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Currently, I am constructed a bayesian multilevel model to analyze a panel data set which now basically looks like the following: $y_{ijt} = \beta_{0ij} + X\beta + \epsilon_{ijt}$. So, now only a individual specific intercept but I want to extend this to other parameters. I estimate the model using bayesian econometrics.

Now, to increase predictive power I want to add a lagged dependent variable in my model, so it looks like this: $y_{ijt} = \beta_{0ij} + X\beta + \rho y_{ij(t-1)} + \epsilon_{ijt}$.

I was wondering whether I should take care of endogeneity by incorporating the lagged dependent variable in my model using the bayesian approach? In the frequentist approach including the lagged dependent variable will lead to severe inconsistency of the parameter $\rho$, so I think that I also have to take this problem into account using bayesian analysis. Could someone give me some explanation about this, since I cannot find any explanation on this subject usinng bayesian analysis.

In this case could someone also help me on how to model the initial value $y_{ij0}$ in this case?

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