Bayesian Structural VAR (SBVAR) with Local Projections (LP)

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A self-contained research framework that estimates a Bayesian Structural VAR (SBVAR) with agnostic identification, and complements it with Local Projections (LP) to assess the robustness of impulse-response dynamics under alternative identification strategies.

This project aims to isolate U.S. monetary policy shocks and evaluate their transmission to Colombian macro-financial variables such as interest rates, inflation, exchange rate, and unemployment.

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🔹 Model Overview

1. Structural Bayesian VAR (SBVAR)

Implements a Bayesian structural model of the form:

$$ A_0 y_t = a_0 + \sum_{i=1}^p A_i y_{t-i} + C \varepsilon_t $$

• Priors:

  • Conjugate Minnesota-type priors for reduced-form parameters $(B, \Sigma)$
  • Soft priors (truncated or skewed-t) for structural coefficients $A_0$ and shock loadings $C$

• Sampling:

  • Gibbs sampling for $B$ and $\Sigma$
  • Metropolis–Hastings for $A_0$ and $C$ with adaptive scaling during warm-up

• Diagnostics:

  • Convergence analysis via R-hat, rolling means, and ESS
  • Posterior trace plots and acceptance-rate monitoring

• Outputs:

  • Posterior draws of $A_0$, $B$, and structural shocks
  • IRFs with 68% and 80% credible intervals
  • FEVDs (Forecast Error Variance Decomposition)

Contemporary system

$$ \begin{aligned} (1);& i^{US}_{5y,t} = \varepsilon^{US}_t \\ (2);& i^{CO}_{5y,t} = a_{21}, i^{US}_{5y,t} + a_{23}, \mathrm{TRM}_t + a_{25}, i^{CB}_t + \varepsilon^{CO}_t \\ (3);& \mathrm{TRM}_t = a_{31}, i^{US}_{5y,t} + a_{32}, i^{CO}_{5y,t} + a_{35}, i^{CB}_t + \varepsilon^{TRM}_t \\ (4);& \pi_t = a_{43}, \mathrm{TRM}_t + a_{46}, u_t + \varepsilon^{\pi}_t \\ (5);& i^{CB}_t = a_{51}, i^{US}_{5y,t} + a_{52}, i^{CO}_{5y,t} + a_{53}, \mathrm{TRM}_t + a_{54}, \pi_t + a_{56}, u_t + \varepsilon^{CB}_t \\ (6);& u_t = a_{62}, i^{CO}_{5y,t} + a_{64}, \pi_t + a_{65}, i^{CB}_t + \varepsilon^{u}_t \end{aligned} $$

Vector of structural shocks

$$ \varepsilon_t = \begin{pmatrix} \varepsilon^{US}_t \\ \varepsilon^{CO}_t \\ \varepsilon^{TRM}_t \\ \varepsilon^{\pi}_t \\ \varepsilon^{CB}_t \\ \varepsilon^{u}_t \end{pmatrix} $$

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2. Local Projections (LP – Jordà, 2005)

Implements two complementary LP methodologies for robustness verification of the SBVAR IRFs:

Method	Description	Folder
LP–Posterior	Runs LP regressions for each posterior draw of the SBVAR, computing percentile-based IRF bands across draws.	src/local_projections_posterior.py
LP–HAC	Estimates a single LP with HAC-robust standard errors, using the structural shock from a representative SBVAR draw.	src/local_projections_HAC.py

Both methods replicate the dynamic responses to U.S. T-bill shocks and allow validation of the shape, timing, and sign consistency of the structural IRFs.

3. MCMC Sampling and Posterior Filtering

The MCMC routine integrates a diagnostic-based chain trimming procedure that ensures the validity of posterior draws used in inference.

• Two-chain sampling: Each MCMC run generates two independent chains with controlled seeds for convergence assessment.
• Diagnostics-based selection: Posterior samples are filtered using rolling diagnostics:
  • Rank R-hat (Gelman–Rubin diagnostic) computed over sliding windows.
  • Cumulative mean stability test to ensure posterior stationarity.
  • Optional validity filters (e.g., positive definiteness, invertibility) applied to each draw before diagnostics.
• Adaptive trimming logic:
  The algorithm selects valid posterior segments through two complementary modes:
  • Global cut (t*) — a single conservative burn-in cutoff based on both R-hat and mean stability.
  • Per-parameter contiguous blocks — identifies stable intervals (segment_draws) for each parameter where chains exhibit convergence and low variance drift.
• Output validation: Only stable and converged posterior segments are used for inference, ensuring that IRFs, FEVDs, and LP regressions are computed from diagnostically consistent samples.

This filtering scheme acts as an automated posterior refinement step, minimizing the inclusion of non-stationary or pre-convergence draws and improving the credibility of structural inference.

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📊 Analytical Outputs

• IRFs (Impulse Response Functions)
  Median responses with 68% and 80% credible or confidence bands.

  

• Local Projections – HAC

  

• FEVD (Forecast Error Variance Decomposition)

  

• Markov Chain Trimming

  

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🧩 Repository Structure

SBVAR/
│
├── experiments/ # Figures and visual outputs
│
├── src/ # Modular source code
│ ├── identification.py # Structural identification priors and A0 routines
│ ├── irfs_fevd.py # IRF and FEVD construction from posterior draws
│ ├── local_projections_HAC.py # LP via HAC standard errors
│ ├── local_projections_posterior.py # LP via posterior draws
│ ├── lp_utils.py # Lag construction and alignment helpers
│ ├── mcmc_advanced.py # MCMC tuning and diagnostics
│ ├── metropolis_sampler.py # Metropolis–Hastings routines
│ ├── posteriors_gibbs.py # Gibbs sampling for B, Σ
│ └── graph_analisis.py # Visualization utilities
│
├── sbar_showcase/
│ └── Bayesian_SVAR_LP.ipynb # Main notebook: SBVAR + LP integration
│
└── Bayesian_SVAR_LP.ipynb # All the code logic in one Jupyter
└── XARIMA13_seasonal_fit # R script for XARIMA13 seasonal

Dependencies

pip install pandas numpy scipy numdifftools matplotlib seaborn arviz xarray ontextlib

🧠 Research Contribution

• Extends standard SBVAR frameworks by integrating Bayesian inference with Local Projections for cross-method consistency analysis.
• Provides a modular and reproducible codebase for structural identification, posterior simulation, and IRF validation.
• Enables robustness verification of structural dynamics under both Bayesian and frequentist perspectives.

References:

J. Jacobo, Una introducción a los métodos de máxima entropía y de inferencia bayesiana en econometría

Contributing

Contributions are welcome! Please open issues or submit pull requests at
https://github.com/pablo-reyes8

License

This project is licensed under the Apache License 2.0.