Waggoner-Zha (2003) Gibbs sampler for a near-VAR

[CRMS]

I have another question regarding the choice of the optimal sampling method for a structural VAR with a near-VAR structure, i.e. where one or a block of variables is treated as exogenous. The RATS program montenearsvar shows how to implement a Metropolis-within-Gibbs sampler for this kind of models, whereas the replication file for Cushman & Zha JME 1997 has a more efficient sampling method that can be used for a specific two-block structure of the model. In both cases, the fact that the marginal likelihood of the covariance matrix has to be recomputed for each draw of the lagged coefficients - which are restricted in these models - was specifically accounted for in the sampling algorithm.

I have know been advised that models of these kind can also be estimated by using the Gibbs sampler provided by Waggoner and Zha in their 2003 JEDC paper "A Gibbs sampler for structural vector autoregressions". One paper that explicitly uses this algorithm for a near-VAR is Bhuiyan, CJE 2012, "Monetary transmission mechanisms in a small open economy". Bhuiyan's model shares some similarities with the original Cushman & Zha model from 1997 and employs the same two-block structure.

What puzzles me about the use of this sampler for a near-VAR is the following: Waggoner and Zha present their algorithm as a very general framework for all kinds of structural VARs. Although they explicitly mention models with restrictions on both the contemporaneous parameter matrix and the lagged coefficients, citing Cushman and Zha as an example (footnote 5, p. 351), the authors don’t seem to imply that their algorithm has to be modified for such a case.

Waggoner and Zha specify their joint likelihood function for the structural coefficients (b) and the lag coefficients (g) as follows:

|det[U1b1|...|Unbn]|^T * exp(-½ Σ(bi'Ui'Y'YUibi - 2gi'Vi'X'YUibi + gi'Vi'X'XVigi))

Implicitly, this still contains the covariance matrix of the residuals from the model estimation. However, if I understand the Waggoner-Zha algorithm correctly, the covariance matrix is only used to initialize the Gibbs sampler by finding the maximum log for the marginal posterior distribution of b. After that, draws for b appear to be made from the posterior distribution without updating for the covariance matrix, and g is drawn conditional on b.

When I run the Waggoner-Zha sampler (implemented in another software) on my own model, the results look rather different to the ones obtained by using the sampling methods recommended by RATS. Would that be due to some misspecification of the Waggoner-Zha sampler for a near-VAR? And do you think it is generally a good idea to use this sampler for this kind of model?

[TomDoan]

First, have you compared with two without the block restriction on the lags?

Waggoner-Zha write the contemporaneous model as y'A=.... while the usual way to write the model in RATS would be Ay=... so the A matrices are transposes of each other.

In the WZ approach, the model produces N(0,I) residuals. The equivalent of the draw for the inverse Wishart for sigma is embedded in their sampling procedure.