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This means that your model is actually two seperate models for bank A and bank B, the only restriction being that the year dummies are the same. Do you get the same problems if you specify independent regressions, i.e. if D and if !D ?
I believe what most people do (if the algorithms are fast enough and time is no issue) is to take the highest boundry, that is, judge the worst-behaving draw (in terms of autocorrelation) by visual inspection, multiply the resulting burn-in by 2, and apply that to all chains.
I am not sure what you are asking here. Usually, the term “random” says something about a way a realization of a variable was achieved and is thus not a property that can be tested. If a variable (or, say, a time series) is the realization of a random process, of course also derived properties such as overlaps will be random.
Ok. I think I understand what you mean. In order to draw from the marg. posterior, I need either grid approx or an analytical solution via integrating over one of the two parameters. In the example of the continuous distribution, the analytical solution of the marignal distr. of σ² is in line 3 of the R code I posted. But an analytical solution of that integral will not always yield something I can directly sample from (i.e., when the marginal posterior does not have a closed form). Then, grid approx (or MCMC) is the only way to go. So, what that the case in the rounded example?
DJE, to me it seems your comment contradicts your answer. If I can draw from the marginal distribution p(σ²|y) or p(µ|y) and then from the conditional distribution p(µ|σ²,y) or p(σ²|µ,y), then I do not need to evaluate the posterior function on a grid or use MCMC in order to obtain draws from the posterior distribution.
