normal_wishart

VAR with Normal-Wishart Prior (NormalWishart)

Overview

The Normal-Wishart VAR implements a Bayesian Vector Autoregression with conjugate Normal-Wishart priors. This specification allows for analytical computation of the posterior distribution, making it computationally efficient while providing a principled way to incorporate prior beliefs about both VAR coefficients and the error covariance matrix.

Methodology

The Normal-Wishart prior assumes:

• VAR coefficients follow a multivariate normal distribution
• The error covariance matrix follows an inverse Wishart distribution
• The conjugate structure enables closed-form posterior computation

This conjugacy property makes the estimator particularly attractive for applications requiring fast computation or when analytical tractability is important.

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

• Computational efficiency: When fast estimation is required
• Analytical tractability: When closed-form posteriors are needed
• Moderate sample sizes: Works well with typical macroeconomic datasets
• Standard applications: Good baseline choice for many VAR applications

When Not to Use

• Time-varying parameters: Use time-varying estimators instead
• Heavy model uncertainty: May want more flexible priors
• Very small samples: May need more informative priors

Practical Considerations

• Start with default parameter values and adjust based on data characteristics
• Monitor stability if using stability threshold
• Consider cross-validation for hyperparameter selection
• The conjugate structure makes this estimator suitable for real-time applications

Computational Notes

• Speed: Very fast due to analytical posterior computation
• Memory: Moderate memory requirements
• Parallelization: Limited parallelization opportunities due to analytical nature
• Numerical stability: Generally stable but monitor condition numbers

References

• Kadiyala, K.R. and S. Karlsson (1997). "Numerical Methods for Estimation and Inference in Bayesian VAR-models"
• Koop, G. and D. Korobilis (2010). "Bayesian Multivariate Time Series Methods for Empirical Macroeconomics"