threshold

VAR with Threshold-Based Regime Switching (Threshold)

Overview

The Threshold VAR implements a non-linear Vector Autoregression where the economy switches between discrete regimes based on the value of a threshold variable. This estimator allows for different VAR dynamics in different regimes, providing a flexible framework for modeling structural breaks, business cycle asymmetries, and other non-linear economic phenomena.

Methodology

The Threshold VAR specification features:

• Multiple VAR regimes with regime-specific coefficients
• Threshold variable determines regime transitions
• Endogenous threshold estimation
• Minnesota-type priors within each regime
• Non-linear dynamics through regime switching

This approach is particularly useful for modeling economies that exhibit different behavior in different states (e.g., recession vs. expansion).

Parameters

Threshold-Specific Parameters

• VarThreshold: Prior variance of the threshold

  • Range: (0, ∞)
  • Default: 10
  • Controls prior beliefs about threshold location uncertainty

• MaxDelay: Maximum delay allowed for threshold variable

  • Range: [1, ∞)
  • Default: 4
  • Maximum lags of threshold variable considered for regime identification

• ThresholdPropStd: Proposal standard deviation for threshold MH algorithm

  • Range: (0, ∞)
  • Default: 0.032
  • Controls acceptance rate in threshold Metropolis-Hastings sampling

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
  • Recommended: Extensive burn-in due to regime switching complexity

• StabilityThreshold: Threshold for maximum eigenvalue magnitude

  • Range: (0, ∞)
  • Default: null (no stability check)
  • Ensures VAR stability in each regime

• MaxNumUnstableAttempts: Maximum number of unstable sampling attempts

  • Range: [1, ∞)
  • Default: 1000
  • Maximum attempts to find stable draw before stopping

Usage Guidelines

When to Use

• Regime switching behavior: When the economy exhibits different dynamics across states
• Business cycle asymmetries: When recession and expansion periods differ substantially
• Structural breaks: When breaks are endogenous to economic conditions
• Non-linear relationships: When linear VARs are inadequate

When Not to Use

• Linear relationships: When linear VARs are sufficient
• Small samples: Insufficient data for regime identification
• Too many regimes: When complexity exceeds data informativeness
• Computational constraints: Very slow estimation process

Practical Considerations

• Requires long time series for reliable regime identification
• Threshold variable selection is crucial - should be economically meaningful
• Monitor regime classification and ensure reasonable regime persistence
• Consider simpler threshold rules if endogenous estimation is problematic

Model Specification

Regime-Dependent VAR

y_t = X_t β^{(j)} + ε_t^{(j)}

where regime j is determined by:

j = 1 if z_{t-d} ≤ τ
j = 2 if z_{t-d} > τ

Threshold Mechanism

• z_t: threshold variable (can be endogenous or exogenous)
• τ: threshold value (estimated)
• d: delay parameter (estimated, ≤ MaxDelay)

Computational Characteristics

Performance

• Speed: Very slow due to regime switching and threshold estimation
• Memory: High memory requirements for regime-specific parameters
• Convergence: Complex convergence due to discrete regime identification
• Numerical issues: Can face identification problems with similar regimes

Implementation Challenges

• Threshold identification can be difficult
• Label switching problems in MCMC
• Sensitivity to prior specifications
• Need for careful initialization

Regime Identification

Threshold Variable Selection

• Should be economically meaningful
• Can be one of the VAR variables or external
• Consider variables known to drive regime changes
• Common choices: interest rates, growth rates, financial stress indices

Diagnostic Considerations

• Plot regime probabilities over time
• Check regime persistence and frequency
• Validate regime classifications against known economic episodes
• Assess parameter differences across regimes

Comparison with Alternative Approaches

vs. Markov Switching VARs

• Threshold mechanism: Observable vs. latent regime variable
• Regime persistence: Can allow for very persistent regimes
• Interpretation: More transparent regime identification
• Estimation: Different computational challenges

vs. Structural Break Models

• Endogeneity: Endogenous vs. exogenous break timing
• Recurrence: Regimes can recur vs. permanent breaks
• Flexibility: More flexible regime switching mechanism
• Complexity: More complex estimation and interpretation

Implementation Notes

Prior Considerations

• Threshold prior variance affects regime identification
• Minnesota priors applied within each regime
• Consider regime-specific prior specifications
• Monitor prior sensitivity for threshold parameters

Convergence Assessment

• Check convergence across all regime-specific parameters
• Monitor threshold value convergence
• Assess mixing between regimes in MCMC
• Use multiple chains with different initializations

References

• Balke, N.S. (2000). "Credit and Economic Activity: Credit Regimes and Nonlinear Propagation of Shocks"
• Hansen, B.E. (1999). "Threshold Effects in Non-Dynamic Panels: Estimation, Testing, and Inference"
• Tsay, R.S. (1998). "Testing and Modeling Multivariate Threshold Models"
• Koop, G. and S.M. Potter (1999). "Dynamic Asymmetries in U.S. Unemployment"