Del Negro-Primiceri (2015) Time-Varying Structural Vector Autoregressions and Monetary Policy: A Corrigendum

Summary

This note corrects a mistake in the MCMC estimation algorithm of Primiceri's (2005) time-varying-parameter structural VAR with stochastic volatility, and shows how to correctly apply the Kim–Shephard–Chib (KSC, 1998) mixture-of-normals procedure to any VAR, DSGE, factor, or unobserved-components model estimated with stochastic volatility. The KSC method approximates each $\log\varepsilon_t^2$ (a $\log\chi^2$ variable) by a mixture of normals, augmenting the Gibbs sampler with mixture-indicator states $s^T$ that select which component applies at each date; conditioning on $s^T$ lets one draw the volatilities with standard Gaussian state-space methods. The correction is subtle: each individual Gibbs step is unchanged — only the ordering of the MCMC steps is wrong in Primiceri (2005), and reordering them makes the sampler a valid draw from the joint posterior. (Review of Economic Studies 82(4): 1342–1345; FRBNY Staff Report No. 619.)

Key Claims

• The model. The TVP-VAR $y_t = c_t + B_{1,t}y_{t-1}+\cdots+B_{k,t}y_{t-k} + A_t^{-1}\Sigma_t\varepsilon_t$, where the time-varying intercepts/coefficients $c_t, B_{i,t}$ and the free elements of the lower-triangular $A_t$ follow random walks and the diagonal standard deviations $\Sigma_t$ follow geometric random walks; all innovations are jointly normal with block-diagonal covariance $V$.
• How KSC enters. Drawing $\Sigma^T$ is hard because volatilities enter multiplicatively; taking logs of the squared, transformed residuals turns the model linear in the log-volatilities but converts $\varepsilon_t$ into $\log\varepsilon_t^2\sim\log\chi^2(1)$. KSC approximate that $\log\chi^2$ density by a 7-component mixture of normals; conditioning on the mixture indicators $s^T$ (which component each observation is assigned to) restores a Gaussian linear state space, so $\Sigma^T$ can be drawn with a simulation smoother.
• Primiceri's Algorithm 1 (the wrong one). It draws (1) $\Sigma^T$ given the indicators, (2) the indicators $s^T$, then (3) the coefficients $\theta\equiv(B^T,A^T,V)$ — where steps 1–2 use the approximate mixture likelihood but step 3 uses the true likelihood.
• Two reasons it's wrong. (i) The sampler alternates between two different likelihood functions — the mixture-of-normals approximation for the volatility/indicator blocks and the exact likelihood for $\theta$ — which is internally inconsistent. (ii) More fundamentally, the blocks as sequenced are not valid full conditionals of a single joint posterior, so the "Gibbs sampler" does not have the intended stationary distribution.
• The fix — reorder, don't rewrite. The corrected algorithm keeps every individual step identical but changes the order in which the volatilities, the mixture indicators $s^T$, and the coefficients $\theta$ are drawn, so that each draw is a genuine full conditional and the sampler targets the correct joint posterior. In particular the indicators must be re-drawn conditional on the current volatilities in the right place in the sweep.
• Scope beyond Primiceri. The same ordering issue afflicts the many empirical macro models (VARs, DSGEs, dynamic factor models, unobserved-components models) that bolt KSC stochastic volatility onto a Gibbs sampler; the corrigendum gives the general recipe for applying KSC correctly.

Concepts Introduced or Extended

• Time-Varying Parameter VAR
• Stochastic Volatility
• Mixture of Normals
• Gibbs Sampler

Entities Mentioned

• Marco Del Negro
• Giorgio E. Primiceri

Quotes

"This note corrects a mistake in the estimation algorithm of the time-varying structural vector autoregression model of Primiceri (2005) … Relative to Primiceri (2005), the main difference in the new algorithm is the ordering of the various Markov Chain Monte Carlo steps, with each individual step remaining the same."

"There are two reasons why this algorithm does not yield draws from the correct posterior distribution … the algorithm alternates between the use of two different likelihood functions … [and] it was conceived as a Gibbs sampler [but the blocks are not valid full conditionals]."

My Take

This is a small but consequential piece of methodological hygiene: because the Primiceri (2005) sampler became the template that hundreds of TVP-VAR-with-stochastic-volatility papers copied, its MCMC ordering bug propagated widely, and Del Negro–Primiceri's fix is now the standard reference for "how to actually run KSC volatility inside a Gibbs sampler." The instructive lesson is that a Gibbs sampler is only valid if every block is drawn from a true full conditional of one coherent joint posterior — mixing an approximate likelihood for some blocks with the exact likelihood for others silently breaks that, and the symptom (a subtly wrong stationary distribution) is invisible without careful derivation. For this wiki it is an essential footnote to the TVP-VAR and stochastic-volatility pages and a reminder that the KSC mixture-of-normals device must be sequenced correctly.