normal_wishart_favar_twostep

Two-Step FAVAR with Normal-Wishart Prior (NormalWishartFAVARTwostep)

Overview

The Two-Step FAVAR with Normal-Wishart Prior combines factor-augmented VAR analysis with conjugate Normal-Wishart priors in a two-step estimation framework. This estimator first extracts factors via principal components, then applies Normal-Wishart priors to the factor-augmented VAR, providing computational efficiency through conjugate priors while leveraging information from large datasets.

Methodology

The Normal-Wishart FAVAR Two-Step specification features:

• Two-step estimation: factor extraction followed by VAR estimation
• Principal component factor extraction in first step
• Conjugate Normal-Wishart priors on VAR parameters in second step
• Analytical posterior computation for computational efficiency
• Combines FAVAR framework with Normal-Wishart tractability

This approach provides computational advantages of conjugate priors while leveraging the information content of large datasets through factor extraction.

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
  • May require 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

• Computational efficiency: When fast FAVAR estimation is required
• Large datasets: When dealing with many observable series
• Analytical tractability: When closed-form posteriors are valued
• Standard FAVAR applications: Good baseline choice for factor-augmented analysis

When Not to Use

• Non-conjugate priors needed: When Normal-Wishart assumptions are inappropriate
• One-step estimation preferred: When joint factor-VAR estimation is desired
• Time-varying parameters: When parameters evolve over time
• Very small datasets: When factor extraction is unreliable

Practical Considerations

• Benefits from conjugate structure providing computational advantages
• Requires sufficient cross-sectional dimension for factor extraction
• Monitor factor interpretability and stability
• Consider factor selection criteria carefully

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: FAVAR with Normal-Wishart Prior

[Y_t]   [Y_{t-1}]
[F_t] = A [F_{t-1}] + ε_t

Conjugate Prior Specification

A | Σ ~ N(M, Σ ⊗ Ω)
Σ ~ IW(S, ν)

where M, Ω, S, ν are hyperparameters following Minnesota-type restrictions.

Posterior Distribution

Analytical posterior due to conjugacy:

A | Σ, Data ~ N(M*, Σ ⊗ Ω*)
Σ | Data ~ IW(S*, ν*)

Computational Characteristics

Performance

• Speed: Very fast due to conjugate structure
• Memory: Moderate requirements for factor-augmented system
• Convergence: Excellent convergence properties
• Scalability: Scales well with number of factors and observables

Conjugate Benefits

• Closed-form posterior distributions
• No MCMC burn-in issues
• Predictable computational time
• Analytical marginal likelihoods for model comparison

Factor-Augmented Analysis

Factor Interpretation

• Extracted factors represent common components in large dataset
• Factor loadings reveal which variables are driven by common factors
• Factor dynamics capture aggregate economic fluctuations
• Factor shocks can be interpreted as aggregate disturbances

Economic Applications

• Factors may represent business cycle, monetary policy, or financial conditions
• Impulse responses to factor shocks show transmission mechanisms
• Forecasting leverages information from entire dataset
• Nowcasting applications using real-time factor updates

Comparison with Related FAVAR Approaches

vs. Flat FAVAR Two-Step

• Priors: Informative Normal-Wishart vs. uninformative flat priors
• Efficiency: More efficient estimation through shrinkage
• Computation: Faster due to conjugate structure
• Performance: Typically better forecasting in small samples

vs. One-Step FAVAR

• Estimation: Two-step vs. simultaneous factor-VAR estimation
• Efficiency: Ignores factor uncertainty but computationally simpler
• Implementation: Easier to implement and debug
• Theoretical: Less theoretically optimal but practically efficient

vs. Minnesota FAVAR Two-Step

• Prior structure: Conjugate vs. hierarchical Minnesota structure
• Computation: Faster analytical vs. simulation-based inference
• Flexibility: Less flexible but more efficient
• Applications: Better for standard applications vs. specialized needs

Implementation Strategy

Factor Extraction Stage

• Standardize data and handle missing observations
• Use information criteria to select number of factors
• Validate factor stability across sample periods
• Assess factor interpretability and loadings

VAR Estimation Stage

• Apply Normal-Wishart priors to factor-augmented system
• Exploit conjugate structure for fast computation
• Validate posterior concentration and stability
• Compare with alternative prior specifications

Model Validation

Factor Assessment

• Examine factor loadings for economic interpretation
• Test factor stability using subsample analysis
• Compare factor forecasting performance
• Assess factor contribution to overall model fit

FAVAR Performance

• Compare forecasting accuracy with standard VAR
• Evaluate impulse response functions and their stability
• Assess structural interpretation through factor identification
• Cross-validate factor selection and prior specification

Posterior Analysis

• Examine posterior concentration of VAR parameters
• Assess model fit through marginal likelihood comparisons
• Validate stability of factor-augmented VAR system
• Check robustness to hyperparameter choices

Applications

Macroeconomic Forecasting

• Nowcasting GDP using large datasets of monthly indicators
• Central bank forecasting with real-time factor updates
• Business cycle analysis using factor dynamics
• Policy impact assessment through factor shocks

Financial Applications

• Asset pricing with macroeconomic factor extraction
• Risk factor identification in large portfolios
• Monetary policy transmission through financial factors
• Stress testing using factor-driven scenarios

References

• Bernanke, B.S., J. Boivin, and P. Eliasz (2005). "Measuring the Effects of Monetary Policy: A Factor-Augmented Vector Autoregressive (FAVAR) Approach"
• 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"
• Banbura, M., D. Giannone, and L. Reichlin (2010). "Large Bayesian Vector Auto Regressions"