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### Sampling strategy and estimation of wedges
- Sample design:
  - Sample all cross-border bank-firm relationships plus at least two domestic relationships for firms borrowing cross-border; within each country, sample a set percentage of domestic bank-firm relationships (example: over 30% of relationships in the whole data are French domestic lending relationships).
- Computational constraints:
  - With 3000 banks and about 6 million firms the full choice set would produce 18 billion observations, making high-dimensional fixed-effect estimation infeasible.
- Solution: stratified sampling with inverse probability weights (Horvitz and Thompson, 1952; Manski and Lerman, 1977):
  - “Certainty stratum” for intensive margin: dyad sampling probability = 1 for existing relationships.
  - Extensive margin: sample “zeros” within firm by randomly sampling banks, targeting approx. 50 observations per firm-time period after combining existing and non-existing relationships and stratifying by country.
  - Dyad sampling probability for never-realized relationships = number of sampled “zeros” divided by total number of banks within country from which zeros were sampled.
  - Final sampling weights = firm sampling probability (intensive margin) × dyad sampling probability (extensive margin).
  - All estimation uses inverse sampling probability weights to recover population parameters in M-estimation methods (Pfeffermann, 1993; Wooldridge, 1999; Solon et al., 2015).

### Results: relationship formation (extensive margin)
- Aggregate cross-border vs domestic relationship likelihood:
  - Cross-border relationships are 3 log points (or, 95%) less likely than domestic relationships (Table 2, Column 2).
  - In levels, the gap is 70 percentage points (pp) lower probability of cross-border relationships (Table 2, Column 4).
- Fixed effects and compositional selection:
  - Firm and bank fixed effects do not materially change wedges relative to a specification with only bank country and firm country fixed effects — interpretation: relationship wedge not primarily driven by compositional selection (larger banks and larger firms unconditionally more likely to form cross-border relationships; Table 7).
- Heterogeneity:
  - Substantial dispersion across country pairs: standard deviation of 24 pp for an unweighted mean of 65 pp; wedges overwhelmingly negative across all country pairs (Appendix Figure 9).
- Selected table summary (Table 2):
  - (1) (PPML): Cross-Border -3.011*** (0.005); N106,584,064; Banks2,673; Firms574,412.
  - (2) (PPML with bank and firm fixed effects): Cross-Border -2.993*** (0.004); N106,584,064; Banks2,673; Firms574,412.
  - (3) (OLS): Cross-Border -0.789*** (0.000); N106,584,064; Banks2,673; Firms574,412.
  - (4) (OLS with more fixed effects): Cross-Border -0.692*** (0.000); N106,584,064; Banks2,673; Firms574,412.
  - Notes: Dependent variable = 1 if firm and bank have outstanding credit; inverse probability sampling weights used; clustered standard errors at bank, firm, and country-pair.

### Results: loan pricing and interest-rate wedges (intensive margin)
- Main finding:
  - Pricing wedges are small relative to relationship wedges.
  - Preferred specification (bank and firm fixed effects, Table 3 Column 2) finds cross-border relationships within the EA feature on average 28 basis points lower interest rates (8.5% of the sample mean) relative to domestic relationships.
- Composition vs within-entity variation:
  - Specifications with only origin and destination fixed effects find cross-border loans even cheaper, underscoring importance of within-bank and within-firm variation to isolate bilateral frictions from compositional differences.
- Heterogeneity:
  - Sizable heterogeneity in pricing frictions across pairs (Appendix Figure 12); wedges take small positive or negative values relative to domestic borrowers, contrasting with uniformly negative relationship wedges.
- Selected Table 3 coefficients and sample details:
  - (1) -0.0083*** (0.0018); N13,303,423; Banks3,000; Firms683,717; Mean Interest Rate0.033.
  - (2) -0.0028*** (0.0007); N13,270,167; Banks2,972; Firms650,482; Mean Interest Rate0.033.
  - (3) -0.0027*** (0.0007); N13,215,479; Banks2,799; Firms646,975; Mean Interest Rate0.033.
  - (4) -0.0026*** (0.0007); N13,268,649; Banks2,968; Firms650,406; Mean Interest Rate0.033.
  - (5) -0.0028*** (0.0007); N10,797,038; Banks2,568; Firms505,965; Mean Interest Rate0.033.
  - Notes: Dependent variable = (log) gross interest rate; Time is a quarter; ILST FE = industry x location x size x time fixed effects; firm-bank size bins = interaction of firm and bank size deciles.

### Results: loan size and intensive margin
- Cross-border loan shares (conditional on existing relationship):
  - Cross-border loan shares are 14% lower than domestic shares (Appendix Table 8, Column 2).
  - Interpretation: firms borrow less from foreign banks even conditional on having a relationship, but gap is small relative to relationship formation frictions.
  - Controlling for within-country bank and firm heterogeneity substantially reduces estimated cross-border distortion.
  - Heterogeneity across pairs present; wedges tilt toward lower cross-border credit across most country pairs (Appendix Figure 12).
- Table 8 summary (selected):
  - Example coefficient: Cross-Border = -0.3193*** (column 1).
  - Samples: N = 13,303,423; 13,270,167; 13,215,479; 13,268,649; 10,797,038 across specifications.

### Risks to identification and robustness checks
- Potential biases and addressed concerns:
  - Omitted variable bias from latent interest rates (Heckman, 1979): unobserved offered-but-rejected interest rates in AnaCredit could bias relationship wedges; bank and firm fixed effects may absorb relevant variation if interest costs reflect bank-specific or firm-specific factors.
  - Interest rate wedges conditional on observed relationships do not capture differences for latent (never-formed) relationships; authors view this as not a major limitation because wedges measure frictions conditional on relationship formation.
  - Residual bank-firm observables could differentially affect cross-border vs domestic credit; relationship-margin concerns subdued given wedges' stability to granular fixed effects.
  - Intensive-margin mitigation: controls for time-varying shocks (bank by time and industry by location by size by time fixed effects), bank-industry specialization (bank by firm sector fixed effects), and size-based assortative matching (bank size decile by firm size decile bins).
- Note: including time-varying fixed effects would cause loss of many observations due to connectedness issues in bipartite settings (Abowd et al., 2002); estimation faced large computational challenges.

### Bank entry wedges: approach and findings
- Econometric framework:
  - Entry indicator Y_git ∈ {0,1} = 1 if group g has entered country i by time t; specification includes origin and destination fixed effects, time FE, cross-border wedges θ^E(o_g,i) interacted with 1{i ≠ o_g}, and controls X^E_gi.
- Identification challenge and solutions:
  - Entry depends on profits in foreign markets, observed only where group operates; approaches:
    - Assume counterfactual profits multiplicatively separable into group push and destination pull factors (absorbed by fixed effects) plus residual uncorrelated with outcomes conditional on controls.
    - Use Section 5 model structure to construct realized profits (where groups operate) and counterfactual profits (where absent).
- Results:
  - Robust negative entry gap of 99 percentage points across specifications (interpreted as average implicit tax on foreign relative to domestic banking).
  - Sizable barriers to cross-border bank entry; bilateral heterogeneity exists but entry barriers large throughout (Appendix Figure 9).
- Table 4 reported estimates:
  - Cross-Border coefficients:
    - (1) -0.988*** (0.005); N748,014.
    - (2) -0.990*** (0.005); N748,014.
    - (3) -0.990*** (0.005); N748,014.
    - (4) -0.988*** (0.005); N748,014.
    - (5) -0.991*** (0.005); N748,014.
  - Notes: Dependent variable = entry indicator; standard errors clustered at parent-country, entry-country, country-pair-level.

### The role of regulatory fragmentation and data sources on banking regulations
- Claims and scope:
  - Estimated cross-border barriers strongly related to policy barriers even after accounting for non-policy determinants; results are descriptive (not claimed causal).
  - Authors build a new dataset of banking policy distances across the euro area because measures of policy distances are not broadly available.
- Data collected and harmonized (high-level):
  - Macroprudential regulation:
    - Governance: CRD IV; buffers include capital conservation buffer, countercyclical capital buffer, systemic risk buffer; borrower-based measures (BoBMs) such as loan-to-value or debt-to-income constraints for specific lending.
    - Source: tabular and textual ESRB data on all measures since 2014; extract statutory buffer values and track exemptions, recommended vs required levels, transitional periods, asset-specific qualifications.
    - Result: yearly country panel; additional yearly bank-level data on realized values of individual buffers for systemic institutions.
  - Microprudential regulation:
    - Governance: CRD IV and CRR IV (liquidity requirements, risk-weight computations, provisioning rules); national deviations via Options & Discretions (O&Ds) reported to EBA.
    - Source: data on take-up of each O&D from EBA disclosures for 2020.

### Data collection, scope, and measuring regulatory fragmentation
- Additional data features:
  - Liquidity and prudential waivers: data available for 2022 and 2023.
  - Deposit Guarantee Systems (DGS): data from EBA, IADI, World Bank BRSS; BRSS extends only through end-2016, supplemented with IADI and manual recoding.
  - Supervision and resolution: granular country-level data from BRSS (end-2016); BRSS treats euro area separately to capture SSM and SRM.
  - Entry, governance, accounting: BRSS data on bank entry policies, accounting, auditing, governance.
  - Corporate bankruptcy laws: OECD data for 2010, 2016, and 2022.
- Measuring fragmentation:
  - Gower (1971) distance metrics applied to regulatory variable vector r_c; continuous variable distance = |r_k^c − r_k^{c′}| / (max_i r_k^i − min_i r_k^i); binary/categorical distance = 1 if differ, 0 if identical.
  - Computed bilateral distances for over 500 variables, averaged to form aggregate distance index bounded between 0 and 1 (interpretable as share of policies that differ).
  - Aggregate statistics: average distance across country pairs in the euro area is 0.24 excluding the diagonal, with standard deviation 0.06.
  - Policy groupings: eight categories — corporate bankruptcy, DGS, entry, governance, macroprudential, microprudential, resolution, supervision.

### Validation of wedges and empirical specification
- Regression framework:
  - τ_{k,ij} = a_i + a_j + β × Policy Distances_{ij} + γ × Non-Policy Distances_{ij} + υ_{ij}, where τ_k_ij are cross-border wedges (k = L, R, E).
- Controls and estimation:
  - Origin and destination fixed effects; non-policy controls from Pellegrino et al. (2025) (geographical, cultural, linguistic distances; shared border, languages, legal origin; statutory bilateral capital tax rates).
  - Relationship wedges estimated from microdata with bank and firm fixed effects via OLS; bank entry wedges estimated with banking group and destination country effects and controlling for model-implied profits via OLS.
- Key empirical findings:
  - Regulatory distances strongly correlate with cross-border relationship wedges (Figure 4, panel a), controlling for origin/destination fixed effects and non-policy barriers.
  - Significant correlations for all regulatory buckets except microprudential and bank entry policies when explaining relationship wedges.
  - Largest correlations with relationship wedges: macroprudential policies, deposit insurance (DGS), bankruptcy fragmentation.
  - Regulatory distances also strongly correlate with cross-border bank entry wedges (Figure 4, panel b); largest correlations for deposit insurance schemes, microprudential policies, macroprudential policies, and national resolution policies.
  - Intensive-margin frictions (loan pricing or loan quantities conditional on existing relationships) show little evidence of correlation with policy distances (Figure 13); harmonization gains likely come primarily via extensive-margin channels (spurring cross-border bank entry and relationship formation).
- Robustness:
  - LPM-based estimates exploit zero-entry bilateral environments; checks include excluding zero-entry pairs and using PPML on non-separated subsets.
  - Conditional on some cross-border bank entry, PPML wedge estimates significantly correlated with regulatory distances in deposit insurance schemes and microprudential policies at the 5% level, and macroprudential policies at the 0.5% level.

### Modeling approach and calibration (transition to quantitative assessment)
- Model environment and agents:
  - Spatial general equilibrium model of the euro area with heterogeneous banks and firms; agents: households H, firms F, banks B across countries i ∈ I; banks belong to groups g ∈ G with origin o_g.
  - Timing: groups decide where to establish branches; banks set gross deposit rates R^D_b = 1 + r^D_b and gross loan rates R^L_bf.
- Key structural mappings to empirical wedges:
  - Relationship shifter: A^L_bf = (1 − τ^L(i_f, i_b)) ˜A^L_bf, where τ^L(i_f, i_b) is cross-border relationship wedge estimated in Section 3.
  - Interest-rate shifter: A^R_bf = (1 + τ^R(i_f, i_b)) ˜A^R_bf, where τ^R(i_f, i_b) is interest-rate cross-border wedge.
  - Branching wedge in entry cost: κ^E_gi = τ^E(i, o_g) + ν_i + μ_g + ε^E_gi.
- Calibration strategy (Section 6.1):
  - Externally calibrated parameters:
    - κ, λ = (0.22, 0.64)
    - δ = 0.05
    - R^* = 1.02
    - σ_L = 1
    - σ_D = 1
    - ψ_1 = 2.5
    - α = 0.04
    - χ_1 = 0.01
  - Internal calibration:
    - Firm fundamentals (z_f, G_f) recovered from AnaCredit micro data via inversion; firms binned into F_i(n_z, n_G).
    - Bank–firm shifters A^L_bF and A^R_bF recovered by inverting loan-share and pricing equations.
    - Deposit shifters A^D_b recovered from observed deposit shares with σ_D = 1.
    - ψ_0 calibrated so aggregate labor supply equals aggregate labor demand; ln χ_0 set so R_W = R^* in baseline steady state.
    - Branching wedges τ^E estimated with linear probability model: τ^E_io_g = X′_{o(g)i} γ where X includes regulatory distance, geographic distance, language similarity, and other bilateral frictions; μ_g and ν_i capture group and destination heterogeneity.

### Counterfactual analysis: fixed bank network (Section 6.2)
- Policy experiment:
  - 10% reduction in cross-border relationship wedges τ^L across all country pairs (equivalent to 10% increase in A^L_bf for cross-border firm-bank pairs), holding banking network fixed.
- Aggregate effects (exact reported outcomes):
  - 10% reduction in τ^L → 1.6% increase in euro area GDP (∆ log(*100) = +1.61).
  - Capital K increases by 2.36% (+2.36).
  - Labor L increases by 1.61% (+1.61).
  - TFP increases by +0.06.
  - Variance of output per hours worked Var(Y/N) falls by −2.71 (percent change).
- Contribution decomposition of GDP gain:
  - Capital: 32.26 percent of the aggregate gain.
  - Labor: 64.03 percent of the aggregate gain.
  - TFP: 3.8 percent of the aggregate gain.
  - Total = 100 percent.
- Financial channel outcomes:
  - Foreign share of bank credit in euro area rises by 1.2 percentage points (euro-area average increase of 1.24 percentage points).
- Mechanism summary:
  - Lower cross-border frictions broaden access to foreign intermediaries → decrease in firms’ effective user costs u_f → scale-up of capital and labor due to complementarities; gains overwhelmingly input-driven rather than allocative-efficiency-driven.

### Heterogeneity, convergence, and distributional implications
- Country heterogeneity:
  - Output gains uneven across countries: Malta, Slovenia, Estonia gain more than average; Netherlands, Luxembourg, Ireland, Belgium show large gains; Spain, Greece, Portugal, Italy gain the least.
  - Ireland and Luxembourg reported as showing largest gains about 8–12% in output (omitted from Figure 5 visualization but reported in text).
  - Smaller economies receive larger increases in cross-border flows; increase in foreign borrowing share correlated with country size and initial exposure to foreign bank credit.
- TFP and allocation:
  - TFP contribution small (3.8% of GDP gain); near-efficient allocation; reforms generate modest reallocation.
  - No clear convergence in marginal products of capital across countries; countries with higher initial user costs do not experience larger declines in average cost of capital.
  - Within-country firm-level dispersion of log user costs increases in almost all countries: firms with already better access to finance benefit most.
- Inequality:
  - Variance of output per hours worked falls by −2.71 percent indicating overall decrease in inequality across countries.

### Counterfactuals with endogenous bank network (Section 6.3)
- Imputation for unobserved counterfactual branches:
  - For group g absent in destination i, impute shifters by geometric average:
    - A^X_{gi,F_j} = [A^X_{g,o_g,F_j} × A^X_{i,F_j}]^{1/2}, X ∈ {L, R}.
    - A^X_{i,F_j} = exp( (1/|B_i|) Σ_{b∈B_i} log A^X_{b,F_j} ).
- Cannibalization test:
  - Compare baseline group profits π_g(I_g) to profits when group opens bank in every missing euro-area country (upper bound on cannibalization).
  - Finding: within-group cannibalization negligible; changes in profits of pre-existing branches very small, supporting independence assumption in entry model.

### Main quantitative findings, interpretation, and policy implications
- Dominant empirical frictions:
  - Barriers to cross-border bank entry and cross-border lending relationships (extensive margin) dominate distortions in interest rates or loan quantities conditional on existing relationships (intensive margin).
- Association with regulatory fragmentation:
  - These extensive-margin barriers are strongly associated with regulatory fragmentation across national frameworks (evidence from policy-distance dataset).
- General equilibrium gains:
  - Partially lifting cross-border barriers yields meaningful but heterogeneous aggregate benefits; majority of gains from increased capital and labor via broader access to foreign finance; limited gains from improved allocation or intensified competition.
- Policy implication emphasized by model:
  - Reducing regulatory barriers and advancing European Banking Union could yield sizeable aggregate benefits, but reforms may not automatically equalize firm-level financing conditions or induce convergence across countries.

### Data appendix and construction highlights
- AnaCredit:
  - Launched September 2018; loan-level reporting threshold 25,000 EUR; each loan record contains ~100 data attributes; cleaning rules applied (e.g., no interest rates < -10% or > 30%).
- RIAD and group-structure algorithm:
  - Uses ownership and branch relations to map branches to ultimate parents and enable nationality-based accounting (remapping increases measured cross-border share from 5% to 14%).
- iBSI:
  - Launched 2014; monthly bank-level balance-sheet indicators covering roughly 95% of euro area banking sector assets; includes IMIR since 2014.
- Regulatory dataset construction:
  - Sources: OECD Insolvency Indicators, ESRB (capital buffers, BoBMs), EBA (O&Ds, liquidity waivers, DGS), WBRSS, IADI.
  - Distance metrics: over 500 variables grouped into eight categories; Table 11 counts: Bankruptcy 13; Deposit Insurance 94; Entry 30; Governance 83; Macroprudential 46; Microprudential 167; Resolution 46; Supervision 53.
- Sampling and estimation details (Appendix A highlights):
  - Relationship regressions use case-control sampling with PPML; firm sampling p_f = 1% in one design; inverse-probability weights specified (equation (42)).
  - Two-way fixed-effects estimator structure and identification via tetrads and connected bipartite network; country-pair coefficients identified up to additive origin and destination shifts.
- Selected descriptive statistics (Table 7 snapshot as of September 2024):
  - Banking Group Size (cross-border subset): N = 897; Mean = 3.40; p50 = 9.000; p90 = 17.000; p99 = 25.000.
  - Total Credit in AnaCredit (cross-border mean): 2.48 (units unspecified in table context).
  - Total Assets (billions) cross-border subset: N = 831; Mean = 78.66; p50 = 167.204; p90 = 675.439; p99 = 1260.637.
  - Firm Total Credit (millions) cross-border: N = 24,628; Mean = 14.07; p50 = 24.986; p90 = 54.884; p99 = 230.960.
  - Number of Bank Relationships (cross-border): N = 24,628; Mean = 2.24; p50 = 5.000; p90 = 6.000; p99 = 10.000.
- Limitations and data coverage caveats:
  - Some country-specific policies over time not tracked beyond prudential requirements and DGS; potential measurement error for recent periods.
  - Microprudential, entry, and auditing requirements considered well-measured; accounting and governance measures may induce overstatement of fragmentation in places due to timing of directives (e.g., IFRS9, 2014 Auditing Directive).
  - Basel III transposition largely synchronized at EU level but divergences can occur via macroprudential buffers, capital phase-ins, and Pillar II discretion.

*Source: https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026123-source-pdf.pdf*

### Appendix A provides further details.

### 2024. We then classify each O&D as a macro or microprudential policy lever.

### 2024. We then classify each O&D as a macro or microprudential policy lever.

### Data collection and scope
- Liquidity and prudential waivers: data available for 2022 and 2023; waivers allow regulatory requirements to be assessed only at the consolidated level and can circumvent solo-level reporting requirements.
- Deposit Guarantee Systems (DGS): data drawn from the EBA, the International Association of Deposit Insurers (IADI), and the World Bank Bank Regulation and Supervision Survey (BRSS); BRSS extends only through end-2016, supplemented with IADI survey data and manual recoding against national legislation to capture changes since 2016.
- Supervision and resolution: granular country-level data from the most recent BRSS wave covering end-2016; BRSS treats the euro area as a separate responding entity to capture the SSM and SRM (Single Supervisory Mechanism and Single Resolution Mechanism).
- Entry, governance, and accounting: BRSS data on bank entry policies, accounting, auditing, and governance (including board composition and compensation, and whether financial statements must be prepared according to international standards).
- Corporate bankruptcy laws and practices: OECD data for 2010, 2016, and 2022 on bankruptcy laws and outcomes.

### Measuring regulatory fragmentation
- Method: Gower (1971) distance metrics applied to a vector of regulatory variables r_c = (r_1^c, ···, r_k^c) across countries; bilateral distance d(r_c, r_c′) = Σ_k d(r_k^c, r_k^c′) where for continuous variables d(r_k^c, r_k^c′) = |r_k^c − r_k^c′| / (max_i r_k^i − min_i r_k^i); for binary/categorical variables distance = 1 if policies differ, 0 if identical.
- Implementation: computed bilateral distances for over 500 variables, then averaged across variables to form an aggregate distance index bounded between 0 and 1 (interpretable as the share of policies that differ between two countries, exact for discrete variables, approximate when continuous variables are included).
- Aggregate statistics: the average distance across country pairs in the euro area is 0.24 excluding the diagonal, with a standard deviation of 0.06.
- Policy groupings: variables grouped into eight categories — corporate bankruptcy, DGS, entry, governance, macroprudential, microprudential, resolution, and supervision.

### Validation of wedges and empirical specification
- Regression framework:
  - τ_k_ij = a_i + a_j + β × Policy Distances_ij + γ × Non-Policy Distances_ij + υ_ij,
  - where τ_k_ij are cross-border wedges (k = L, R, E), a_i and a_j are origin and destination fixed effects, and υ_ij is a residual.
- Controls: specifications include origin and destination fixed effects and non-policy controls from Pellegrino et al. (2025) (geographical, cultural, linguistic distances; shared border, languages, or legal origin indicators; statutory bilateral capital tax rates).
- Estimation: relationship wedges estimated from microdata with bank and firm fixed effects via OLS; bank entry wedges estimated with banking group and destination country effects and controlling for model-implied profits via OLS.

### Key empirical findings
- Regulatory distances strongly correlate with cross-border relationship wedges (Figure 4, panel a), controlling for origin/destination fixed effects and non-policy barriers.
- Significant correlations found for all regulatory buckets except microprudential and bank entry policies when explaining relationship wedges.
- The largest correlations with relationship wedges are for:
  - macroprudential policies,
  - deposit insurance (DGS),
  - bankruptcy fragmentation.
- Regulatory distances also strongly correlate with cross-border bank entry wedges (Figure 4, panel b).
  - Largest correlations for bank entry wedges are differences in:
    - deposit insurance schemes,
    - microprudential policies,
    - macroprudential policies,
    - national resolution policies.
  - Corporate bankruptcy, domestic supervision practices, and bank governance rules also correlate negatively with entry wedges but are smaller and less statistically significant.
- Intensive-margin frictions (loan pricing or loan quantities conditional on existing relationships) show little evidence of correlation with policy distances (Figure 13); harmonization gains likely come primarily via extensive-margin channels (spurring cross-border bank entry and relationship formation).
- Robustness notes:
  - When correlating branching wedges with bilateral distance measures, LPM-based estimates exploit zero-entry bilateral environments; robustness checks include excluding zero-entry pairs and using PPML on non-separated subsets.
  - Conditional on some cross-border bank entry, PPML wedge estimates are significantly correlated with regulatory distances in deposit insurance schemes and microprudential policies at the 5% level, and macroprudential policies at the 0.5% level.

### Modeling approach (transition to quantitative assessment)
- The next step described is building a quantitative spatial general equilibrium model of the euro area with heterogeneous banks and firms to quantify macroeconomic gains from lifting regulatory barriers.
- The model incorporates frictions to bank entry, loan demand, and interest rate setting that map directly to the empirically estimated wedges.

*Source: wpiea2026123-source-pdf - 2024.*

### Section 3.  We then use this model to run counterfactuals that quantify the effect

### wpiea2026123-source-pdf - Section 3.  We then use this model to run counterfactuals that quantify the effect

### Environment and agents
- Economy populated by households H, firms F, and banks B across countries i ∈ I; households h ∈ H_i, firms f ∈ F_i, banks b ∈ B_i.
- Banks belong to groups g ∈ G; B = ∪_{g∈G} B_g and groups have origin countries o_g.
- Timing: groups decide where to establish branches; banks set gross deposit rates R^D_b = 1 + r^D_b and gross loan rates R^L_bf, taking others as given.

### Households and deposits
- Preferences: GHH preferences with unitary risk aversion; labor supply n_h from intratemporal condition n_h = (w/ψ_0)^{1/ψ_1}, where ψ_1 > 0 and ψ_0 > 0.
- Household budget constraint: c_h = (1 + r^D_b) d_h + w n_h + T.
- Households choose bank b to maximize r^D_b × ω^D_bh; ω^D_bh i.i.d. Extreme Value type II with location A^D_bi and scale σ_D > 0.
- Deposit choice probability:
  - p^D_bi = A^D_bi (r^D_b)^{σ_D} / Σ_{b'} A^D_{b'i} (r^D_{b'})^{σ_D}.

### Firms and demand for loans
- Production: y = z_f k^κ n^λ, with firm-specific productivity z_f and κ + λ < 1.
- Financing constraint: Σ_b ℓ_bf = q k_f where q is unit cost of capital.
- Firm non-default profit: π^ND_f = y_f − w n_f − Σ_b R^L_bf ℓ_bf + q(1−δ)k_f, with δ ∈ (0,1).
- Default: firm repays only (1−α) of loan if default; idiosyncratic default cost ψ_f ∈ {0, ψ̄} with ψ̄ > α; firm defaults iff α > ψ_f so default probability G_f = G_f(α) (denoted G_f).
- Ex post profit with default: π^ND_f + (α − ψ_f) Σ_b R^L_bf ℓ_bf · 1_{Default=1}.
- Firms partition capital into P projects, each funded by a loan ℓ_bf = ℓ_f / P and face bank-project-specific service shifters ω^L_bfp drawn from Fréchet with location A^L_bf and common scale σ_L.
- Bank choice probability for firm f:
  - p^L_bf = A^L_bf (u_bf)^{−σ_L} / Σ_{b'} A^L_{b'f} (u_{b'f})^{−σ_L},
  where u_bf ≡ q [R^L_bf (1 − α G_f) + δ − 1]; composite user cost u_f given by equation (19).

### Cross-border relationship and interest-rate wedges
- Relationship shifter decomposition:
  - A^L_bf = (1 − τ^L(i_f, i_b)) ˜A^L_bf, where τ^L(i_f, i_b) is the cross-border relationship wedge (Section 3 estimated these wedges).
- Interest-rate-related shifter decomposition:
  - A^R_bf = (1 + τ^R(i_f, i_b)) ˜A^R_bf, where τ^R(i_f, i_b) is the interest rate cross-border wedge.
- Interpretation: τ^L and τ^R capture policy and non-policy barriers (regulatory differences, geographic/language/cultural frictions).

### Banks, rates, and branching decisions
- Bank balance sheet: L_b = W_b + D_b where L_b = ∫_f ℓ_bf df, D_b = ∫_h d_h × p^D_bh dh, and W_b is wholesale funding (positive if borrowing, negative if investing).
- Wholesale gross interest rate R_W determined endogenously (large open economy) by ln R_W = ln R^* + χ_1 ln(1 + B) + ln χ_0, where B = ∫_b W_b db, R^* is world risk-free rate, and χ_0, χ_1 > 0.
- Banks perfectly price-discriminate ex post; bank-specific loan offer to firm f: R^L_bf = R^L_b × A^R_bf / (1 − α G_f).
- Optimal deposit and lending rates (mark-down/mark-up form):
  - R^L_b = [1 / (1 + ε^{-1}_{L/R^L_b})] R_W,
  - R^D_b = [1 / (1 + ε^{-1}_{D/R^D_b})] R_W,
  where ε_{L/R^L_b} < 0 and ε_{D/R^D_b} > 0 are elasticities of loan demand and deposit supply to bank rates.
- Bank profits expression (with optimal rates):
  - π^B_bi = R_W [ (1 − 1/(1 − ε^{-1}_{L/R^L_b})) L_b + (1 / (1 + ε^{-1}_{D/R^D_b})) D_b ].

### Branching decision and branching wedges
- Group entry cost into destination i: κ^E_gi = τ^E(i, o_g) + ν_i + μ_g + ε^E_gi, where τ^E(i, o_g) is branching wedge, ν_i destination fixed effect, μ_g group fixed effect, and ε^E_gi uniform error.
- Group enters destination i if additional profits exceed κ^E_gi. Under uniform ε^E, entry probability linear in expected profits minus bilateral barriers:
  - P(1_{gi} = 1) = F_ε(π^B_gi − τ^E(i, o_g) − ν_i − μ_g).
- Empirical/structural assumption: entry decisions independent across destinations; supported by calibrated model evidence.

### Market clearing and equilibrium
- Labor market: ∫_f n_f df = ∫_h n_h dh.
- Loan and deposit markets clear via definitions of L_b and D_b; wholesale market clears through R_W function.
- Final goods market clears:
  - ∫_f y_f + (1 − δ) k_f / a_k df = ∫_h c_h dh + ∫_g ∫_i κ^E_gi 1_{gi} dg di.
- Equilibrium: policy rules for households, firms, banks, groups and prices (w, R_W) such that agents optimize and markets clear.

### Calibration strategy (Section 6.1)
- Combines external calibration and internal estimation (invert model to match micro moments).
- Externally calibrated parameters (Table 5 values preserved):
  - κ, λ = (0.22, 0.64)
  - δ = 0.05
  - R^* = 1.02
  - σ_L = 1
  - σ_D = 1
  - ψ_1 = 2.5
  - α = 0.04
  - χ_1 = 0.01
- Interpretation/choices:
  - λ = 0.64 (labor share consistent with EU), κ = 0.21 (implies total returns to scale 0.85).
  - α = 4% matches median estimated loss given default in euro area.
  - χ_1 = 0.01 implies ~10 basis points increase in R_W for 10% of total deposits additional lending (illustrative calculation provided).
- Internal calibration:
  - Firm fundamentals (z_f, G_f) recovered from AnaCredit micro data via inversion of optimal conditions; firms binned by deciles of productivity and default risk into F_i(n_z, n_G).
  - Bank–firm shifters A^L_bF and A^R_bF recovered by inverting loan-share equation (18) and pricing condition (37).
  - Deposit shifters A^D_b recovered from observed deposit shares and σ_D = 1.
  - Labor supply parameter ψ_0 calibrated so aggregate labor supply equals aggregate labor demand: ψ_0 = w × [∫ n_f df]^{−ψ_1}.
  - ln χ_0 set so R_W = R^* in baseline steady state.
  - Branching wedges τ^E estimated with linear probability model: τ^E_io_g = X′_{o(g)i} γ where X includes regulatory distance, geographic distance, language similarity, and other bilateral frictions; μ_g and ν_i capture group and destination heterogeneity.

### Counterfactual analysis with fixed bank network (Section 6.2)
- Policy experiment: 10% reduction in cross-border relationship wedges τ^L across all country pairs (equivalent to 10% increase in A^L_bf for cross-border firm-bank pairs), holding banking network fixed.
- Aggregate effects:
  - 10% reduction in τ^L → 1.6% increase in euro area GDP (∆ log(*100) = +1.61).
  - Capital K increases by 2.36% (+2.36).
  - Labor L increases by 1.61% (+1.61).
  - TFP increases by +0.06 (less than 0.1%).
  - Variance of output per hours worked Var(Y/N) falls by −2.71 (percent change).
- Contribution decomposition of the GDP gain:
  - Capital: 32.26 percent of the aggregate gain.
  - Labor: 64.03 percent of the aggregate gain.
  - TFP: 3.8 percent of the aggregate gain.
  - Total = 100 percent.
- Financial channel outcomes:
  - Foreign share of bank credit in euro area rises by 1.2 percentage points (euro-area average increase of 1.24 percentage points; Figure 5b).
- Mechanism summary:
  - Lower cross-border frictions broaden access to foreign intermediaries → decrease in firms’ effective user costs u_f → scale-up of capital and labor due to complementarities.
  - Gains are overwhelmingly input-driven rather than allocative-efficiency-driven.

### Heterogeneity and (lack of) convergence
- Country heterogeneity:
  - Output gains uneven across countries; Malta, Slovenia, Estonia gain more than average; Netherlands, Luxembourg, Ireland, Belgium show large gains; Spain, Greece, Portugal, Italy gain the least.
  - Ireland and Luxembourg omitted from Figure 5 visualization but reported as showing largest gains about 8–12% in output.
  - Increase in foreign borrowing share correlated with country size and initial exposure to foreign bank credit; smaller economies receive larger increases in cross-border flows.
- TFP and allocation:
  - TFP contribution small (3.8% of GDP gain). Near-efficient allocation, output gap depends on weighted variance of user costs; reforms generate modest reallocation.
  - No clear convergence in marginal products of capital across countries; countries with higher initial user costs do not experience larger declines in average cost of capital (Appendix Figure D.4a).
  - Within-country firm-level dispersion of log user costs increases in almost all countries (Appendix Figure D.4b): firms with already better access to finance benefit most, implying limited reach of integration to the most constrained firms.
- Inequality:
  - Variance of output per hours worked falls by −2.71 percent indicating overall decrease in inequality across countries.

### Counterfactuals with endogenous bank network (Section 6.3)
- Imputation for unobserved counterfactual branches:
  - For group g absent in destination i, impute shifters for a counterfactual branch by geometric average of group fundamentals in origin o_g and average fundamentals among banks in destination i:
    - A^X_{gi,F_j} = [A^X_{g,o_g,F_j} × A^X_{i,F_j}]^{1/2}, X ∈ {L, R}.
    - A^X_{i,F_j} = exp( (1/|B_i|) Σ_{b∈B_i} log A^X_{b,F_j} ).
- Cannibalization test:
  - Compare baseline group profits π_g(I_g) to profits when group opens bank in every missing euro-area country (upper bound on cannibalization).
  - Finding: within-group cannibalization negligible; changes in profits of pre-existing branches are very small (Appendix Figure D.5), supporting independence assumption in entry model.

### Main quantitative findings and interpretation
- Dominant frictions empirically: barriers to cross-border bank entry and cross-border lending relationships (extensive margin) dominate distortions in interest rates or loan quantities conditional on existing relationships.
- These extensive-margin barriers are strongly associated with regulatory fragmentation across national frameworks (evidence from policy-distance dataset).
- General equilibrium gains from partially lifting cross-border barriers are meaningful but heterogeneous: majority of gains from increased capital and labor via broader access to foreign finance; limited gains from improved allocation or intensified competition.
- Policy implication emphasized by model: reducing regulatory barriers and advancing European Banking Union could yield sizeable aggregate benefits, but reforms may not automatically equalize firm-level financing conditions or induce convergence across countries.

*Source: wpiea2026123-source-pdf - Section 3.  We then use this model to run counterfactuals that quantify the effect*

### References

### References

### Bibliographic sources
- Comprehensive list of citations for the paper, including methodological references (e.g., Abowd et al. (1999, 2002), Angrist and Pischke (2009), Gourieroux et al. (1984)), empirical studies of cross-border banking and capital allocation (e.g., Acharya et al. (2014), Cetorelli and Goldberg (2012), Gopinath et al. (2017), Niepmann (2015, 2023)), datasets and institutional reports (AnaCredit; iBSI; ESRB; EBA; OECD; IADI; World Bank Group), and recent working papers and technical reports (e.g., Beck et al. (2024), Bertaut et al. (2024), Pellegrino et al. (2025)).
- Key econometric and sampling references cited: Horvitz and Thompson (1952); Manski and Lerman (1977); Pfeffermann (1993); Wooldridge (1999); Solon et al. (2015); Correia et al. (2020).

### Empirical Appendix (A) — measurement, sampling, and estimation details
- A.1 Operational geography
  - Presents alternative mappings of cross-border positions: residency-based (geography) vs nationality-based (group-structure).
  - Remapping from geography to nationality increases the cross-border share of lending from 5% to 14%.
  - Two notable patterns from remapping:
    - The diagonal remains overwhelmingly larger than off-diagonal terms, confirming strong domestic concentration.
    - The bilateral matrix becomes sparser for many smaller countries; many banking branches that appeared as cross-border lenders by geography are owned by parent groups domiciled elsewhere, leading many smaller debtor countries to borrow over 50% from foreign banking groups.
  - Larger economies (France, Germany, Spain) remain below the euro area average in cross-border shares; Italy shows cross-border shares around 20% after remapping.

- A.2 Remapping with group structures
  - Uses the group structure described in subsection B to aggregate bilateral positions at the banking-group level (Figure 7 shown in source).
  - Interpretation: nationality-based accounting raises measured integration but does not overturn documented home bias and low cross-border branching.

- A.3 Cross-border entry: branches versus subsidiaries
  - Definitions: branch = part of parent bank; subsidiary = host-country legal entity.
  - Key institutional differences: deposit protection, supervision/resolution, capital and liquidity binding at subsidiary level but not for branches.
  - Empirical patterns (Figure 8):
    - Banking groups from Austria, France, Germany, Luxembourg, and the Netherlands primarily expand through branching.
    - Banking groups from Belgium and Spain expand across branches and subsidiaries evenly.
    - Banking groups from Italy tend to favor subsidiaries.
  - Estimated entry wedge in main text is an average over legal forms of entry.

- A.4 Sampling designs
  - Interest rate regressions (seed sampling and expansion)
    - Step 1: seed sampling of positive dyads. For cross-border country pairs (i_b ≠ i_f) sample the universe of relationships; for domestic pairs (i_b = i_f) draw a 5% simple random sample of dyads from D^+_bc. For some smaller countries, also sample the universe of domestic relationships.
    - Step 2: time expansion — for each sampled dyad (b,f) recover the full time series (deterministic).
    - Step 3: firm-level snowballing — augment sample by including all other banks b′ such that dyad (f,b′) is observed as positive in at least one period.
  - Relationship regressions (case-control sampling with PPML)
    - Step 1: firm sampling. Country-stratified sample of firms with inclusion probability p_f; set p_f = 1%.
    - Step 2: zero subsampling within firm-time risk sets. For each sampled firm-period (f,t), draw enough zeros to reach 50 observations per firm. Let π_bft denote the conditional inclusion probability of zero dyad (b,f,t).
    - Inclusion probability for a retained zero dyad: Pr{(b,f,t) is observed in the estimation sample} = p_f π_bft.
    - Inverse-probability weights (equation (42)):
      - w_ijt = 1/p_f, if Y_bft > 0
      - w_ijt = 1/(p_f π_bft), if Y_bft = 0 and (b,f,t) is sampled.
    - Estimation implemented by weighted PPML on the sampled data; weighted estimator targets the same population conditional mean as full-universe estimator under known sampling probabilities.

- A.5 Estimation details
  - Two-way fixed-effects estimator structure (outcome for bank–firm edge):
    - y_bf = α_b + δ_f + θ_{c(b),c(f)} + u_bf  (equation (43))
  - Identification intuition via a four-edge tetrad:
    - Define foreign-minus-domestic gaps ∆_b for banks and difference of gaps yields (θ_HF − θ_HH) − (θ_FF − θ_FH) + ũ (equations (44)-(45)).
    - Fixed effects cancel in differences; estimator aggregates many local difference-in-differences (tetrads and longer even cycles) across the connected bipartite bank–firm network (references: Abowd et al. (2002); Bonhomme (2020); Kline (2024)).
  - Normalization caveat:
    - Country-pair coefficients θ_cd are identified only up to additive origin- and destination-country shifts; meaningful objects are normalized deviations (e.g., deviations after removing origin and destination means or differences relative to domestic lending) (equation (46)).

- A.6 Additional estimates and figures
  - Relationship wedges (θ_L), quantity wedges (loan shares conditional on relationships), interest rate wedges (θ_R), and bank entry wedges (θ_E) are displayed as debtor–creditor country matrices (Figures 9–12).
  - Table 8: Cross-Border Wedges in Credit Quantities — OLS estimates of log share of borrowing on cross-border indicator with various fixed effects; sample sizes:
    - N = 13,303,423; 13,270,167; 13,215,479; 13,268,649; 10,797,038 across columns (different specifications).
    - Banks: 3,000; 2,972; 2,799; 2,968; 2,568.
    - Firms: 683,717; 650,482; 664,975; 650,406; 505,965.
    - Average Share: 0.438; 0.437; 0.435; 0.437; 0.399.
    - Example coefficient: Cross-Border = -0.3193*** (column 1).
  - Table 9: Cross-Border Wedges in Bank Entry — PPML estimates; sample N = 748,014 across columns. Example coefficient: Cross-Border = -4.635*** (column 1).
  - Table 10: Regulatory Drivers of Cross-Border Frictions — OLS estimates of bilateral wedges on regulatory frictions; sample N = 400; 324; 324; 400 across panels. Example pooled coefficient (Panel A, column 1): Pooled = -1.134*** (standard error (0.153)).
  - Figure 13: Correlations of cross-border barriers with policy distances — coefficient plots for policy variables on equation (4) for quantity wedges and interest rate wedges (policy variables jointly included; origin and destination fixed effects; non-policy controls from Pellegrino et al. (2025)).

### Data Appendix (B) — data sources, construction, and cleaning
- B.1 AnaCredit
  - AnaCredit launched in September 2018; loan-level reporting threshold 25,000 EUR.
  - Each loan record contains approximately 100 data attributes (creditor, debtor, instrument, credit risk measures).
  - Cleaning rules applied to ensure sensible reported quantities (e.g., no interest rates < -10% or > 30%; correction of decimal reporting errors).

- B.2 RIAD and group-structure algorithm
  - RIAD's statistical unit: organisational unit (OU) covering legal entities and affiliates (branches, subsidiaries).
  - Relevant RIAD relationships:
    - Is Owner: ownership link with capital share and control flag.
    - Is Branch: links non-resident branch to its headquarters (treated as 100% control).
    - Is Direct Parent: derived from ownership links with effective control.
    - Is Ultimate Parent: entity with no parent undertaking.
  - Control-based group structure: head is an OU not controlled by any other entity; members are OUs directly/indirectly controlled by the head; unique placement in tree (no cycles).
  - Mapping branches to ultimate parents:
    - Step 1: identify branch–headquarter via Is Branch.
    - Step 2: climb control hierarchy to top entity that is a deposit-taking MFI (S122) and has no parent undertaking; assign UP(b) = p^(k)(h(b)) (formal definition in source).
  - Purpose: reassign branch-level activity to the parent banking group for nationality-based accounting.

- B.3 iBSI
  - iBSI launched in 2014; monthly bank-level balance-sheet indicators covering roughly 95% of euro area banking sector assets.
  - Includes end-of-month outstanding amounts and monthly transactions; since 2014 includes individual MFI interest rate (IMIR) data; sample extended in 2015 and expanded to nearly all European banks since late 2019.

- B.4 Regulatory dataset construction
  - Sources and processing for regulatory fragmentation measures include OECD Insolvency Indicators (2010–2016, 2022), ESRB datasets on capital buffers and borrower-based measures, EBA supervisory disclosure (options, discretions, liquidity waivers, deposit guarantee schemes), World Bank Regulation and Supervision Survey (WBRSS), and IADI survey data.
  - B.4.1 OECD Insolvency Indicators: added flags for Europe and EEA membership.
  - B.4.2 ESRB data:
    - Combined Capital Buffer Requirements: quarter panel covering 31 countries over 2013–2024; when buffer rates reported as ranges, handled as described (examples: Liechtenstein SyRB “1% for exposures secured by real estate property” treated as 0–1%; Norway SyRB 3–5%; Romania midpoints assigned where bank-level ranges reported).
    - Capital Conservation Buffer (CCoB): dummy exemption variable constructed; panel quarterly over 2013–2024.
    - Countercyclical Capital Buffer (CCyB): cleaned to retain effective policy actions; aggregated to country–quarter; final dataset covers 30 countries over 2014–2024 (Iceland had two CCyB changes in 2020Q1).
    - Reciprocation of macroprudential buffers: reshaped from matrix form; merged with ESRB Capital Overview to identify measure status as of December 16, 2025.
    - Borrower-Based Measures: constructed dummy for implementation and dummy for exemptions; legal status variable for recommendations created but not used in analysis.
  - B.4.3 EBA data:
    - Liquidity waivers from Supervisory Disclosure Annex 4 Part 6 Revisions 1 and 2 (survey years 2022 and 2023) — reports number of institutions receiving specified waivers.
    - Options and discretions dataset based on Supervisory Disclosure Annex 2 Part 1 Revision 2 (published 19 November 2024) and Annex 2 Part 1 (26 February 2020). Where Article 493 counterparts exist, split into “a” (national competent authority) and “b” (government) variables. Dataset covers 32 countries over 2 survey periods (2020 and 2024).
    - Deposit Guarantee Schemes (DGS) data aggregated 2015–2024 at country-year level.
    - Note: one data amendment for Poland (index 019) where a percentage value reported as 976 treated as missing.
  - B.5 World Bank Regulation and Supervision Survey (WBRSS)
    - Covers 2011–2016 publicly; extracted variables on bank entry, ownership, capital, governance, liquidity, deposit insurance, asset classification, provisioning; final coverage restricted to EU countries.
  - B.6 International Association of Deposit Insurers (IADI)
    - Survey data 2014–2024 compiled and harmonized; hand-checked for consistency and discrepancies resolved via original documentation and official sources; source notes that artificial intelligence was used to assist verification when discrepancies were detected.

- Data and tables included in appendix (selected descriptive statistics)
  - Table 7: Descriptive statistics for banks and firms in AnaCredit-iBSI-RIAD integrated dataset as of September 2024. Notable entries (bank-level, cross-border relationships subset):
    - Banking Group Size: N = 2,291 (all relationships); cross-border N = 897; Mean (cross-border) = 3.40; p50 = 9.000; p90 = 17.000; p99 = 25.000.
    - Total Credit in AnaCredit: Mean (all) = 1.27; Mean (cross-border) = 2.48; p99 (all) = 21.864; p99 (cross-border) = 41.944.
    - Total Assets (billions) all: N = 2,103; Mean = 43.56; p50 = 47.504; p90 = 168.004; p99 = 1184.911. Cross-border subset: N = 831; Mean = 78.66; p50 = 167.204; p90 = 675.439; p99 = 1260.637.
  - Firm-level descriptive statistics (selected):
    - Total Firm Credit (millions): All relationships N = 3,503,981; Mean = 0.83; p50 = 1.031; p90 = 2.274; p99 = 10.945. Cross-border N = 24,628; Mean = 14.07; p50 = 24.986; p90 = 54.884; p99 = 230.960.
    - Number of Bank Relationships: All relationships N = 3,503,981; Mean = 1.32; p50 = 2.000; p90 = 3.000; p99 = 5.000. Cross-border N = 24,628; Mean = 2.24; p50 = 5.000; p90 = 6.000; p99 = 10.000.
  - Note: All values in the bank liability panel are given in billions of Euros; dataset snapshot dated as of September 2024; cross-border defined based on operational geography.

*Source: wpiea2026123-source-pdf - References*

### references were corrected to align with the relevant article.  For example, in row 44, Article 400(2),

### wpiea2026123-source-pdf - references were corrected to align with the relevant article.  For example, in row 44, Article 400(2),

### Data corrections and availability
- Corrections: references were corrected to align with the relevant article. For example, in row 44, Article 400(2), point (i) and 493(3), point (i) (column D), the narrative should read: “Yes on a case-by-case basis for Article 400(2)(i) subject to fulfillment of certain criteria N for Article 493(3)(i)” instead of the existing narrative within the raw data.
- Nature of corrections: these corrections reflect minor clerical errors in the source file and do not alter the substantive coding of the measures.
- Documentation: All modifications to the raw data are documented, and a complete spreadsheet detailing these changes is available from the authors upon request.

### B.7 Compiling the dataset
- Structure:
  - The cleaned individual datasets are consolidated into a single harmonized file for the computation of distance metrics.
  - Data organization: matrix format, with each row representing a variable–year observation and each column representing an EU member state.
- Variable categorization:
  - Each individual variable is assigned to one of eight categories: supervision, microprudential, macroprudential, resolution, entry, governance, deposit insurance, and bankruptcy.
  - Rationale: seven categories capture key dimensions of banking regulations, requirements, and their implementation; the eighth category, bankruptcy, is included given its importance for credit outcomes.
  - For datasets covering multiple categories, variables were ascribed following consensus (example: for each Option & Discretion in the EBA data, policies were read and assigned to micro vs macroprudential depending on whether the policy related mostly to exemptions to a micro or macroprudential requirement).

### B.8 Potential limitations
- Coverage limits:
  - Beyond prudential requirements and DGS, country-specific policies over time were not tracked. For more recent time periods, some measures of regulatory fragmentation may thus suffer from measurement error.
- Areas with limited measurement concerns:
  - Microprudential, entry, and auditing requirements: considered not to have considerable measurement concerns.
    - Microprudential and entry requirements are stringently dictated by the CRR; deviations occur mainly through the take-up of O&Ds or waivers, both measured.
    - Auditing: the 2014 Auditing Directive had to be completely phased-in by mid-2016.
- Residual concerns:
  - Accounting: measures may overstate fragmentation due to the adoption of IFRS9 standards effective January 2018.
  - Governance: the 2013 CRD IV tasked the EBA with developing internal governance guidelines, released in 2017 but only effective June 2018.
- Basel III transposition and divergences:
  - Basel III iterations have been transposed at the EU level since the CRD IV (e.g., CRR2/CRD5).
  - Using the BIS adoption trackers, adoption of each standard has been verified as synchronized between the EU and the individually tracked countries (Germany, France, Italy, Spain and the Netherlands).
  - Remaining divergences across countries due to Basel III could occur because of macroprudential buffers, capital phase-ins, and Pillar II requirements.
    - The authors actively track national macroprudential buffers over time.
    - Capital phase-ins would induce transitory variation around the CRR-defined anchor until full convergence.
    - Supervisory discretion over Pillar II requirements for non-significant institutions supervised by national competent authorities is a potential source of concern; direction of bias is unclear.

### B.9 Distance metrics by categories — key statistics
- Visualization: Figure 14 reproduces Figure 3, breaking down the distance index into its eight sub-components: (a) Bankruptcy, (b) Deposit Insurance, (c) Entry, (d) Governance, (e) Macroprudential, (f) Microprudential, (g) Resolution, and (h) Supervision.
  - These panels plot bilateral regulatory distance metrics across euro area country pairs for each policy category, constructed from the novel dataset of national regulatory frameworks described in the paper and Appendix D.
- Table 11: Number of Variables, by Category
  - Bankruptcy: 13
  - Deposit Insurance: 94
  - Entry: 30
  - Governance: 83
  - Macroprudential: 46
  - Microprudential: 167
  - Resolution: 46
  - Supervision: 53
- Note: A full list of the names of the variables can be found at the authors’ websites.

### C Selected model proofs — substantive derivations and expressions
- C.1 Proof of Bank Choice Probability: Equation (8)
  - Utility specification for household i choosing bank b:
    - u_bi = σ_D ln(r^D_b) + ln(A^D_bi) + ε_bi, with r^D_b > 0, A^D_bi > 0 ∀ b,i.
  - ε_bi distributed iid standard Type I Extreme Value (Gumbel) with CDF F(ε) = exp(−exp(−ε)) and density f(ε) = exp(−ε) exp(−exp(−ε)).
  - Deterministic utility: V_bi ≡ σ_D ln(r^D_b) + ln(A^D_bi).
  - Resulting choice probability:
    - p^D_bi = A^D_bi (r^D_b)^{σ_D} / Σ_{b′} A^D_b′i (r^D_b′)^{σ_D}.
  - Intermediate expression: exp(V_bi) = (r^D_b)^{σ_D} A^D_bi.

- C.2 Proof of Elasticities: (26) and (27)
  - Expected profits with W_b = (L_b − D_b):
    - π^B_b = [R^L_b − R^W] L_b − [R^D_b − R^W] D_b.
  - First order conditions:
    - (R^L_b − R^W) ∂L_b/∂R^L_b + L_b = 0.
    - (R^D_b − R^W) ∂D_b/∂R^D_b + D_b = 0.
  - Markup and markdown conditions:
    - R^L_b = R^W − R^L_b ε^{-1}_{L/R^L_b}.
    - R^D_b = R^W − R^D_b ε^{-1}_{D/R^D_b}.
  - Rearranged:
    - R^L_b = 1 / (1 + ε^{-1}_{L/R^L_b}) R^W.
    - R^D_b = 1 / (1 + ε^{-1}_{D/R^D_b}) R^W.

- C.3 Proof of Optimal Interest Rate Conditions: Equation (24)
  - Firm-specific loan condition:
    - No-arbitrage: R^L_bf × (1 − α G_f) = R^L_b × A^R_bf.
  - Optimal firm-specific rate:
    - R^L_bf = R^L_b A^R_bf / (1 − α G_f), with α ∈ [0,1], G_f = probability of default, A^R_bf bank-firm wedge.

- C.4 Expressions for Elasticities — detailed derivatives
  - Derivative of loan demand L_b to interest rate R^L_b:
    - ∂L_b/∂R^L = ∫_f [∂p_bf/∂R^L_b ℓ_f + ∂ℓ_f/∂R^L_b p_bf] df.
    - ∂p_bf/∂R^L_b derivation leads to expressions involving σ_L, R^L_bf, (1−α G_f), δ^{-1}, p_bf(1−p_bf), and A^R_bf.
    - ∂ℓ_f/∂R^L_b derivation involves q, λ, κ, k_f, A^L_bf, u_f, u_bf, σ_L, and A^R_bf.
  - Final expression for elasticity of loans:
    - (R^L_b / L_b) ∂L_b/∂R^L_b = −σ_L ∫_f [ℓ_f / L_b R^L_b R^L_bf (1−α G_f) + δ^{-1} p_bf(1−p_bf) A^R_bf] df − q^2 ∫_f [ (1−λ)/(1−λ−κ) k_f u^{σ_L}_f (u_bf)^{−1−σ_L} R^L_b / L_b A^L_bf A^R_bf p_bf ] df.
  - Derivative and elasticity of deposits D_b to deposit rate R^D_b:
    - ∂D_b/∂R^D_b = σ_D 1 / r^D_b ∫_h p^D_bh (1 − p^D_bh) d_h dh.
    - Elasticity:
      - (R^D_b / D_b) ∂D_b/∂R^D_b = σ_D [1 + r^D_b / r^D_b ∫_h d_h R p^D_bh d_h dh / ∫_h p^D_bh (1−p^D_bh) dh ] (expression presented in the source).

### D Additional model results — counterfactuals and figures
- Figures and noted results:
  - Figure D.1: Increase in Productive Capital
    - Shows the counterfactual change in productive capital across euro area countries resulting from the removal of estimated cross-border barriers. Changes are computed using the calibrated structural model.
  - Figure D.2: Increase in Lending
  - Figure D.3: Increase in Borrowing
    - Notes: Figures C.6 and C.7 (referenced as D.2 and D.3 here) illustrate the estimated changes in cross-border lending and borrowing volumes resulting from the reduction of regulatory barriers and financial fragmentation.
  - Figure D.4: Effects of Banking Integration on User Costs of Capital
    - Scenario: a 10% reduction in cross-border relationship wedges (τ_L) across all country pairs.
    - Panel (a): percent change in average user costs for each euro area country, relative to the baseline calibration.
    - Panel (b): percent change in the variance of log(user costs), relative to the baseline calibration.
  - Figure D.5: Cannibalization Test: Change in Banking Groups’ Profits

*Barriers to a European Banking Union — Working Paper No. WP/2026/123*

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_Source: https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026123-source-pdf.pdf_