ind_normal_wishart_favar_twostep

Two-Step FAVAR with Independent Normal-Wishart Prior (IndNormalWishartFAVARTwostep)

Overview

The Two-Step FAVAR with Independent Normal-Wishart Prior combines factor-augmented VAR analysis with independent Normal-Wishart priors in a two-step estimation framework. This estimator first extracts factors via principal components, then applies equation-by-equation independent Normal-Wishart priors to the factor-augmented VAR, providing computational efficiency while allowing for different prior specifications across equations.

Methodology

The Independent Normal-Wishart FAVAR Two-Step specification features:

• Two-step estimation: factor extraction followed by VAR estimation
• Principal component factor extraction in first step
• Equation-by-equation independent Normal-Wishart priors
• Separate prior specification for each VAR equation
• Analytical posterior computation for each equation independently

This approach provides flexibility in prior specification across equations while maintaining computational advantages of conjugate priors.

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
  • Minimal burn-in due to conjugate structure

• 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-specific priors: When different equations may require different prior specifications
• Computational efficiency: When fast FAVAR estimation with some flexibility is needed
• Heterogeneous variables: When variables in the system have very different characteristics
• Robustness: When protecting against misspecification in cross-equation restrictions

When Not to Use

• Cross-equation restrictions desired: When system-wide restrictions are theoretically motivated
• Small samples: When cross-equation information sharing is crucial for efficiency
• Uniform prior beliefs: When all equations should be treated identically
• Maximum computational speed: When fastest possible estimation is required

Practical Considerations

• Allows for heterogeneous treatment of different variables
• More flexible than system-wide priors but less efficient
• Good balance between flexibility and computational speed
• May lose efficiency from not pooling information across equations

Model Specification

Two-Step Procedure

Step 1: Factor Extraction

F_t = Λ X_t + e_t

Factors extracted via principal components from observable dataset X_t.

Step 2: Independent FAVAR Equations Each equation estimated separately:

y_{i,t} = α_i + Σ_j β_{ij} y_{j,t-1} + Σ_k γ_{ik} f_{k,t-1} + ε_{i,t}

Independent Prior Specification

For each equation i:

β_i | σ²_i ~ N(m_i, σ²_i Ω_i)
σ²_i ~ IG(s_i, ν_i)

where hyperparameters can differ across equations.

Independent Posteriors

Each equation has analytical posterior:

β_i | σ²_i, Data ~ N(m*_i, σ²_i Ω*_i)
σ²_i | Data ~ IG(s*_i, ν*_i)

Computational Characteristics

Performance

• Speed: Very fast due to equation-by-equation conjugate structure
• Memory: Moderate requirements, scales linearly with number of variables
• Convergence: Excellent convergence properties for each equation
• Scalability: Scales well, can be parallelized across equations

Independence Benefits

• Each equation can be estimated separately and in parallel
• No cross-equation MCMC needed
• Analytical posteriors for all equation-specific parameters
• Robust to misspecification in individual equations

Factor-Augmented Analysis

Factor Treatment

• Same factors used in all equations but with equation-specific coefficients
• Factor loadings can vary freely across equations
• Each equation can have different sensitivity to factor dynamics
• Robust to equation-specific relationships with factors

Cross-Equation Independence

• Residual correlations captured through factor structure rather than direct modeling
• Each equation optimized independently given the extracted factors
• Potential loss of efficiency from not modeling cross-equation correlations directly
• More robust to outliers or structural breaks in individual equations

Comparison with Related FAVAR Approaches

vs. System Normal-Wishart FAVAR

• Prior structure: Independent vs. system-wide Normal-Wishart priors
• Efficiency: Less efficient but more robust to misspecification
• Flexibility: More flexible treatment of individual equations
• Computation: Similar speed, potentially parallelizable

vs. Flat FAVAR Two-Step

• Priors: Informative Minnesota vs. uninformative flat priors
• Shrinkage: Better performance through equation-specific shrinkage
• Forecasting: Typically superior forecasting performance
• Overfitting: Less prone to overfitting in each equation

vs. One-Step Independent FAVAR

• Estimation: Two-step vs. simultaneous factor-VAR estimation
• Computational cost: Much faster than one-step approach
• Factor uncertainty: Ignores factor uncertainty but computationally efficient
• Implementation: Much easier to implement and diagnose

Implementation Strategy

Factor Extraction Stage

• Standardize data and handle missing observations
• Use information criteria to select number of factors
• Validate factor stability and interpretability
• Assess factor loadings for economic meaning

Independent Equation Estimation

• Estimate each VAR equation separately using extracted factors
• Apply equation-specific Normal-Wishart priors
• Exploit analytical posteriors for fast computation
• Consider parallel estimation across equations

Prior Specification

• Can customize priors for each equation based on economic reasoning
• May use different Lambda parameters for different types of variables
• Consider equation-specific treatments for exogenous variables
• Monitor prior sensitivity across equations

Model Validation

Equation-Specific Diagnostics

• Examine posterior properties for each equation separately
• Assess parameter estimates and their uncertainty
• Check equation-specific stability and fit
• Compare equation-specific forecasting performance

System-Level Assessment

• Combine equation-specific results for system analysis
• Assess overall VAR stability using eigenvalue analysis
• Evaluate system-wide forecasting performance
• Compare with system-wide estimation approaches

Factor Analysis

• Examine factor loadings across all equations
• Assess consistency of factor interpretation across equations
• Test factor importance variation across different variables
• Validate robustness of factor extraction methodology

Applications

Heterogeneous Variable Systems

• Mixed-frequency data with different variable characteristics
• Financial and macroeconomic variables requiring different treatments
• International data where country-specific effects are important
• Sectoral analysis where industries have different dynamics

Robustness Applications

• When uncertainty about cross-equation restrictions exists
• Applications requiring robustness to outliers in specific variables
• Exploratory analysis before imposing system restrictions
• When different equations may have different structural breaks

Advantages and Limitations

Key Advantages

• Flexibility in treating different equations/variables
• Computational efficiency through equation independence
• Robustness to equation-specific misspecifications
• Potential for parallelization across equations

Main Limitations

• Loss of efficiency from not pooling cross-equation information
• Does not directly model residual cross-correlations
• May miss important cross-equation restrictions
• Can lead to less coherent system-wide interpretation

Economic Interpretation

Factor Loading Differences

• Different variables may respond differently to common factors
• Equation-specific factor sensitivities reveal heterogeneous transmission
• Can identify which variables are most factor-driven
• Helps understand differential impacts of factor shocks

Heterogeneous Dynamics

• Variables may have different persistence patterns
• Cross-variable spillovers can vary in strength and timing
• Allows for variable-specific adjustment speeds
• Can capture asymmetric relationships across the system

References

• Bernanke, B.S., J. Boivin, and P. Eliasz (2005). "Measuring the Effects of Monetary Policy: A Factor-Augmented Vector Autoregressive (FAVAR) Approach"
• Litterman, R.B. (1986). "Forecasting with Bayesian Vector Autoregressions-Five Years of Experience"
• Kadiyala, K.R. and S. Karlsson (1997). "Numerical Methods for Estimation and Inference in Bayesian VAR-models"
• Stock, J.H. and M.W. Watson (2002). "Forecasting Using Principal Components From a Large Number of Predictors"