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Econ-338 · economics · Economics

International business-cycle transmission

LLM status Open unverified

Statement

Original subject group: Business cycles and macro dynamics

Audit classification: Background; proposed extension. The reference models inflation spillovers; the output-correlation network intervention is a new specification. Verification concerns the cited version, not first priority or proof that this exact editorial specification remains unsolved on October 8, 2026.

Definitions and precise research target

Editorial mathematical specification.
Fix countries $c=1,\ldots,C$ with measured sectoral input–output links, bilateral trade costs, cross-border assets, and country-specific policy rules. Let $\Omega$ denote the global production-input matrix, partitioned into domestic and international blocks. A structural model $M$ maps sectoral and financial innovations into country log outputs $y_c$. Construct $\Omega^0$ by replacing imported intermediate inputs with specified domestic substitutes, holding initial final-demand shares and aggregate financial shocks fixed; all resulting equilibrium prices and quantities must adjust within the model. For horizon $H$, define the network contribution to cycle synchronisation between countries $c$ and $d$ as \[ \Delta_{cd}(H)=\operatorname{Corr}_M(\Delta_H y_c,\Delta_H y_d;\Omega) -\operatorname{Corr}_M(\Delta_H y_c,\Delta_H y_d;\Omega^0), \] where $\Delta_H y_c=y_{c,t+H}-y_{c,t}$, both correlations use the same declared innovation distribution, and each output change has strictly positive finite variance under both networks. Identify $\Delta_{cd}(H)$, or a sensitivity set, using shocks with independently supported geographic incidence. Distinguish propagation from common shocks and endogenous network formation; specify whether national policy rules remain fixed or optimally respond. Validate predicted transmission against held-out trade disruptions. Removing international links is a model-based intervention whose substitute technology must be disclosed; its correlation effect cannot be read directly from observed trade–cycle correlations.

The matrix $\Omega$ must refer either to fixed technology coefficients or to observed benchmark expenditure shares; in the latter case the model updates shares after intervention. A valid replacement $\Omega^0$ specifies domestic suppliers, resource use, substitution possibilities, and trade costs, so that removed foreign inputs are not replaced for free. At a fixed initial law define $U_c^v=y_{c,t+H}^v-y_{c,t}^v$, $v\in\{0,1\}$, using common primitive shocks. Each correlation is $\operatorname{Cov}(U_c^v,U_d^v)/(\operatorname{Var}(U_c^v)\operatorname{Var}(U_d^v))^{1/2}$. The identified set consists of $\Delta_{cd}(H)$ over models matching the same observable trade, price, and output law and restrictions. This identifies a chosen network intervention, not a source-independent percentage of international synchronization.

Variants and assumption changes

The following are editorial variants, not additional published open conjectures.
  1. Shock propagation. Replace correlations by a foreign-output impulse response to a valid country-specific productivity shock. Hold the shock variance fixed and compare domestic-only versus international intermediate-input links; this avoids contamination from varying common-shock variance.
  2. Financial links. Retain trade links and instead remove cross-border credit using a specified domestic financing replacement. Compare trade-only, finance-only, and joint interventions, allocating their interaction explicitly.

References and source audit

The reference models inflation spillovers; the output-correlation network intervention is a new specification.
  1. Julian di Giovanni; Şebnem Kalemli-Özcan; Alvaro Silva; Muhammed A. Yıldırım. 2023. Pandemic-Era Inflation Drivers and Global Spillovers. Abstract and Section1, printed pp.1--3; inflation-spillover model, not output-synchronization identification. https://www.newyorkfed.org/medialibrary/media/research/staff_reports/sr1080.pdf. DOI: https://doi.org/10.59576/sr.1080.

Integrated refinement: ECON-083

Multinational firms and cross-border shock transmission

This refinement adopts the additional source conventions (audited:conventions), subject to its local restrictions. Ownership-mediated multinational transmission. Distinguish multinational ownership links $G^O$ from supplier links $G^S$. For a parent-country shock $\epsilon$ and a fixed host-country firm population, identify host impulse responses \[ \Psi_r^Y(h)=\left.\frac{\partial}{\partial\epsilon} \sum_{j\in r}Y_{j,t+h}(\epsilon)\right|_{\epsilon=0}, \qquad Y\in\{Q,L,B\}, \] for output, employment and financing with their own units. State shock timing, support, normalization and regularity, or use finite shock contrasts. Separate ownership financing and knowledge channels from ordinary trade links, local demand and common shocks; allow domestic-firm spillovers and endogenous entry. Link-specific removal experiments require explicit replacement technologies or financing and a common initial state, as in the existing transmission exercise.

Supplied source audit for ECON-083

Local [S1], [S2], etc. in this audit and the references immediately below belong to ECON-083, independently of the original entry's reference numbering. [S1]'s open-questions chapter and [S2] explicitly support multinational transmission through international networks. The impulse-response and channel definitions are editorial.

References for integrated refinement ECON-083

[S1] Stefania Garetto, Nina Pavcnik, Natalia Ramondo, Vanessa Alviarez, Jingting Fan, Nitya Pandalai-Nayar, Nicola Limodio, Isabela Manelici, Nicolas Morales, Evangelina Dardati, Ezequiel Garcia-Lembergman, Grace Weishi Gu, Galina Hale, David Hémous, Ralf Martin, Farid Farrokhi, Heitor S. Pellegrina, Pierre-Louis Vézina, Laura Boudreau, and Jose P. Vasquez (2025). Foreign Direct Investment and Development. VoxDevLit 13(1), February 2025.

Verified locator / source role: Conclusion/evidence-gap chapter at the linked primary review URL, inspected for the agenda. See the audit conclusion for distinctions between direct agenda support and editorial extensions.

https://voxdev.org/voxdevlit/foreign-direct-investment/fdi-advances-open-questions

[S2] Oliver Hanney / VoxDev (living compilation). Evidence gaps: Key unanswered questions in development economics. Accessed 8 October 2026. The page currently covers 20 topics; the "16-topics URL is a historical slug.

Verified locator / source role: Topic heading: Foreign Direct Investment and Development. Supports the broad research agenda; this is not a verbatim quotation or an attribution of the displayed mathematical target.

https://voxdev.org/topic/evidence-gaps-key-unanswered-questions-across-16-topics-development-economics

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Common conventions: Additional-source status and mathematical conventions

Of the 100 supplied ECON entries, 65 distinct problems appear here; 32 close matches refine earlier entries, and three duplicate questions map to existing entries. Appendix app:audited-crosswalk lists every destination. The attachment's P/R/S/U distinctions and source audits are retained. Its precise mathematical formalizations are editorial unless the individual audit says otherwise. Candidate subsidy targets ECON-004--006 retain their U flags; the resolved approval-core question ECON-007 updates OP-0202 above. The following supplied conventions govern both this part and the attributed ECON refinements elsewhere in the catalogue, subject to local restrictions.

Parameterized questions. A problem family lists inputs that must be fixed before an instance is evaluated: population, horizon, policies, information, data law, model class and welfare weights. These are required choices, not quantities the citations are assumed to specify. Undefined applications should not be treated as identified estimands.

Indices and integrability. Sets of agents, firms and regions are finite unless a population measure is explicitly used. $i,j$ index units, $r$ regions, $t$ dates and $h$ horizons. $\mathbb E$ and $\Pr$ refer to the declared population and law; $\mathbf1\{A\}$ is an indicator. Every displayed expectation and discounted sum needs existence in the stated domain. Infinite horizons use a discount in $(0,1)$ and a convergence condition; finite horizons may use discount one.

Optimization and existence. A displayed $\max$, $\min$, or $\arg\max$ is conditional on attainment. For finite feasible menus this is immediate; general spaces need continuity and compactness/coercivity or another existence argument. Without attainment, the target is the supremum/infimum and arbitrarily near-optimal feasible choices. $\inf\varnothing=+\infty$. Empty feasibility and an unidentified target are different issues.

Potential outcomes. $Y_i(z)$ is the outcome under a specified intervention, not the observed conditional mean among people choosing $z$. In binary comparisons $\Delta Y_i=Y_i(1)-Y_i(0)$. Specify assignment, treatment versions, compliance, information and observation horizon. Interference is allowed under a full policy vector; compressed exposure notation needs a justified mapping. No interference, consistency and overlap are substantive assumptions, not automatic consequences of notation.

Populations and observation. Fix baseline eligibility and population weights. Conditioning on post-policy employment, survival, relocation or program uptake can change the population being compared. Death is not ordinary loss to follow-up; outcomes truncated by death need a composite convention, joint survival target, or explicitly defined survivor stratum. Fertility-changing interventions need a population functional rather than an unexplained fixed descendant cohort. Measurement and missingness models must be supplied.

Identification. A structural model $m\in\mathcal M$ implies observable law $P_m$ and target $\theta(m)$. Identification means \[ P_{m_1}=P_{m_2}\ \Longrightarrow\ \theta(m_1)=\theta(m_2) \quad\text{for all }m_1,m_2\in\mathcal M . \] Without identification, the target is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Writing $\theta(P)$ before establishing identification can hide incompatible latent models. Confidence sets additionally need a sampling law, confidence level, asymptotic sequence and uniformity class. Point identification, coverage and informative precision are separate properties.

Mediation. Total effects of randomized assignment are distinct from cross-world natural indirect effects. Belief/effort-channel effects require a meaningful mediator intervention and additional measurement, exclusion or mediation assumptions. Randomization of the initial policy alone does not supply those assumptions. Report bounds or interventional alternatives when the stronger target is unsupported.

Equilibrium. A counterfactual $W(z)$, $p(z)$, or $\mu_t^z$ requires individual optimization, feasibility, government/resource budgets, market clearing and state-transition laws. State equilibrium existence and selection, or report ranges over compatible equilibria. Initial-state integration includes subsequent endogenous transitions; current-state integration must use the policy-dependent distribution. Reduced-form invariance after policy is not assumed.

Welfare accounts. Scores, utility, health, dollars and ecological quantities require explicit conversion before addition. Fees, taxes, subsidies, wages and extortion payments are transfers between parties; distributional weights can make them welfare relevant, but their face value is not automatically a resource loss. State ownership, geographic boundaries, foreign-income weights and fiscal finance. Do not add profits to household welfare already including dividends, or count land capitalization and the underlying benefit twice. A benefit-cost expression must distinguish gross benefits from net welfare and subtract every real cost exactly once.

Derivatives and risk. Log derivatives require positive arguments; local responses require admissible perturbations and specified equilibrium branches. Differentiation under expectations needs regularity/domination, and boundaries may allow only one-sided derivatives. Risk uses a specified law; ambiguity uses a declared nonempty law set on a common history space. Adaptive policies must be nonanticipative. A robust criterion is an editorial choice unless attributed explicitly.

Scope overlaps. Allocation, collective choice, contracts, household bargaining and coordinated policy overlap economics with optimization and game theory. They are retained because they appeared in the original catalogue; no claim of a strictly game-theory-free selection is made.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-338-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-338

International business-cycle transmission: a research attempt

Status: unresolved. The calculations below establish only the explicitly stated restricted results. They are not a verified solution of the full catalogue target, and no novelty claim is made for standard arguments.

Target and a common-shock linear benchmark

The target compares output-change correlations under an observed international input network and a domestic replacement network, with identical primitive shock laws, explicit substitute technologies and equilibrium adjustment. A further refinement separates multinational ownership from supplier transmission. Neither correlation nor an input matrix alone provides these counterfactuals.

Let a specified linearized equilibrium imply country output changes $U^v=L_v\epsilon$, where $v=0,1$ denotes the replacement and observed networks, and $\epsilon$ has mean zero and covariance $\Sigma$. Country aggregation and horizon responses are already included in $L_v$. Then \[ C_v=L_v\Sigma L_v^{\mathsf T},\qquad \Delta_{cd}=\frac{(C_1)_{cd}}{\sqrt{(C_1)_{cc}(C_1)_{dd}}} -\frac{(C_0)_{cd}}{\sqrt{(C_0)_{cc}(C_0)_{dd}}}, \] assuming positive diagonal terms. This is a conditional answer: identifying $C_1$ is insufficient to identify $L_0$. A domestic substitute must determine the counterfactual technology and hence $L_0$, rather than merely deleting observed expenditure shares.

Propagation versus correlated innovations: exact equivalence

Consider the two-country response system \[ U_1=aU_2+\epsilon_1,\qquad U_2=bU_1+\epsilon_2, \quad 0\leq a,b<1. \] Thus $L=(1-ab)^{-1}\begin{pmatrix}1&a\\b&1\end{pmatrix}$. For independent unit-variance shocks, \[ \operatorname{Corr}(U_1,U_2) =\frac{a+b}{\sqrt{(1+a^2)(1+b^2)}}. \] In the declared replacement response system set $a=b=0$ with the same innovation law. This yields zero correlation and the displayed correlation is its network contribution. The replacement is a reduced-form response assumption; in an economic application its domestic technology must separately justify it.

Now fix an observed Gaussian covariance $C=\begin{pmatrix}1&r\\r&1\end{pmatrix}$, with $0<r<1$. Model A has $a=b=0$ and primitive covariance $\Sigma_A=C$, so all observed synchronization comes from common shocks and removing links changes nothing. Model B sets \[ a=b=\frac{1-\sqrt{1-r^2}}r\in(0,1),\qquad \Sigma_B=(I-A)C(I-A)^{\mathsf T}, \quad A=\begin{pmatrix}0&a\\a&0\end{pmatrix}. \] The off-diagonal of $\Sigma_B$ is $r(1+a^2)-2a=0$. Its diagonal is positive because $C$ is positive definite and $I-A$ invertible. Thus Model B has independent Gaussian primitive innovations and produces exactly the same observed joint Gaussian law as Model A. Under its link-removal response it has network contribution $r$, versus zero in Model A.

The observed coefficient of correlation therefore cannot separate propagation from common shocks even with unlimited Gaussian data. If input coefficients themselves are observed and structurally pinned to $A$, this particular equivalence can be excluded, but counterfactual substitute technology and shock identification still need restrictions. The example is a warning about a reduced-form identification claim, not a demonstration that any fully observed structural network is unidentified.

What externally identified shocks would recover

Suppose $K$ instruments $Z$ satisfy $\mathbb E[\epsilon Z^{\mathsf T}]=B$, where the incidence and scale matrix $B$ is known and has full row rank. Then \[ \mathbb E[UZ^{\mathsf T}]=LB, \qquad L=\mathbb E[UZ^{\mathsf T}]B^{\mathsf T}(BB^{\mathsf T})^{-1}. \] If $L$ is square and invertible, $\Sigma=L^{-1}\operatorname{Var}(U)L^{-\mathsf T}$ is identified as well. The proof is matrix multiplication and the full-row-rank right inverse. If instruments only isolate one geographic shock, only the corresponding response column is recovered; filling the other columns with zeros is not justified. Instrument relevance without known scale identifies impulse responses only under a normalization.

To compute $L_0$ one additionally needs a known structural map $L(\Omega,\chi)$, where $\chi$ includes domestic replacement costs, finance and policy behavior. Identification of $L_1$ need not identify $\chi$: many primitive technologies can produce the same local response at one network while disagreeing after a large replacement. Held-out disruptions that change relevant links are useful precisely because they test this transport step.

Correlation normalization can reverse intuitive comparisons

For a differentiable network parameter $a$, write $v_c=C_{cc}$, $v_d=C_{dd}$, $k=C_{cd}$ and $r=k/\sqrt{v_cv_d}$. Then \[ dr=\frac{dk}{\sqrt{v_cv_d}} -\frac r2\left(\frac{dv_c}{v_c}+\frac{dv_d}{v_d}\right). \] A propagation mechanism can increase covariance while reducing correlation if it raises one country's variance sufficiently. This is why impulse responses and correlations answer different questions. Uniform continuity and informative confidence bounds also require variances bounded away from zero; the canonical strict-positivity condition alone supplies no uniform numerical stability near zero.

In the two-country independent-shock example with unequal variances, covariance is proportional to $b\sigma_1^2+a\sigma_2^2$. Reporting a larger observed covariance without holding these shock variances fixed cannot attribute a larger propagation coefficient. The correlation formula removes a scale but does not solve this composition problem.

Ownership and supplier channels on one path

Let $V_{os}$ be a scalar target with ownership links at setting $o\in\{0,1\}$ and supplier links at $s\in\{0,1\}$, all under specified replacements and the same primitive shocks. An order-averaged attribution is \[ \Phi_O=\tfrac12\{V_{10}-V_{00}+V_{11}-V_{01}\},\qquad \Phi_S=\tfrac12\{V_{01}-V_{00}+V_{11}-V_{10}\}. \] Their sum equals $V_{11}-V_{00}$ by cancellation, so the interaction is allocated equally. This is an accounting convention, not an identification theorem. Each of the four counterfactuals requires its own feasible financing and production replacement. For host output, employment and financing responses, apply it separately in the relevant units rather than add unlike quantities. Entry-induced changes require a fixed baseline firm population or an explicitly expanded population functional.

Completion Estimate

39%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt derives exact linear covariance comparisons, propagation-common-shock observational equivalence, instrument-rank identification, and order-averaged ownership-supplier accounting. It does not identify the domestic replacement equilibrium or full trade and multinational transmission channels with feasible technology and financing substitutes and endogenous policy responses.

Assessment and checked sources

The exact covariance formulas, observational equivalence and instrument rank condition are established. Recovering the intervention-specific replacement equilibrium and identifying ownership versus supplier channels in the full model remain unresolved.

Julian di Giovanni, Şebnem Kalemli-Özcan, Alvaro Silva and Muhammed A. Yıldırım, Pandemic-Era Inflation Drivers and Global Spillovers, New York Fed Staff Report 1080, current linked PDF abstract and introduction inspected; its inflation model is background, not a theorem about the present correlation target: https://www.newyorkfed.org/medialibrary/media/research/staff_reports/sr1080.pdf. Foreign Direct Investment and Development, VoxDevLit, research-agenda chapter inspected for multinational transmission questions: https://voxdev.org/voxdevlit/foreign-direct-investment/fdi-advances-open-questions.

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