Shift-Share IV with Exogenous Shocks: Identification and Inference
The same shift-share IV ratio admits a different identifying argument: hold exposure shares fixed and regard the aggregate shocks as quasi-random. Shares may then be correlated with outcome errors. (Borusyak, Hull, and Jaravel 2022) rewrites the location-level moment as a shock-level sum, making clear why shock dispersion matters for both consistency and inference.
1. Aggregate the moment, not the estimated coefficient
Let \(e_\ell\) be observation weights, \(s_{\ell k}\ge0\) exposure shares, and \(B_\ell=\sum_ks_{\ell k}g_k\). Residualize \(Y\) and \(X\) against the same controls using weights \(e_\ell\). Define
and define \(\overline X_k^\perp\) similarly. Ignore shocks with zero aggregate exposure.
Proposition 1. The location-level IV ratio is exactly
The last ratio is weighted shock-level IV. Intercepts and shock-level controls must correspond to the exposure-weighted controls included in the original regression.
Switching the order of summation proves the identity for any data array. Identification requires assumptions on the shocks beyond this algebra.
2. The consistency argument
Suppose the conditional mean of \(g_k\) is \(q_k'\mu\), given exposures, shock-level controls, and aggregated outcome errors. Include \(\sum_ks_{\ell k}q_k\) among the location controls. The centered shocks \(\widetilde g_k=g_k-q_k'\mu\) then satisfy a conditional mean-zero restriction.
The IV numerator at the true coefficient becomes
If the centered shocks are conditionally uncorrelated, its conditional variance is
After normalizing \(\sum_ks_k=1\), bounded moments and \(\sum_ks_k^2\to0\) make this variance vanish. Together with the paper’s control-estimation conditions and a first-stage denominator converging to a nonzero limit, this proves consistency.
The quantity \(1/\sum_ks_k^2\) is an effective number of shocks. Adding locations without dispersing aggregate exposure does not necessarily increase it. If shocks are correlated within groups, replace the dispersion requirement by one on aggregate exposure across independent shock groups, with the corresponding moment restrictions.
3. Heterogeneity and interpretation
In a simple reduced-form model \(Y_\ell=Y_\ell(0)+\sum_ks_{\ell k}g_k\theta_{\ell k}\), centered independent shocks yield weights proportional to \(e_\ell s_{\ell k}^2\operatorname{Var}(g_k)\) for the regression on exposure. For an IV treatment effect, first-stage responses enter the weights as well. A convex-average interpretation therefore needs sign restrictions on those responses; it does not follow from nonnegative shares alone.
Conditioning on composition permits endogenous shares while placing the identifying restriction on shocks. This differs from share exogeneity, and it changes which observations supply independent variation for the standard error.
4. The aggregation calculation with controls
For any residualized variable \(V_\ell^\perp\),
The first-stage and reduced-form moments obey the same identity, so their ratio does too. Residualizing \(B\) is unnecessary inside a moment with \(V^\perp\), because the fitted part of \(B\) is in the control span and is orthogonal to \(V^\perp\) under the same weights.
If shock means depend on \(q_k\), then their location-level mean is \(\sum_ks_{\ell k}q_k'\mu\). Including the exposure-weighted \(q_k\) among location controls removes this term from the moment. An arbitrary location intercept only removes it when the relevant exposure sum is constant. Incomplete shares therefore require particular care with the exposure-sum control.
5. Why a shock variance includes cross-location products
Suppress weights for one display and write \(R_k=\sum_\ell s_{\ell k}\varepsilon_\ell\). Conditional on baseline exposures and errors, the centered score is \(\sum_k\widetilde g_kR_k\). Independent centered shocks give
Squaring the aggregate retains every pair of locations exposed to the shock. With two equal exposures and residuals \(u\), the contribution is \(4\sigma_k^2u^2\); squaring residuals separately gives \(2\sigma_k^2u^2\). Opposite residual signs would instead give zero. The missing cross-products can therefore change the variance in either direction.
Equivalently, the location instruments have conditional covariance
Similar exposures can link geographically distant locations through the same shocks. Spatial distance alone need not remove this covariance. The inference arguments in (Adão, Kolesár, and Morales 2019; Borusyak, Hull, and Jaravel 2022) account for these shared contributions.
6. From dispersed shocks to a central limit theorem
Theorem 2. Under the papers’ conditional many-shock CLT, moment, approximation-error, and variance-estimation conditions, with a stable nonzero first-stage denominator \(D=\sum_\ell e_\ell B_\ell^\perp X_\ell\) and \(\widehat V/V\to_p1\),
A sufficient Lyapunov condition for the ideal independent-shock score, for some \(\eta>0\), is
This condition controls large normalized contributions, rather than simply counting shocks. For equal aggregate shares \(s_k=1/K\), the exposure concentration is \(\sum_ks_k^2=1/K\). If one shock instead retains share \(c_0>0\), this concentration is at least \(c_0^2\), however many tiny shocks are added. Increasing the number of locations exposed to that same shock cannot supply the missing independent variation.
Vanishing variance of the unnormalized score gives consistency under the first-stage conditions. A normal approximation also needs control after normalization by that variance. Exposure dispersion alone does not establish all required moment and nondegeneracy conditions.
A fourth-moment calculation links the no-dominant-shock condition to variance estimation. In the ideal conditional experiment, treat \(R_k\) as fixed and let the centered shocks be independent with variances \(\sigma_k^2>0\) and \(\mathbb{E}[\widetilde g_k^4]\leq C\sigma_k^4\). The oracle variance estimate
has conditional expectation \(V=\sum_kR_k^2\sigma_k^2\). Independence yields
When the largest shock’s variance share vanishes, Chebyshev’s inequality gives \(\widetilde V/V\to_p1\). The displayed ratio also bounds the fourth-moment Lyapunov expression. Thus the same concentration calculation supports both the score CLT and the oracle variance estimate.
For the feasible residual aggregate \(\widehat R_k=R_k+\Delta_k\), the variance difference is
Cauchy–Schwarz bounds the first term in magnitude by \(2\sqrt{\widetilde V}\sqrt{\sum_k\widetilde g_k^2\Delta_k^2}\). Thus a sufficient residual-estimation condition is \(\sum_k\widetilde g_k^2\Delta_k^2=o_p(V)\), together with control of estimated shock means. Merely having each residual converge pointwise would not provide this variance-weighted bound.
Aggregation forms one \(R_k\) from all locations exposed to a shock before squaring. This retains their shared variation. The shocks, or independent groups of shocks, then supply the independent terms in the CLT and variance calculation.
7. Feasible variance and correlated shock groups
Implementation uses exposure weights to aggregate residualized outcomes and treatments, followed by the corresponding weighted shock-level IV regression. Feasible score calculations substitute estimated residuals and centered shocks. Their total variance error must be negligible relative to \(V\); pointwise convergence alone is insufficient.
If shocks are independent only across groups \(h\), the variance becomes
The within-group covariance terms remain in this expression. Dispersion assumptions and variance estimation must therefore use independent groups as their units. Treating each shock separately would omit those terms.
Finally, \(D\) is the IV first-stage moment. It equals a residualized instrument sum of squares only in a regression directly on that instrument. A Wald interval also requires \(D\) to stay away from zero at the appropriate scale. A null-imposed score test can avoid that division when inverted, connecting this issue to weak-instrument inference. Neither standard-error construction repairs failure of the shock exogeneity condition.