Stationarity of coefficients when sampling VAR by Gibbs sampler I am using Gibbs Sampler for VAR and have noticed that some researchers check the stationarity of $\beta$ coefficients while drawing. I am not sure why do they do that? Bayesian VARs do not require stationarity. Probably it is connected somehow to Gibbs Sampler? 
I saw previous posts here and here but still cannot understand the issue. Is it a requirement or just a convenience? I am interested in IFRs and FEVDs.
 A: You should think of the restriction as part of the prior. Since you're Gibbs sampling, you probably have a Gaussian prior for $\beta$
$$
\tilde{p}(\beta)=\textrm{N}(\beta;\,\bar{\beta}_0,\,\boldsymbol{\Phi}_0),
$$
which means you have a Gaussian conditional posterior for $\beta$
$$
\tilde{p}(\beta\,|\,\mathbf{y}_{1:T},\,...)=\textrm{N}(\beta;\,\bar{\beta}_T,\,\boldsymbol{\Phi}_T).
$$
But because the researcher is checking stationarity inside the Gibbs sampler, the prior for $\beta$ that they're really using is
$$p(\beta)\propto\tilde{p}(\beta)\cdot\mathbf{1}_{\mathcal{R}}(\beta),$$
where $\mathcal{R}$ is the stationary region of the parameter space and $\mathbf{1}(\cdot)$ is an indicator function. So it's the usual Gaussian prior, but truncated to the stationary region.
This means that the conditional posterior is
$$
p(\beta\,|\,\mathbf{y}_{1:T},\,...)\propto\tilde{p}(\beta\,|\,\mathbf{y}_{1:T},\,...)\cdot\mathbf{1}_{\mathcal{R}}(\beta).
$$
To sample from this conditional posterior distribution in the Gibbs sampler, you can simulate the "unrestricted" posterior $\beta^*\sim\tilde{p}(\beta\,|\,\mathbf{y}_{1:T},\,...)$, and then accept/reject the draw depending on whether or not $\beta^*\in\mathcal{R}$. So the fact that they're doing that inside the Gibbs sampler means that from the outset, they've written down a prior for $\beta$ that is truncated to the stationary region of the parameter space (whether they've explicitly acknowledged it or not).
As for why you would do this, you are correct in pointing out that you do not have to. But in some applications it might simply make sense. We don't observe that the data have blown up, so why should we entertain values for $\beta$ that imply that it would? And it can be a convenience for forecasting. We might want to simulate from the posterior predictive distribution $p(\mathbf{y}_{T+1:T+H}\,|\,\mathbf{y}_{1:T})$, which can be done by simulating the VAR for each posterior draw of $\beta$. If some of your $\beta$ draws are explosive, your forecasts could explode, which might not make sense in your application, and could also cause numerical problems.
