beta_tv

Time-Varying VAR (BetaTV)

Overview

The Time-Varying VAR (BetaTV) implements a Bayesian Vector Autoregression where coefficients evolve over time following random walk processes. This estimator allows for gradual structural changes in the relationships between variables while maintaining constant error covariance, making it suitable for capturing slowly evolving economic relationships.

Methodology

The BetaTV specification:

• VAR coefficients follow time-varying random walk processes
• Error covariance matrix remains constant over time
• Uses state-space representation with Kalman filtering
• Employs MCMC methods for posterior simulation
• Focuses on parameter drift rather than volatility changes

This approach is particularly useful for modeling gradual structural changes in economic relationships while maintaining computational tractability.

Parameters

MCMC Settings

• Burnin: Number of burn-in draws

  • Range: [0, ∞)
  • Default: 0
  • Recommended: Several thousand for proper convergence

• 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

• Gradual structural change: When economic relationships evolve slowly over time
• Parameter instability: When coefficient stability tests suggest time-variation
• Long time series: When sufficient data exists to identify time-varying patterns
• Policy regime changes: When gradual policy shifts affect relationships

When Not to Use

• Volatility changes: When error variance changes are the primary concern (use stochastic volatility models)
• Abrupt breaks: When structural changes are sudden rather than gradual
• Short time series: When insufficient data exists for time-varying estimation
• Computational constraints: When fast computation is essential

Practical Considerations

• Requires longer time series than constant-parameter models
• Monitor convergence carefully due to high-dimensional parameter space
• Consider starting values and prior specifications carefully
• May need substantial burn-in period for MCMC convergence

Model Specification

State Equation

Coefficients evolve according to:

β_t = β_{t-1} + ν_t

where ν_t ~ N(0, Q) represents coefficient innovations.

Observation Equation

Standard VAR structure with time-varying coefficients:

y_t = X_t β_t + ε_t

where ε_t ~ N(0, Σ) with constant covariance.

Computational Characteristics

Performance

• Speed: Slow due to MCMC and state-space filtering
• Memory: High memory requirements for storing time-varying parameters
• Convergence: Requires careful monitoring and adequate burn-in
• Parallelization: Limited opportunities due to sequential nature

Diagnostics

• Monitor parameter trace plots for convergence
• Check effective sample sizes for all parameters
• Assess autocorrelation in MCMC chains
• Validate stability of time-varying coefficients

Comparison with Related Models

vs. Constant Parameter VAR

• Flexibility: More flexible, allows parameter change
• Complexity: Much more complex estimation
• Data requirements: Needs longer time series
• Interpretation: Time-varying coefficients provide richer insights

vs. General Time-Varying VAR

• Scope: Coefficients only vs. both coefficients and volatility
• Complexity: Less complex than full time-varying specification
• Computational: Faster than models with both parameter and volatility variation
• Use case: When parameter change is primary concern

Implementation Notes

Prior Considerations

• Coefficient random walk variance (Q matrix) is crucial
• Initial state priors affect early period estimates
• Consider informative priors for stability
• Monitor prior sensitivity

Convergence Assessment

• Use multiple chains with different starting values
• Assess Gelman-Rubin statistics
• Check effective sample sizes exceed 1000
• Monitor acceptance rates in MCMC

References

• Cogley, T. and T.J. Sargent (2005). "Drifts and Volatilities: Monetary Policies and Outcomes in the Post WWII US"
• Primiceri, G.E. (2005). "Time Varying Structural Vector Autoregressions and Monetary Policy"
• Koop, G. and D. Korobilis (2013). "Large Time-Varying Parameter VARs"