[Paper Review] Bayesian Inference for Structural Vector Autoregressions Identified by Markov-Switching Heteroskedasticity

Helmut Lütkepohl, Tomasz Woźniak|arXiv (Cornell University)|Nov 20, 2018

Monetary Policy and Economic Impact47 references23 citations

TL;DR

This paper develops a full Bayesian framework for structural vector autoregressions (SVARs) identified via Markov-switching heteroskedasticity, enabling formal statistical testing of identification conditions through Savage-Dickey density ratios. It establishes global and partial identification via parametric restrictions, supports model comparison via marginal data density computation, and finds strong empirical support for including Divisia money in the monetary policy reaction function.

ABSTRACT

In this study, Bayesian inference is developed for structural vector autoregressive models in which the structural parameters are identified via Markov-switching heteroskedasticity. In such a model, restrictions that are just-identifying in the homoskedastic case, become over-identifying and can be tested. A set of parametric restrictions is derived under which the structural matrix is globally or partially identified and a Savage-Dickey density ratio is used to assess the validity of the identification conditions. The latter is facilitated by analytical derivations that make the computations fast and numerical standard errors small. As an empirical example, monetary models are compared using heteroskedasticity as an additional device for identification. The empirical results support models with money in the interest rate reaction function.

Motivation & Objective

• To develop a full Bayesian inference framework for SVAR models identified through Markov-switching heteroskedasticity, overcoming limitations of frequentist approaches.
• To enable formal statistical testing of over-identifying restrictions by making the data informative on identification conditions.
• To support model comparison and selection using marginal data densities (MDD) and Savage-Dickey density ratios (SDDR).
• To allow for partial identification, focusing on specific equations (e.g., interest rate reaction function) without requiring full identification of all shocks.
• To provide fast MCMC sampling and analytical computation of SDDR and MDD for efficient posterior inference and model assessment.

Proposed method

• Derives parametric restrictions for global and partial identification of structural parameters under Markov-switching heteroskedasticity.
• Proposes a Savage-Dickey density ratio (SDDR) procedure using a generalized beta-F compound distribution to test identification conditions.
• Develops a fast, analytical MCMC sampler for the posterior of structural parameters, reducing numerical standard errors.
• Computes marginal data densities (MDD) analytically to enable full Bayesian model comparison and selection.
• Uses hierarchical priors with data-driven shrinkage to allow the data to inform identification, minimizing prior dominance.
• Applies the framework to a monetary policy model with Divisia money aggregates, comparing identification schemes via MDD and SDDR.

Experimental results

Research questions

• RQ1 Can over-identifying restrictions in SVAR models be formally tested when identification is based on Markov-switching heteroskedasticity?
• RQ2 What parametric conditions ensure global or partial identification of structural parameters in SVAR-MSH models?
• RQ3 How can the Savage-Dickey density ratio be applied to test identification conditions in a Bayesian framework with heteroskedasticity?
• RQ4 Does the inclusion of Divisia money in the interest rate reaction function receive stronger empirical support when heteroskedasticity is used for identification?
• RQ5 Can Bayesian model comparison via marginal data density detect differences in model fit when only partial identification is achieved?

Key findings

• The SDDR results provide strong evidence (log SDDR ≈ -22.85) that the variance of the fourth structural shock (interest rate equation) differs from all others, supporting its identification via heteroskedasticity.
• For the full sample (1967Q1–2013Q4), models with Divisia money in the interest rate equation yield the highest marginal data density (MDD = -1632.5), indicating superior model fit.
• In the shorter sample (1967Q1–2007Q4), the MDD for the Taylor Rule with Money model is -1453.7, outperforming alternatives, though differences are small when all rows of A0 are restricted.
• The MDD for the unrestricted interest rate equation is -1643.6 (full sample), and the model with money in the interest rate equation has a higher MDD, indicating better fit.
• The SDDR values for hypotheses like σ²₄/σ²ⱼ = 1 are very low (e.g., -22.85), indicating strong statistical evidence against equality, confirming the interest rate equation is well-identified.
• The framework allows formal testing of identification conditions even for partially identified models, avoiding the need to condition on additional restrictions as in frequentist approaches.