reduced_form_estimators

Reduced-Form Estimators in BEAR

This document provides a technical overview of the reduced-form estimators available in the BEAR toolbox, organized by category as they appear in the graphical user interface.

Plain Estimators

These are the foundational VAR estimators with various prior specifications but no time-varying parameters or cross-unit dependencies.

Ordinary (Ordinary)

• Description: Ordinary least squares
• Technical Details: Classical OLS estimation without priors
• Use Case: Baseline estimation for comparison; suitable when prior information is minimal

VAR with Flat Prior (Flat)

• Description: VAR with flat (uninformative) prior
• Technical Details: Uses diffuse priors with minimal restriction on parameters
• Use Case: When minimal prior information should influence the estimation

VAR with Minnesota Prior (Minnesota)

• Description: VAR with Minnesota prior
• Technical Details: Shrinks VAR coefficients toward random walk for own lags, zero for cross-variable lags
• Use Case: Standard choice for macroeconomic VARs; good balance between flexibility and shrinkage

VAR with Normal-Wishart Prior (NormalWishart)

• Description: VAR with Normal-Wishart prior
• Technical Details: Conjugate prior specification allowing analytical posterior computation
• Use Case: When computational efficiency and analytical tractability are important

VAR with Independent Normal-Wishart Priors (IndNormalWishart)

• Description: VAR with independent Normal-Wishart priors
• Technical Details: Uses independent priors across equations, reducing cross-equation dependence
• Use Case: When equations are believed to be largely independent

VAR with Normal-Diffuse Prior (NormalDiffuse)

• Description: VAR with Normal-Diffuse prior
• Technical Details: Combines informative priors on coefficients with diffuse priors on error covariance
• Use Case: When coefficient shrinkage is desired but error variance should be data-driven

Time-Varying Estimators

These estimators allow for time-varying parameters and/or stochastic volatility in the VAR system.

Time-Varying VAR (BetaTV)

• Description: Time-varying VAR with time-varying coefficients
• Technical Details: Allows VAR coefficients to evolve over time following random walk processes
• Use Case: When structural relationships change gradually over time

VAR with Time-Varying Parameters and Stochastic Volatility (GeneralTV)

• Description: VAR with time-varying parameters and stochastic volatility
• Technical Details: Combines time-varying coefficients with time-varying error covariances
• Use Case: Most flexible time-varying specification for capturing both parameter and volatility changes

Cogley-Sargent Stochastic-Volatility VAR (CogleySargentSV)

• Description: Cogley-Sargent stochastic-volatility VAR
• Technical Details: Implements the Cogley-Sargent approach with stochastic volatility
• Use Case: Standard time-varying VAR with focus on evolving volatility patterns

Carriero Stochastic Volatility VAR (CarrieroSV)

• Description: Carriero stochastic volatility VAR
• Technical Details: Uses Carriero et al. specification for stochastic volatility
• Use Case: Alternative stochastic volatility specification with different identification assumptions

CCMM Stochastic Volatility VAR with Random Walk Heteroscedasticity (CCMMSV)

• Description: CCMM stochastic volatility VAR with random walk heteroscedasticity
• Technical Details: Chan, Koop, Poirier, and Tobias specification with random walk volatility
• Use Case: When volatility follows smooth, persistent changes

CCMM Stochastic Volatility VAR with Outliers (CCMMSVO)

• Description: CCMM stochastic volatility VAR with outliers
• Technical Details: Extends CCMMSV to handle occasional large deviations (outliers)
• Use Case: When data contains occasional extreme observations that should be downweighted

CCMM Stochastic Volatility with Jumps and Large Shocks (CCMMSVOT)

• Description: CCMM stochastic volatility with jumps and large shocks
• Technical Details: Most flexible CCMM specification allowing for jumps in volatility
• Use Case: When volatility exhibits both smooth evolution and occasional jumps

Stochastic Volatility VAR for Large Shocks (LargeShockSV)

• Description: Stochastic volatility VAR for large shocks
• Technical Details: Specifically designed to handle periods with unusually large innovations
• Use Case: When the sample includes crisis periods or structural breaks

General Stochastic Volatility VAR for Large Shocks (GenLargeShockSV)

• Description: General stochastic volatility VAR for large shocks
• Technical Details: Extended version of large shock SV with additional flexibility
• Use Case: Most general specification for handling large shocks and volatility changes

Stochastic Volatility with Random Inertia (RandomInertiaSV)

• Description: Stochastic volatility with random inertia
• Technical Details: Incorporates time-varying persistence in the VAR dynamics
• Use Case: When the persistence of shocks varies over time

Panel Estimators with Separable Units

These estimators handle panel VAR models where units can be treated separately with some form of information sharing.

Hierarchical Panel (HierarchicalPanel)

• Description: Hierarchical panel VAR
• Technical Details: Uses hierarchical Bayesian approach to share information across units
• Use Case: When units are heterogeneous but share common distributional characteristics

Mean OLS Panel (MeanOLSPanel)

• Description: Mean OLS panel VAR
• Technical Details: Pools information across units through mean restrictions
• Use Case: Simple panel approach when units are relatively homogeneous

Normal-Wishart Panel (NormalWishartPanel)

• Description: Normal-Wishart panel VAR
• Technical Details: Applies Normal-Wishart priors in panel setting
• Use Case: Standard Bayesian panel VAR with conjugate priors

Zellner-Hong Panel (ZellnerHongPanel)

• Description: Zellner-Hong panel VAR
• Technical Details: Implements Zellner-Hong approach to panel VAR estimation
• Use Case: Alternative panel specification with specific information sharing structure

Panel Estimators with Cross-Unit Dependence

These estimators explicitly model dependence between units in panel VAR models.

Static Cross-Unit Panel (StaticCrossPanel)

• Description: Static cross-unit panel VAR
• Technical Details: Models constant cross-unit correlations and spillovers
• Use Case: When spillovers between units are important but constant over time

Dynamic Cross-Unit Panel (DynamicCrossPanel)

• Description: Dynamic cross-unit panel VAR
• Technical Details: Allows for time-varying cross-unit dependencies
• Use Case: When spillovers between units evolve over time

One-Step Factor-Augmented Estimators

These estimators implement Factor-Augmented VAR (FAVAR) models using one-step estimation where factors and VAR parameters are estimated simultaneously.

One-Step FAVAR with Flat Prior (FlatFAVAROnestep)

• Description: One-step FAVAR with flat prior
• Technical Details: Simultaneous estimation of factors and VAR with uninformative priors
• Use Case: FAVAR estimation with minimal prior restrictions

One-Step FAVAR with Minnesota Prior (MinnesotaFAVAROnestep)

• Description: One-step FAVAR with Minnesota prior
• Technical Details: Applies Minnesota-type shrinkage in one-step FAVAR framework
• Use Case: Standard FAVAR with Minnesota shrinkage

One-Step FAVAR with Normal-Wishart Prior (NormalWishartFAVAROnestep)

• Description: One-step FAVAR with Normal-Wishart prior
• Technical Details: Uses conjugate priors in one-step FAVAR estimation
• Use Case: Computationally efficient FAVAR with analytical posterior

One-Step FAVAR with Independent Normal-Wishart Prior (IndNormalWishartFAVAROnestep)

• Description: One-step FAVAR with independent Normal-Wishart prior
• Technical Details: Independent priors across equations in one-step FAVAR
• Use Case: FAVAR with reduced cross-equation dependence

One-Step FAVAR with Normal-Diffuse Prior (NormalDiffuseFAVAROnestep)

• Description: One-step FAVAR with Normal-Diffuse prior
• Technical Details: Combines coefficient shrinkage with diffuse error variance priors
• Use Case: FAVAR with selective prior information

Two-Step Factor-Augmented Estimators

These estimators implement FAVAR models using two-step estimation where factors are first extracted, then used in VAR estimation.

Two-Step FAVAR with Flat Prior (FlatFAVARTwostep)

• Description: Two-step FAVAR with flat prior
• Technical Details: Principal component factor extraction followed by flat prior VAR
• Use Case: Simple two-step FAVAR approach

Two-Step FAVAR with Minnesota Prior (MinnesotaFAVARTwostep)

• Description: Two-step FAVAR with Minnesota prior
• Technical Details: Standard two-step approach with Minnesota prior on VAR
• Use Case: Most common FAVAR specification in practice

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

• Description: Two-step FAVAR with Normal-Wishart prior
• Technical Details: Two-step estimation with conjugate priors
• Use Case: Computationally efficient two-step FAVAR

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

• Description: Two-step FAVAR with individual Normal-Wishart prior
• Technical Details: Independent priors across equations in two-step framework
• Use Case: Two-step FAVAR with equation independence

Two-Step FAVAR with Normal-Diffuse Prior (NormalDiffuseFAVARTwostep)

• Description: Two-step FAVAR with Normal-Diffuse prior
• Technical Details: Selective prior information in two-step FAVAR
• Use Case: When coefficient priors are informative but error variance should be data-driven

Time-Varying Factor-Augmented Estimators

These estimators combine FAVAR models with time-varying parameters and/or stochastic volatility.

Two-Step FAVAR with Time-Varying Coefficients (BetaTVFAVAR)

• Description: Two-step FAVAR with time-varying coefficients
• Technical Details: Combines two-step FAVAR with time-varying VAR parameters
• Use Case: When factor relationships evolve over time

Two-Step FAVAR with Time-Varying Parameters and Stochastic Volatility (GeneralTVFAVAR)

• Description: Two-step FAVAR with time-varying parameters and stochastic volatility
• Technical Details: Most flexible time-varying FAVAR specification
• Use Case: When both factor loadings and volatility change over time

Two-Step FAVAR with Cogley-Sargent Stochastic Volatility (CogleySargentSVFAVAR)

• Description: Two-step FAVAR with Cogley-Sargent stochastic volatility
• Technical Details: Applies Cogley-Sargent SV framework to FAVAR
• Use Case: Time-varying volatility in factor-augmented setting

Two-Step FAVAR with Carriero Stochastic Volatility (CarrieroSVFAVAR)

• Description: Two-step FAVAR with Carriero stochastic volatility
• Technical Details: Alternative SV specification for FAVAR models
• Use Case: Different volatility dynamics in factor models

Two-Step FAVAR with Random-Inertia Stochastic Volatility (RandomInertiaSVFAVAR)

• Description: Two-step FAVAR with random-inertia stochastic volatility
• Technical Details: Time-varying persistence in FAVAR framework
• Use Case: When factor model persistence varies over time

Specialized Estimators

These estimators implement specific methodologies that don't fit into the main categories above.

VAR with Threshold-Based Regime Switching (Threshold)

• Description: VAR with threshold-based regime switching
• Technical Details: Non-linear VAR with discrete regime changes based on threshold variable
• Use Case: When the economy switches between distinct regimes

Mixed-Frequency VAR (MixedFrequency)

• Description: Mixed-frequency VAR
• Technical Details: Handles variables observed at different frequencies within single model
• Use Case: When combining high-frequency and low-frequency economic indicators

Technical Notes

Computational Considerations

• Plain estimators are computationally fastest and most stable
• Time-varying estimators require MCMC and are computationally intensive
• Panel estimators scale with number of cross-sectional units
• FAVAR estimators depend on the number of factors and observables

Prior Selection Guidelines

• Minnesota priors work well for macroeconomic data
• Flat priors are appropriate when prior information is limited
• Normal-Wishart priors offer computational advantages
• Time-varying specifications should be used when structural change is suspected

Model Selection

The choice of estimator should be based on:

1. Data characteristics (time-varying volatility, structural breaks)
2. Sample size (complex models require larger samples)
3. Research objectives (forecasting vs. structural analysis)
4. Computational constraints (time-varying models are resource-intensive)