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Banque Nationale De Belgique Sa — National Bank of Belgium — Technical Presentation (February 2026)

36 slides from Banque Nationale De Belgique Sa's technical presentation, Feb 5, 2026

1. Slide 1: Group Local Projections

   Group Local Projections Macroeconomic Seminar, The National Bank of Belgium Jiaming Huang (Hong Kong University of Science and Technology) February 5, 2026

2. Slide 2: Heterogeneity in Empirical Macro

   Heterogeneity in Empirical Macro • Unit-specific IRs: ▷ low efficiency: large confidence bands • Parametric heterogeneity on IRs ▷ FAVAR: strong factor structure ▷ Global VAR: a-priori determined interdependencies across units ▷ Panel VAR: shrinkage priors Question: Can we improve efficiency while all

3. Slide 3: Discretized Heterogeneity

   Discretized Heterogeneity • Grouped panels are popular when modeling heterogeneity ▷ Add interaction terms ▷ Group-by-group estimation • Example 1: marginal propensity to consume (poor vs. rich households) • Example 2: investment response to monetary policy (constrained- vs. unconstrained firms) Pre

4. Slide 4: This Paper

   This Paper • Data driven methods to recover latent groups • Develop asymptotic theory ▷ consistent group membership and group IRs ▷ asymptotic normal distribution of group IRs • Study US housing price responses to monetary shocks ▷ appreciation after contractionary shocks in an economically large cl

5. Slide 5: Literature

   Literature 1 Heterogeneous treatment effects ▷ Group structure in macro: Cloyne and Surico (2017); Cloyne et al. (2020); Ottonello & Winberry (2020) ▷ VAR: Bernanke et al. (2005); Dees et al. (2007); Canova et al. (2012) ▷ Panel: Ramaswamy et al. (1993); Lin & Ng (2012); Bonhomme & Manresa (2015); S

6. Slide 6: Group Local Projection Estimator

   Group Local Projection Estimator Group Local Projection Estimator Asymptotic Theory Simulation Application: US Housing Markets 5

7. Slide 7: Group Local Projections

   Group Local Projections outcome at horizon h endogenous policy variables yi,t+h = xi,t ′ βγi ,h + c′i,t ϕi,h + ϵi,t+h , h = 0, . . . , H grouped impulse responses • We have an L × 1 instrument zi,t 6

8. Slide 8: Group Local Projections

   Group Local Projections outcome at horizon h endogenous policy variables yi,t+h = xi,t ′ βγi ,h + c′i,t ϕi,h + ϵi,t+h , h = 0, . . . , H grouped impulse responses • Example (Jordà and Taylor, 2016) ▷ yi,t+h (cumulative) change in log real GDP ▷ xi,t change in cyclically adjusted primary balance ▷ zi

9. Slide 9: Multiple-equation GMM

   Multiple-equation GMM The GLP estimator is obtained by minimizing GMM criterion 1 ∑∑ ′ {β̂, ϕ̂, γ̂} = arg min Q̂NT (β, ϕ, γ), Q̂NT (β, ϕ, γ) = m̄i,h Ω̂i m̄i,h β∈ΘG ,ϕ∈ΦN ,γ∈G N i h 1∑ m̄i,h = zi,t (yi,t+h − xi,t βγi ,h − c′i,t ϕi,h ) T t • We rely on individual-level moment conditions E [m̄i,h ] = 0

10. Slide 10: Algorithm: Iterative Procedure

    Algorithm: Iterative Procedure 1 Initialization. Obtain initial group-level responses. Example: sampling from individual local projections estimates β̂ (0) and ϕ̂(0) . 2 Estimate Group Assignment. 3 Estimate Group-level Coefficient. 4 Check convergence. Repeat Steps 2-3 until estimates converge. Repe

11. Slide 11: Algorithm: Iterative Procedure

    Algorithm: Iterative Procedure 1 Initialization. 2 Estimate Group Assignment. Given β̂ (r) and ϕ̂(r) , solve ∑ ( ) ( ) m̄′ih β̂γ(r)i ,h ,ϕ̂(r) (r+1) ĝi (β̂ (r) , ϕ̂(r) ) = arg min i,h Ω̂i m̄ih β̂γ(r)i ,h ,ϕ̂(r) i,h γi ∈{1,...,G} h Unit-by-unit GMM evaluation 3 Estimate Group-level Coefficient. 4 Che

12. Slide 12: Algorithm: Iterative Procedure

    Algorithm: Iterative Procedure 1 Initialization. 2 Estimate Group Assignment. (r) (r) 3 Estimate Group-level Coefficient. Given the group partition γ̂ (r) = (ĝ1 , . . . , ĝN ), solve { } β̂ (r+1) , ϕ̂(r+1) = arg min Q̂NT (β, ϕ, γ̂ (r) ) β∈ΘG ,ϕ∈ΦN Group-level GMM estimation 4 Check convergence. Rep

13. Slide 13: Algorithm: Iterative Procedure

    Algorithm: Iterative Procedure 1 Initialization. 2 Estimate Group Assignment. 3 Estimate Group-level Coefficient. 4 Check convergence. Repeat Steps 2-3 until estimates converge. Repeat Steps 1-3 for different initializations and select the best solution. 8

14. Slide 14: Asymptotic Theory

    Asymptotic Theory Group Local Projection Estimator Asymptotic Theory Simulation Application: US Housing Markets 9

15. Slide 15: Assumption

    Assumption • Valid LP-IV at individual-level ▷ rank condition, (lead-lag) exogeneity condition (Stock and Watson, 2018) GMM • Uniform convergence at panel-level ▷ Strong mixing with exponential decay rate (Hahn and Kuersteiner, 2011) Mix • Identifiability at group-level ▷ Separable group IRs & non-n

16. Slide 16: Consistency

    Consistency Under the above conditions, we have for any δ > 0 ( ) P sup | γ̂i − γi0 |> 0 = op (1) + o(NT−δ ) (1) i∈{1,...,N} β̂j − β̃j = op (NT−δ ), j = 1, . . . , G0 . (2) where β̃j is the infeasible coefficient estimator where the true grouping γ 0 is known. • GLP can consistently recover the laten

17. Slide 17: Asymptotic Normality

    Asymptotic Normality • Assumptions: ▷ Cross-sectional independence ▷ Large N and T: Nj /T → κj ∈ [0, ∞) as N and T go to infinity • Under the regularity conditions, we can establish √ d √ 0 Nj T(β̂j,h − βj,h ) → Σ−1 j,h N( κj B, Ψj,h ), j = 1, . . . , G0 (3) ▷ The bias come from the incidental param

18. Slide 18: Number of Groups

    Number of Groups • In practice the number of groups can be selected by an information criterion IC(G) = Q̂NT,G + ϱN,T G(H + 1)Q̂NT,Gmax (4) ▷ ϱN,T → 0 and there exists some constant a > 0 such that ϱN,T Ta → ∞ • Under some regularity conditions, as N, T → ∞, we have ( ) 0 p P min IC(G) > IC(G ) → 1

19. Slide 19: Simulation

    Simulation Group Local Projection Estimator Asymptotic Theory Simulation Application: US Housing Markets 14

20. Slide 20: Simulation Design

    Simulation Design ( ) (( ) ( )) yi,t = µi + ργi yi,t−1 + δγi xi,t + ϵi,t ϵi,t i.i.d. 0 1 0.3 , ∼ N , . xi,t = µi + πz̃i,t + ui,t ui,t 0 0.3 1 G0 = 2 G0 = 3 δ (3, 3) (3, 3, 3) Design 1 ρ (0.2, 0.6) (0.2, 0.6, 0.9) δ (1, 2) (1, 2, 3) Design 2 ρ (0.5, 0.5) (0.5, 0.5, 0.5) IV π 0.7 0.7 Fraction (0.5, 0.

21. Slide 21: Result: Accuracy, CB Ratio, and RMSE

    Result: Accuracy, CB Ratio, and RMSE G0 = 2 G0 = 3 BR (%) RMSE (×100) BR (%) RMSE (×100) Design N T AC (%) GLP IGLP GLP PAN IND IGLP AC (%) GLP IGLP GLP PAN IND IGLP 100 100 84.1 15.8 15.7 29.6 33.5 63.8 10.0 78.3 20.3 19.3 52.7 81.7 77.5 15.3 100 200 92.5 15.0 14.8 19.1 33.0 43.4 6.5 90.3 18.8 18.0

22. Slide 22: Result: Accuracy, CB Ratio, and RMSE

    Result: Accuracy, CB Ratio, and RMSE BR (%) RMSE (×100) Design N T AC (%) GLP IGLP GLP PAN IND IGLP 100 100 84.1 15.8 15.7 29.6 33.5 63.8 10.0 100 200 92.5 15.0 14.8 19.1 33.0 43.4 6.5 100 300 96.2 14.7 14.5 13.6 32.8 35.0 5.1 200 100 84.1 11.2 11.2 29.0 33.2 63.9 7.9 1 200 200 92.7 10.6 10.6 18.4 3

23. Slide 23: Result: Coverage Rates

    Result: Coverage Rates Design N T h=0 h=1 h=2 h=3 h=4 h=5 h=6 100 100 92.4 90.6 88.1 86.4 86.7 88.2 89.0 100 200 94.1 94.0 94.9 94.4 92.0 93.3 92.3 100 300 94.6 94.3 95.2 94.6 94.4 93.6 93.8 200 100 90.2 86.2 78.7 77.2 80.2 83.9 84.8 1 200 200 92.9 91.7 92.6 92.0 91.3 91.7 91.5 200 300 93.2 93.6 95.

24. Slide 24: Result: RMSE (Unknown Number of Groups)

    Result: RMSE (Unknown Number of Groups) Design N T PAN Ĝ = 2 Ĝ = 3 Ĝ = 4 Ĝ = 5 Ĝ = 6 Ĝ = 7 Ĝ = 8 IND IC 100 100 81.7 56.7 58.2 59.2 60.0 61.0 62.1 62.5 77.5 2.0 100 200 81.2 40.2 39.2 39.7 40.4 41.1 41.7 42.1 52.5 2.7 100 300 81.0 33.0 27.1 30.3 31.1 31.9 32.4 32.9 42.2 3.0 200 100 81.5 56.1

25. Slide 25: Result: RMSE (No Group Structure)

    Result: RMSE (No Group Structure) N 100 100 100 200 200 200 300 300 300 T 100 200 300 100 200 300 100 200 300 PAN 45.7 45.2 45.7 45.6 45.4 45.4 45.8 45.5 45.5 Ĝ = 2 39.9 39.1 39.0 39.7 39.1 39.0 39.8 39.2 39.2 Ĝ = 3 37.4 34.0 33.6 36.7 34.2 34.1 36.8 34.6 34.4 Ĝ = 4 36.6 31.6 30.3 35.7 31.6 30.3

26. Slide 26: Application: US Housing Markets

    Application: US Housing Markets Group Local Projection Estimator Asymptotic Theory Simulation Application: US Housing Markets 20

27. Slide 27: Empirical Framework

    Empirical Framework • Monthly MSA house prices from 1991:5 to 2007:12 (N=382, T = 224) • Instrument FFR with high-frequency monetary surprises (Miranda-Agrippino and Ricco, 2018) • Controls: FRM30, IP, PCE, REALLN plus 4 lags • Mimics Del Negro and Otrok (2007) 21

28. Slide 28: Grouping Housing Responses

    Grouping Housing Responses Figure 1: Group IR (G=3) 22

29. Slide 29: Heterogeneous Dynamics: Economic Significance

    Heterogeneous Dynamics: Economic Significance Table 1: MSA Groups Economic Profile GDP Income Population Employment Regulation Elasticity Debt (Low) Debt (High) count mean std mean std mean std mean std mean std mean std mean std mean std 1 24 53686 22898 49063 14082 1486 2768 885 1782 0.513 0.396 1

30. Slide 30: External Grouping Criteria: Income Level

    External Grouping Criteria: Income Level (a) The Richest 10% MSAs (b) Poorest 10% MSAs in Group 1 Figure 2: External Grouping Criterion (Income Level) 24

31. Slide 31: Alternative: FAVAR

    Alternative: FAVAR (a) Five Factor (b) Eight Factors Figure 3: Housing Responses: FAVAR with Different Factors 25

32. Slide 32: Takeaways

    Takeaways • Methodologically: ▷ Computationally efficient framework to model heterogeneous IRs ▷ Potential extension: time-varying group membership, spillovers, etc • Empirically: ▷ Positive IRs to contractionary monetary shocks in economically important regions - suggestive of housing bubbles ▷ Alte

33. Slide 33: Thank you!

    Thank you! 26

34. Slide 34: Assumption: GMM

    Assumption: GMM 1 The parameter space Θ and Φ are compact. 2 For all i, t, E[zi,t w′i,t ] has full column rank and E|zi,t,j wi,t,k |4(1+δ) < ∞ for j = 1, . . . , L and k = 1, . . . , K and some δ > 0. 3 For all i, t and h, E[zi,t ϵi,t+h ] = 0 and E|zi,t,j ϵi,t+h |4(1+δ) < ∞ for j = 1, . . . , L and

35. Slide 35: Assumption: Mixing and Tail Conditions

    Assumption: Mixing and Tail Conditions 1 For all i, {(yi,t , w′i,t , z′i,t )′ : 1 ⩽ t ⩽ T} is a strictly stationary α-mixing sequence of random vectors with mixing coefficients ai (·) such that a(·) = maxi ai (·) satisfies a(τ ) ⩽ exp (−c1 τ c2 ) where c1 , c2 are positive constants. 2 For each i and

36. Slide 36: Assumption: Group Structure

    Assumption: Group Structure ∑ 1 For all j = 1, . . . , G0 , plimN→∞ N1 N 0 i=1 1{γi = j} = πj > 0. ∑H 0 0 2 For all j ̸= k, h=0 ∥βj,h − βk,h ∥ > 0. 3 The true number of groups G0 is finite and known. back

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