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I am currently working on hierarchical models and try to get my head around the following question: What influence has the prior choice of the covariance matrix in the 2nd stage, especially when comparing a diagonal matrix of some form against a full VCV matrix?

In the former case I am assuming independence across regressors, while in the latter I allow for correlations across them. But what implications does either choice have? It is clear that it depends on the application at hand, but right now I think about this in a more general sense. I'll write down the model, so that it's clear what I mean. So let's assume I have a dependent variable $\{y_i\}_{t=1}^T$, then

\begin{equation} \begin{aligned} y_i | X_i, \beta_i, \sigma^2 &\sim N(X_i \beta_i, \sigma^2 I_T),\\ \beta_i | b, \Lambda &\sim N(b, \Lambda), \end{aligned} \end{equation} where the remaining prior distributions are fairly normal (IG for $\sigma^2$, N for $b$). So finally, I have to specify a prior distribution for $\Lambda$. There are basically two options,

\begin{equation} \begin{aligned} (1) \quad \Lambda &\sim IW(v, S) \\ (2) \quad \Lambda &= diag(\lambda_1,...,\lambda_k) \\ &\lambda_j \sim IG(c_0, d_0) \text{ for } j=1,...,k \end{aligned} \end{equation}

I am aware of more sophisticated prior specifications (half-t distribution following Gelman (2006, Bayesian Analysis) and ist multivariate extension by Huang and Wand (2013, Bayesian Analysis) or the seperation strategy by Barnard, McCulloch and Weng (2000, Statistica Sinica)), but that is not the main point I want to make here. Instead I would like to discuss the general implications when assuming either independence or allowing for correlations of regressors.

Thanks for any clarifications in advance.

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