normal_diffuse

VAR with Normal-Diffuse Prior (NormalDiffuse)

Overview

The Normal-Diffuse VAR implements a Bayesian Vector Autoregression that combines informative Normal priors on coefficients with diffuse (uninformative) priors on the error covariance matrix. This hybrid approach allows for selective incorporation of prior information: strong beliefs about coefficients while letting the data determine the error structure.

Methodology

The Normal-Diffuse prior specification:

• VAR coefficients follow informative Normal priors (Minnesota-type)
• Error covariance matrix receives diffuse/uninformative priors
• Balances prior information with data-driven error covariance estimation
• Maintains computational efficiency while providing flexibility on error terms

This approach is particularly useful when researchers have strong beliefs about coefficient relationships but want to remain agnostic about error correlations and volatilities.

Parameters

Minnesota-Style Prior Parameters

• Autoregression: Prior on first-order autoregression

  • Range: [0, 1]
  • Default: 0.8
  • Controls the belief in random walk behavior for own lags

• Lambda1: Overall tightness of priors

  • Range: (0, ∞)
  • Default: 0.1
  • Smaller values impose tighter priors (more shrinkage)

• Lambda2: Variable weighting

  • Range: (0, ∞)
  • Default: 0.5
  • Controls relative weights between own-variable and cross-variable lags

• Lambda3: Lag decay

  • Range: (0, ∞)
  • Default: 1
  • Controls how quickly prior importance decays with lag length

• Lambda4: Exogenous variable tightness

  • Range: (0, ∞)
  • Default: 100
  • Controls prior tightness on exogenous variable coefficients

• Lambda5: Block exogeneity shrinkage

  • Range: (0, ∞)
  • Default: 0.001
  • Controls shrinkage when block exogeneity is imposed

Advanced Options

• Exogenous: Priors on exogenous variables flag

  • Default: false
  • Enables specialized priors for exogenous variables

• BlockExogenous: Block exogeneity flag

  • Default: false
  • Imposes block exogeneity restrictions

MCMC Settings

• Burnin: Number of burn-in draws

  • Range: [0, ∞)
  • Default: 0
  • May require burn-in depending on posterior complexity

• StabilityThreshold: Threshold for maximum eigenvalue magnitude

  • Range: (0, ∞)
  • Default: null (no stability check)
  • Ensures VAR stability by rejecting draws with eigenvalues above threshold

• MaxNumUnstableAttempts: Maximum number of unstable sampling attempts

  • Range: [1, ∞)
  • Default: 1000
  • Maximum attempts to find stable draw before stopping

Usage Guidelines

When to Use

• Selective prior information: When confident about coefficients but uncertain about error structure
• Heteroskedasticity concerns: When error variance may vary significantly
• Model comparison: As intermediate between informative and uninformative approaches
• Robust error modeling: When wanting data to determine error covariance structure

When Not to Use

• Full conjugacy needed: When analytical tractability is essential
• Strong error structure beliefs: When you have informative priors on error covariance
• Very small samples: May need more informative priors throughout

Practical Considerations

• Monitor MCMC convergence more carefully than fully conjugate approaches
• Consider diagnostics for error covariance estimation
• Compare with fully informative and uninformative alternatives
• Useful when transitioning between different prior specifications

Comparison with Related Estimators

vs. Normal-Wishart

• Coefficients: Similar informative priors
• Error covariance: Diffuse vs. informative Wishart prior
• Flexibility: More flexible on error structure
• Computation: Slightly more complex due to non-conjugacy

vs. Flat Prior

• Coefficients: Informative vs. uninformative priors
• Error covariance: Both use diffuse priors
• Shrinkage: Provides coefficient shrinkage, flat prior does not
• Performance: Better with limited data due to coefficient priors

Computational Notes

• Speed: Moderate, faster than fully non-conjugate but slower than conjugate
• Memory: Moderate memory requirements
• Convergence: Monitor MCMC chains for proper mixing
• Numerical stability: Generally stable but monitor error covariance estimation

References

• Litterman, R.B. (1986). "Forecasting with Bayesian Vector Autoregressions"
• Doan, T., R. Litterman, and C. Sims (1984). "Forecasting and Conditional Projection Using Realistic Prior Distributions"
• Kadiyala, K.R. and S. Karlsson (1997). "Numerical Methods for Estimation and Inference in Bayesian VAR-models"