does K_{\text{eff}} tell you about which sample can honestly invoke the BHJ “many shocks” story?
(d) (3 pts) Maya’s full bank sample has only six banks, two of which hold most deposits region-by-region, while the labor-demand version of the design she read about had several hundred industries. State which identification view — GPSS or BHJ — she can honestly invoke in her bank setting, and why the other view fails in her data. (One or two sentences; tie it to K_{\text{eff}} and to “how many shocks.”)
(e) (2 pts) Name the natural BHJ-style diagnostic Maya would report to defend the shift side (other than K_{\text{eff}}), and say in one sentence what it is the shock-level analogue of on the share side.
Problem 5 — Critique a described design (16 points)
A team studying the 2008–09 financial crisis writes: “We estimate the effect of local credit-supply contraction on small-business closures across 700 U.S. commuting zones. We instrument realized local credit growth with a shift-share instrument: baseline 2006 deposit shares of each of the 30 largest U.S. bank holding companies, times each holding company’s national (leave-one-region-out) change in lending from 2007 to 2009. We report a strong first stage (effective F = 48) and cluster our standard errors by state.”
(a) (4 pts) Classify this design. Roughly how many shocks does it have, and how dispersed is exposure likely to be? Based on that, which identification view (GPSS or BHJ) is the more natural primary defense here, and which is the backup? Justify in two or three sentences.
(b) (4 pts) State the shift-side credibility argument and one concrete threat to it. Specifically: for the quasi-random shifts assumption to hold, the national lending changes of these 30 holding companies must be as-good-as-random with respect to which regions are exposed to them. Give one concrete reason this could fail during 2007–09 (think about what was systematically true of the banks that contracted lending most, and whether their pre-crisis footprints were random across regions).
(c) (4 pts) State the share-side credibility argument and one concrete threat to it. Specifically: for the exogenous shares assumption, 2006 deposit shares must be unrelated to the regions’ other determinants of small-business closure. Give one concrete back-door story (think: were regions banking heavily with the most crisis-exposed institutions also regions with, e.g., the frothiest housing markets or weakest pre-crisis business balance sheets?).
(d) (4 pts) Propose two concrete diagnostics the team should run — one suited to the share defense and one suited to the shift defense — and say in one sentence each what a failing result would look like. (Hint: one is a balance/pre-trend test on the shares plus a Rotemberg-weight ranking; the other is an effective-number-of-shocks count plus a shock-balance check.)
Problem 6 — Connecting back to IV: relevance, exclusion, and honest inference (16 points)
Map the whole shift-share machine onto the Ch 3.4 IV vocabulary.
(a) (4 pts) Translate relevance and exclusion into shift-share terms for Maya’s bank design. For each: write the condition involving B_r, and say in one sentence what it requires of the bank-mix prediction. Which one is testable and which must be argued — and is that the same testable/untestable split you learned for a generic instrument in Ch 3.4?
(b) (4 pts) Explain the chapter’s central unification in two or three sentences: why does the product structure B_r=\sum_k s_{rk}\,g_k give you “two shots” at the single untestable exclusion restriction? State the algebraic fact about products and errors that makes “defend the shares OR defend the shifts” a legitimate either/or rather than a both-required.
(c) (4 pts) Adão, Kolesár and Morales (2019) showed that standard shift-share standard errors are too small. Explain the mechanism in two or three sentences: why are the regression errors of two commuting zones on opposite coasts correlated when both are heavily exposed to the same bank, even though they share no geographic cluster — and what does that do to the team’s by-state clustering from Problem 5? Then name what kind of standard errors fix this, and one alternative route (from BHJ) that gets honest inference “for free.”
(d) (4 pts) Synthesize the chapter’s verdict in your own words as a four-line checklist a referee would demand of any shift-share paper: (i) the relevance check you always report; (ii) which exclusion defense you name given your data regime, and the diagnostic that backs it; (iii) the other diagnostic you’d add if your data instead favored the other regime; (iv) the inference fix that keeps your standard errors honest, with the authors who derived it. Be specific and name names.
End of PS 4.5. Solutions in book/appendices/E-solutions-manual/E-w4-ps4.5-solutions.md. This set closes Week 4 and the full Weeks 3–4 causal-inference arc. The reflex it drills — name the threat, name which exogeneity assumption your data lets you defend, separate what you can test from what you must argue, and bring the matching diagnostic — is exactly the lens Weeks 5–6 hand you for reading frontier empirical papers tables-first, including ones that say “we instrument local exposure with a Bartik shift-share design.” When you meet that sentence in a published paper, you will not nod along; you will ask whether they defended the shares or the shifts, whether they showed Rotemberg weights, how many effective shocks they had, and whether their standard errors are AKM-robust.