ind_normal_wishart

VAR with Independent Normal-Wishart Priors (IndNormalWishart)

Overview

The Independent Normal-Wishart VAR implements a Bayesian Vector Autoregression using independent Normal-Wishart priors across equations. Unlike the standard Normal-Wishart approach, this specification treats each equation independently, reducing cross-equation dependence in the prior while maintaining computational efficiency through conjugate priors.

Methodology

The Independent Normal-Wishart prior assumes:

• Each VAR equation has its own Normal-Wishart prior
• Cross-equation correlations are minimized in the prior specification
• Conjugate structure allows efficient posterior computation
• Independence assumption can be beneficial when equations are believed to be largely separate

This approach strikes a balance between computational tractability and reduced prior dependence across equations.

Parameters

Prior Configuration

• Sigma: Method of calculating priors on covariance matrix
  • Values: "ar" (autoregressive) or "eye" (identity-based)
  • Default: "ar"
  • Controls how the prior covariance matrix is constructed

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
  • Note: Analytical posterior may not require burn-in

• 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

• Equation independence: When VAR equations are believed to be largely independent
• Reduced cross-equation correlation: When minimizing prior cross-equation dependence
• Computational efficiency: Still benefits from conjugate structure
• Robustness concerns: When worried about strong cross-equation prior assumptions

When Not to Use

• Strong cross-equation relationships: When variables are highly interdependent
• Small sample sizes: May lose efficiency compared to pooled approaches
• Time-varying parameters: Use time-varying estimators instead

Practical Considerations

• Compare results with standard Normal-Wishart to assess impact of independence assumption
• Monitor whether independence assumption is appropriate for your data
• Consider this approach when standard Normal-Wishart seems too restrictive
• Still maintains computational advantages of conjugate priors

Differences from Standard Normal-Wishart

Key Distinctions

• Independence: Each equation treated separately in prior specification
• Cross-equation correlation: Reduced prior correlation between equations
• Flexibility: More flexible than standard Normal-Wishart in some dimensions
• Efficiency: May be less efficient if true cross-equation relationships exist

Performance Characteristics

• Computational speed: Similar to standard Normal-Wishart
• Memory usage: Comparable memory requirements
• Numerical stability: Generally stable with similar considerations

Computational Notes

• Speed: Very fast due to analytical posterior computation
• Memory: Moderate memory requirements, similar to standard Normal-Wishart
• Parallelization: Can potentially parallelize across equations
• Numerical stability: Generally stable but monitor condition numbers

References

• Carriero, A., G. Kapetanios, and M. Marcellino (2009). "Forecasting Exchange Rates with a Large Bayesian VAR"
• Koop, G. and D. Korobilis (2010). "Bayesian Multivariate Time Series Methods for Empirical Macroeconomics"