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\footnotetext[1]{Luo: Department of Economics, Virginia Tech, [email removed]. Tsang: Department of Economics, Virginia Tech, [email removed]. Yang: Wenlan School of Business, Zhongnan University of Economics and Law, [email removed].} \setcounter{footnote}{0}
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Keywords: production networks, spatial general equilibrium, partial identification, transportation polytopes, local shocks
JEL Codes: C67, D57, E23, R12
Regional production-network calculations require assumptions about who buys from whom across space. A national input-output table tells us how much each industry buys from each other industry. Regional activity data tell us where industries are located. They do not tell us which supplier states sell intermediate inputs to which buyer states. The missing matrix is a bilateral state-sector buyer-seller matrix.
This paper studies what can be learned about that matrix before imposing a regionalization rule. We focus on the intermediate-input spatial kernel, the joint distribution of supplier and buyer locations for purchases from each supplier industry. Public data restrict its origin and destination margins only after the researcher chooses proxy mappings from regional activity to intermediate supply and demand. They do not identify the cells of the joint distribution. A proportional allocation, gravity completion, or support rule is therefore a maintained restriction on an unidentified coupling, not an observed regional IO table.
We develop a partial-identification approach to this problem. The admissible set consists of all nonnegative spatial kernels that satisfy maintained proxy margins, support restrictions, and auxiliary shipment moments. For linear one-step exposure statistics, the sharp identified interval is computed by transportation linear programs. This lets us separate exposure claims that survive the admissible set from claims that are imposed by a particular completed regional network.
The distinction matters because the same missing links are harmless for some propagation questions and first order for others. A pure national industry shock has no within-industry geographic variation, so the spatial kernel cancels in first-round exposure and in the Leontief accounting multiplier. This does not imply that industry shocks have small aggregate effects. It means that they cannot identify the spatial kernel. Local shocks are different. For regional and region-sector shocks, the kernel determines where downstream exposure lands, even when aggregate exposure looks stable.
We make the point in a regional trade-production model. The national input-output table gives the industry input share. What it does not give is the spatial sourcing share that says which supplier locations sell to which buyer locations. In an unrestricted regionalization, the regional intermediate-spending coefficient is
The national IO table identifies \(\omega_{ji}\), but not \(\pi^{ij}_{r|s}\). The empirical application imposes a lower-dimensional supplier-sector kernel, so buyer industries share the same spatial sourcing pattern within a supplier sector. This reduces the dimensionality of the missing bilateral matrix, but it does not make the remaining supplier-sector coupling identified.
The empirical application uses U.S. states, sixteen sectors, national IO coefficients, QCEW wage-bill and employment proxies for state-sector activity, and the 2017 Commodity Flow Survey. The CFS moments show that the proportional conditional-independence completion is too spatially diffuse in shipment-covered sectors. The average within-state shipment share is 0.506 in the CFS, compared with 0.028 under conditional independence and 0.384 under a structured gravity completion. Conditional independence also has a mean distance-bin total variation gap of 0.578, compared with 0.123 for structured gravity.
These shipment moments restrict the admissible set, but they do not identify local-shock incidence. For a Gulf regional shock, conditional independence lies outside the final sharp exposure interval for the directly shocked state of Louisiana: its point exposure is 7.5e-05, below the final sharp lower endpoint of 0.0009. Thus the proportional completion is not simply one admissible point inside the final set. The same calculation also shows the limit of the available public data. Adding bilateral CFS cell bands lowers the median state exposure interval width by 11.9 percent relative to home-share and distance-bin moments. The final exact-margin specification still determines only 0.6 percent of sharp state-pair exposure rankings, and only 1.0 percent of pairs involving the 20 states with the largest upper endpoints. No state is guaranteed to be in the top exposure decile, and all 51 states remain possible top-decile states.
The residual interval width is not concentrated only in the shipment-covered goods sectors. Services and other pooled sectors account for 53.5 percent of total state interval width, with FIRE and professional and business services alone accounting for 26.5 and 19.4 percent. CFS-covered goods and logistics account for 38.7 percent. Additional goods-shipment information therefore narrows the set, but it cannot by itself resolve state-level incidence when large service-sector sourcing relationships remain weakly observed. When wage-bill and employment margins are treated as bands rather than exact margins, aggregate exposure is no longer pinned down by exact-margin aggregation. Its width is 0.0019, while the median state width remains 0.0036.
We make three contributions. First, we characterize sharp bounds for unknown regional production-network couplings. The endpoints are transportation linear programs, and the dual variables summarize which margins or moments bind a given bound. Second, we compare common regionalization restrictions using CFS shipment moments and public state-sector activity proxies. The comparison covers conditional independence, structured gravity completion, local support restrictions, bilateral CFS cell bands, and banded margins. Third, we apply the admissible set to local-shock exposure and show which state-incidence conclusions survive. The analysis does not estimate a full regional IO table, employment effect, output effect, or welfare effect. A full counterfactual still requires final-demand sourcing, behavioral parameters, and equilibrium closure. A causal event study also needs a design for the outcome response.
The closest substantive literature is macroeconomic work on production networks and spatial propagation. Input-output linkages shape the transmission of shocks across sectors and, in regional models, across locations Hulten1978,LongPlosser1983,Horvath1998,Horvath2000,Gabaix2011,FoersterEtAl2011,AcemogluEtAl2012,Atalay2017,CaliendoEtAl2018,BaqaeeFarhi2019. A natural structural reference point is CaliendoEtAl2018, who specify a quantitative spatial model to study regional and sectoral productivity shocks, equilibrium reallocation, aggregate effects, and welfare. AdaoArkolakisEsposito2020 provide a complementary reduced-form bridge between shift-share designs and spatial general equilibrium effects by estimating bilateral reduced-form elasticities across local labor markets. We ask a different question. We study what public data identify about one input into these calculations before imposing the trade, substitution, labor-market, final-demand, and market-clearing structure needed for a full counterfactual.
The analysis is also related to evidence on firm-level production networks. Firm-to-firm data show that actual buyer-seller links matter for shock transmission BarrotSauvagnat2016,BoehmEtAl2019,CarvalhoEtAl2021. Those papers use more detailed link-level information than is available in public regional IO construction. We ask what remains identifiable when the researcher has national industry linkages, state-sector proxy margins, and shipment moments, but not buyer-seller intermediate-input links.
The paper also uses partial identification and data combination. The identified-set logic follows the partial-identification perspective of Manski2003, Tamer2010, and Molinari2020. The empirical problem also resembles data-combination settings in which separate sources identify different margins or moments of an unobserved joint distribution CrossManski2002,RidderMoffitt2007. We do not report confidence intervals for the identified set. If sampling inference over estimated bounds were added, the relevant econometric issues would be those studied by ImbensManski2004 and Stoye2009.
Finally, the computation uses the same mathematical structure as discrete optimal transport with fixed marginals. We use that structure as an identification and accounting tool, not as a behavioral transport-cost model. The linear-program representation is standard in optimal transport Galichon2016. The regional IO literature has long developed non-survey, partial-survey, location-quotient, and balancing methods for constructing regional and interregional tables MillerBlair2009,FleggEtAl1995,FleggWebber2000,BoeroEtAl2018. We do not propose a new regionalization formula. We provide an identified-set analysis of the intermediate-input spatial kernel that such calculations often take as an input.
The remainder of the paper proceeds as follows. Section (ref) defines the spatial-kernel coupling problem, the maintained proxy margins, and the sharp linear-programming bounds. Section (ref) explains which exposure and multiplier calculations depend on the kernel before structural closure. Section (ref) describes conditional independence, structured gravity completion, support restrictions, and CFS moment bands. Section (ref) presents the U.S. state-sector evidence. Section (ref) reports the identified-set exposure results. Section (ref) compares the Gulf regional shock with a national manufacturing shock. Section (ref) concludes. The appendix gives proofs, data details, nonlinear multiplier calculations, and robustness tables.
The empirical problem is a missing joint distribution. A national input-output table gives supplier-industry spending shares \(\omega_{ji}\) and buyer industry intermediate-input intensities \(\mu_j\). Their product is
The missing component is the spatial coupling for intermediate purchases of a fixed supplier sector \(i\). Let \(K^i_{rs}\) be the share of supplier-\(i\) intermediate purchases that pairs supplier state \(r\) with buyer state \(s\). The destination-conditional sourcing share is
where \(b^i_s\) is the destination margin. The regional input coefficient is
Thus the national IO block identifies industry linkages, while \(K^i\) allocates those linkages across space.
The baseline uses maintained proxy margins. The origin margin is the state-sector activity share,
with \(x_{ri}\) measured by the relevant state-sector activity proxy. The destination margin measures where supplier-\(i\) inputs are used:
These margins are not observed bilateral intermediate-input flows. They are maintained mappings from public state-sector activity data and the national IO block.
This creates two layers of non-identification. First, state-sector activity does not separately identify output sold to intermediate users, household final demand, and residual final use. Second, even if the relevant origin and destination totals were known, the bilateral matrix matching supplier states to buyer states would remain unidentified. The baseline therefore studies the narrower intermediate-input kernel. Final-demand sourcing and equilibrium closure are left to Appendix (ref).
A two-region example makes the non-identification concrete. Let the origin and destination margins both be \((1/2,1/2)\). The two couplings \[
\qquad and \qquad
\] match the same margins. The first is entirely local sourcing, while the second is entirely cross-region sourcing. National IO coefficients and regional margins alone cannot distinguish them.
For a generic finite coupling problem, let \(K\) be a nonnegative matrix with row margin \(a\) and column margin \(b\). Support restrictions set selected entries to zero. Auxiliary moments have the form \[ m_h(K)=\sum_{x,y}g_{h,xy}K_{xy}. \] With a support set \(\mathcal S\) and a moment set \(\mathcal M\), the admissible set is
Conditional independence is one feasible point when \(K_{xy}=a_xb_y\). Structured gravity is another point completion. Support restrictions and CFS moment bands define subsets of the transport polytope.
When auxiliary moments are estimated or reconciled from imperfect data, we use moment bands:
For exposure, the target is linear. Let \(Q(E)=\sum_{s,j}q_{sj}E_{sj}\). For a shock \(z\),
The theorem and propositions are the main tools used below. They also define the limits of the analysis. The bounds are sharp for linear exposure given the maintained margins and moment bands. They do not identify behavioral elasticities, final-demand substitution, factor adjustment, or welfare.
What does the missing kernel affect? Given an admissible intermediate-input kernel \(K\), define the regional input matrix
The first-round exposure of destination node \((s,j)\) to a shock vector \(z\) is
This one-step exposure is linear in \(K\). With a unit shock, a value of 0.001 means one-tenth of one percentage point in this accounting exposure index before equilibrium responses. The corresponding Leontief accounting multiplier is
whenever \(\rho(A^K)<1\). A Domar-style accounting exposure index is
These are accounting quantities. Employment, output, and welfare responses require final demand, prices, factor adjustment, financing, and market clearing.
For a pure industry shock, \(z_{ri}=z_i\) for every origin \(r\), the first-round exposure of buyer node \((s,j)\) is
because the sourcing shares sum to one. The same cancellation holds for Leontief accounting exposure because each term in the Leontief series maps a vector that is constant across origins within an industry into another vector with the same property. Pure industry shocks therefore cannot validate a spatial regionalization rule.
For a pure regional shock, \(z_{ri}=z_r\), the kernel generally determines where exposure lands. One aggregate incidence measure still cancels. If exposure is first aggregated over destination regions using the supplier-sector destination margins \(b^i_s\), then
The proposition gives the paper's boundary result. Industry shocks can answer industry propagation questions, but they say little about regional incidence. Regional and region-sector shocks require the spatial kernel for local incidence.
Leontief exposure is harder to bound sharply because \(M^K\) is nonlinear in \(K\). Exact global bounds are nonlinear optimization problems. We therefore use one local outer-bound calculation only as an appendix result. Appendix (ref) gives the sensitivity formula and the conservative remainder bound used in the nonlinear multiplier table. The main empirical results below are sharp for linear one-step exposure, not for the full nonlinear multiplier.
Common regionalization rules are restrictions on the feasible coupling set. Some select a point completion. Others define an admissible subset over which linear exposure can be bounded. The question is whether the restriction is admissible relative to maintained margins, support restrictions, and auxiliary shipment moments.
The proportional completion fills in the missing joint coupling as
In destination-conditional form, every buyer region sources supplier-\(i\) inputs from the same origin distribution. Conditional independence is also the maximum-entropy completion, maximizing
This makes conditional independence a useful benchmark, not an identified coupling. It rules out home bias, distance decay, corridor structure, and other dependence between supplier and buyer locations after conditioning on supplier industry.
We also use a structured gravity completion as a point comparison. For each supplier industry \(i\), the unbalanced kernel is
It is rebalanced by iterative proportional fitting to match the maintained margins.
The empirical analysis uses different restrictions by sector. For shipment-covered sectors, we estimate a gravity-style point completion,
where origin and destination fixed effects absorb the marginals. For mixed sectors, we pool spatial parameters. For local sectors, we impose support restrictions rather than treating shipment-like observations as sectoral intermediate-input flows. The baseline local support includes same-state pairs, adjacent-state pairs, and the minimum additional nearest-state radius needed for feasibility.
Let \(\mathcal S_i\) be a support set and let \(\mathcal K_i(a^i,b^i,\mathcal S_i)\) be the nonnegative matrices that match the margins and place zero mass outside \(\mathcal S_i\). For any linear exposure functional \(L(K^i)=\sum_{r,s}\ell_{rs}K^i_{rs}\), sharp bounds are
These are sharp under the maintained support restriction because the feasible set is exactly the transport polytope matching \(a^i\), \(b^i\), and \(\mathcal S_i\).
All restrictions are fixed before propagation outcomes are inspected. This differs from a standard regional IO construction because a completed matrix can fit shipment geography and still reveal little about local-shock incidence.
We next compare common spatial-kernel restrictions with observed shipment geography. The empirical application uses U.S. states and a sixteen-sector aggregation. The national input-output block provides the industry shares \(\omega_{ji}\) and the intermediate-input intensities \(\mu_j\). The baseline origin margins, destination margins, and exposure weights use 2019 QCEW wage-bill shares. The destination marginal is constructed from destination activity, sector intermediate-input intensities, and the national IO matrix, as in equation (ref). It is a proxy for intermediate-demand geography, not regional final expenditure. Wage bills are the baseline because they are a public, consistently available measure of state-sector economic activity and they weight high-productivity state-sector cells more than headcount alone. QCEW employment shares are the first robustness margin, and model-output margins are reported only in robustness. State-to-state shipment flows come from the 2017 Commodity Flow Survey. State distances are computed from Census state centroids.
We therefore interpret the empirical inputs as maintained proxy measures. The spatial kernel is the unknown intermediate-input coupling. Conditional independence is the proportional benchmark. Structured gravity completion is a low-dimensional point completion estimated from shipment geography. One-step exposure is an accounting exposure measure before employment responses, price responses, and welfare effects.
CFS moments are auxiliary shipment moments, not the full intermediate-input coupling. The distinction matters for interpretation. First, CFS shipments include final goods as well as goods that may become intermediate inputs. Second, CFS commodity classifications do not map one-for-one into the production industries in the IO table. Third, wholesale shipments and re-shipments can break the link between the producing origin and the intermediate-input seller relevant for a buyer. Fourth, services are missing or weakly covered, which matters because many local and business-service sectors are large in the IO block. Finally, CFS observes goods movement. It does not observe the buyer-seller use of intermediate inputs by destination industry. For this reason, the CFS restrictions below restrict shipment geography within selected sectors, but they do not convert the spatial kernel into an observed matrix.
We use point completions only as comparisons. Conditional independence is the proportional benchmark. Structured gravity, with pooled variants where data are thin, is the shipment-informed comparison. For local sectors, the relevant calculation is not a point completion but the support-restricted admissible set. Lower and upper support-bound kernels summarize feasible ranges. All comparisons hold fixed the same national IO block and the same intermediate-demand marginals. They differ only in the intermediate-input spatial coupling.
Figure (ref) shows the main empirical fact. In the four strict shipment-covered sectors, observed state-to-state CFS flows display large within-state shares. The average observed CFS home share is 0.506. Conditional independence implies 0.028. The structured gravity completion implies 0.384. Thus conditional independence does not merely smooth bilateral flows. It removes most of the same-state mass observed in shipment data. Appendix Table (ref) reports the corresponding home-share and held-out RMSE values.
Table (ref) reports admissibility frontiers using the same evidence. For each restriction, we ask how much the CFS moment restrictions must be relaxed before that restriction becomes admissible. A larger tolerance means the restriction is farther from the shipment evidence. Formally, for a restriction \(R\), let \[ \tau_R=\inf\{\tau:m(K_R)\in\mathcal M(\tau)\} \] be the smallest tolerance under which the restriction is admissible for the maintained moment. For the shipment-covered sectors, the moments are the CFS home share, the CFS distance-bin distribution, and held-out flow fit. Conditional independence requires an average home-share tolerance of 0.478, while sector gravity requires 0.122. The corresponding distance-bin total variation gaps are 0.578 and 0.123. This should not be read as a formal statistical rejection of conditional independence, because it does not use CFS sampling variances. It shows that conditional independence is admissible only under a much looser moment set than sector gravity.
Local support restrictions are different from the shipment-covered point-completion comparisons in Table (ref). They define feasible sets rather than CFS flow-fit statistics. Appendix Figure (ref) reports the corresponding local-sector home-share intervals. The mean interval width is 0.714.
The gap is economically large. The proportional completion does more than smooth flows at the margin. It almost eliminates home bias in the sectors where shipment data show home bias most clearly. In manufacturing, observed CFS flows imply a within-state share of 0.371, while the CI completion implies 0.026. In wholesale, the corresponding numbers are 0.606 and 0.028. Gravity is closer on both home shares and held-out flow fit, but it remains a maintained low-dimensional restriction rather than an identified intermediate-input spatial kernel.
This distinction motivates the sector treatment in Table (ref): sector-specific CFS restrictions for shipment-covered sectors, pooled parameters where shipment evidence is thin, and support-restricted bounds for local sectors.
We now report sharp bounds for one-step exposure over the admissible set of intermediate-input kernels. The bounds are not ranges across selected point completions. They are the minimum and maximum exposure values attainable by any kernel satisfying the maintained proxy margins, support restrictions, and CFS moment bands. The calculations hold fixed the national IO table, intermediate-demand proxy margins, sectoral intermediate-input intensities, and state-sector wage-bill exposure weights. Only the intermediate-input spatial coupling changes.
The results show that the public data determine some incidence comparisons but leave many state rankings unresolved. The calibrated economy has 51 states and 16 sectors. For each coupling \(K\), we construct the regional input matrix in equation (ref) using \(B_{ji}=\mu_j\omega_{ji}\), with \(\mu_j=\text{intermediate}_j/\text{output}_j\). The maximum spectral radius across all calibrated and comparator matrices is 0.4483, so the accounting inverse exists in every reported case.
We first study a Gulf regional shock that hits Louisiana and Mississippi in all supplier sectors. This shock has geographic content, so the spatial kernel matters for where exposure lands. At the same time, the exact-margin aggregate target has a cancellation property. Because the aggregate weights and destination margins use the same wage-bill proxy, the unknown bilateral kernel collapses to maintained origin margins after destination aggregation. The zero aggregate width in the exact-margin baseline should therefore not be read as evidence that bilateral sourcing is precisely identified. We therefore focus on state-level incidence.
Table (ref) reports how the Gulf exposure set changes as information is added. The first four rows add exact proxy margins, local-sector support restrictions, CFS home-share bands, and CFS distance-bin bands. The fifth row adds selected bilateral CFS cell bands for shipment- covered sectors. The final two rows replace exact wage-bill margins with bands whose lower and upper endpoints are the QCEW wage-bill and employment shares. The mean home-share feasibility tolerance is 0.035, and the maximum is 0.141. Appendix Table (ref) reports the sector-specific tolerances.
The top-decile classification is deliberately conservative, so we also solve sharp pairwise bounds. For each unordered state pair, we bound \(E_s-E_{s'}\). One state is classified as dominating the other only when the sharp lower bound is positive or the sharp upper bound is negative. Under the final exact-margin bilateral-CFS set, only eight of 1,275 state pairs are determined, or 0.6 percent. Only one state has any robust dominance relation. The determined share among pairs involving the 20 states with the largest upper endpoints is 1.0 percent. This pattern is visible in Figure (ref): except for Louisiana, most lower endpoints among high-upper-bound states remain close to zero.
Figure (ref) shows why many rankings remain unresolved. Most lower endpoints among the high-upper-bound states are close to zero, and many intervals overlap even after CFS restrictions. The figure also shows that conditional independence is outside the final sharp interval for one directly shocked state. For Louisiana, conditional independence gives exposure 7.5e-05, below the sharp lower endpoint 0.0009. Structured gravity remains inside all displayed intervals.
The sector decomposition in Table (ref) explains where residual uncertainty comes from. The decomposition is exact: the state exposure interval width is the sum of sector-level endpoint differences because the transport problem separates by supplier sector. FIRE, professional and business services, manufacturing, wholesale, and transportation are the largest contributors to median state width. Grouping the sectors shows that services and other pooled sectors account for 53.5 percent of total state interval width. FIRE and professional and business services alone account for 26.5 and 19.4 percent. CFS-covered goods and logistics account for 38.7 percent. The residual uncertainty is therefore not only a problem of missing goods-shipment cells. It also reflects large service and mixed sectors whose intermediate-input geography is not directly observed in CFS.
The bilateral CFS cells lower the median state width by 11.9 percent, so they add information, but the gain is modest relative to the remaining interval width. Appendix Table (ref) reports which selected bilateral cells bind in the Gulf exposure endpoint problems.
The banded-margin rows in Table (ref) evaluate the role of exact wage-bill margins. In those rows, each sector's total mass remains one, but origin and destination shares are allowed to range between the QCEW wage-bill and employment shares. Under banded margins with bilateral CFS restrictions, aggregate exposure width is 0.0019 and median state width is 0.0036. These rows should be interpreted as a separate admissible-set layer, not as a nested refinement of the exact wage-bill baseline.
Appendix Table (ref) reports the CFS moment-band calculation. Appendix Table (ref) repeats the identified-set calculation with wage-bill, employment, mixed, and model-output margins. The model-output row is a robustness specification rather than the baseline. Across those margin constructions, all states remain possible top-decile exposure states. Appendix Table (ref) also shows that excluding mining CFS moments leaves the median and p90 state interval widths at 0.0031 and 0.0042. The exact-margin aggregate cancellation is therefore a property of the maintained aggregate target. Weak identification of regional incidence is not.
Appendix Table (ref) reports the largest nonzero LP moment shadows for state-level Gulf exposure bounds. The rows are not whole-state exposure endpoints. They are sector-specific dual results inside the state-level bounds. They show which CFS moment restrictions bind particular sector-level endpoints, while the maximum primal-dual gap in the full dual output is only 0.
Nonlinear Leontief multiplier exposure is harder to bound sharply. Appendix Table (ref) reports the perturbation outer-bound calculation. In the Gulf application, the first-order LP interval is much narrower than the final outer interval because the conservative perturbation radius, 2.947, is above one. The resulting outer interval is valid and contains the reported point-completion multiplier values, but it is too wide to support a sharp nonlinear identified-set claim in this application. Without additional structure, nonlinear Leontief exposure is much harder to sharply bound than linear one-step exposure. For this reason, the empirical results in the main text are interpreted as sharp bounds for one-step exposure, not for full nonlinear propagation.
The preceding results imply a simple distinction. A shock can have economic effects without identifying the spatial kernel. If the shock is national within a supplier industry, every origin in that industry is hit and the destination-conditional sourcing shares sum out. A shock with geographic content is different because the spatial kernel determines where downstream exposure lands.
Table (ref) applies this distinction to two shock designs. The first is a Katrina-style Gulf regional shock: \[ z_{ri}^{Gulf} = \mathbf{1}\{r\in\{\mathrm{LA},\mathrm{MS}\}\}. \] The shock hits Louisiana and Mississippi in every supplier sector. Its geographic content is sharp while its industry content is broad. The second is a national manufacturing-input shock: \[ z_{ri}^{Mfg} = \mathbf{1}\{i=\mathrm{manufacturing}\}. \] It is best interpreted here as an industry-shock limiting case rather than a detailed tariff counterfactual.
For the Gulf regional shock, the point-completion aggregate difference is not negligible. The Domar-weighted loss under conditional independence is 0.860 of the structured-gravity loss. The regional allocation also changes substantially: state-level allocation TV is 0.137, and top-decile overlap is 0.598. Thus the proportional completion changes both aggregate accounting loss and which downstream places are classified as highly exposed. For the national manufacturing shock, all metrics are invariant up to numerical precision, as predicted by Proposition (ref).
The ranking changes are most visible outside the directly shocked states. Louisiana and Mississippi remain the two most exposed destination states under both completions. But Arkansas is ranked 3 under structured gravity and 36 under conditional independence. California moves from rank 10 to 3, and New York moves from rank 20 to 5, under conditional independence. These are point-completion comparisons, not sharp identified rankings, but they show why regional incidence cannot be inferred from aggregate exposure alone.
Regional production-network calculations require assumptions about who buys from whom across space. Standard data provide national industry IO tables and regional sectoral activity, but they do not observe the bilateral state-sector buyer-seller matrix. This paper shows that those data identify an admissible set of intermediate-input spatial kernels, not a unique regional network or a full regional IO table. A completed regional IO matrix should therefore be interpreted as a maintained restriction on the missing coupling, not as an observed input.
This distinction matters because the missing spatial kernel is not equally relevant for all propagation questions. Pure national industry shocks are invariant to the kernel and therefore provide a limiting case for regionalization assumptions. Local regional and region-sector shocks are different because the kernel determines where downstream exposure lands. For linear one-step exposure, the admissible set delivers sharp transportation-program bounds. For nonlinear Leontief accounting exposure, the same problem becomes a nonlinear identified-set problem. The perturbation calculation provides conservative outer bounds rather than sharp global bounds.
The U.S. state-sector application shows that the issue is quantitatively relevant. CFS shipment moments imply substantially more home bias and distance concentration than conditional independence, so the proportional completion is too spatially diffuse in shipment-covered sectors. At the same time, these moments do not identify the full intermediate-input coupling or the state-level incidence of local shocks. For a Gulf regional shock, selected bilateral CFS cells narrow the state exposure intervals, but they determine few state-pair rankings. Banded wage-bill and employment margins remove the exact-margin aggregate cancellation, yet state-level rankings remain weakly identified.
These bounds can restrict quantitative spatial models, but they do not replace them. A full counterfactual still requires final-demand sourcing, household-demand geography, behavioral elasticities, factor adjustment, financing, and market clearing. The point is to separate the intermediate-input spatial network features supported by proxy margins and auxiliary shipment moments from those imposed by regionalization assumptions. Rather than evaluating a structural model at a single proportional regionalization, researchers can ask whether its exposure and counterfactual conclusions survive over the admissible set of spatial kernels.
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