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Consider the following m regression equation system:

$$r^i = X^i \beta^i + \epsilon^i \;\;\; \text{for} \;i=1,2,3,..,T$$

where $r^i$ is a $(T\times 1)$ vector of the T observations of the dependent variable, $X^i$ is a $(T\times k)$ matrix of independent variables, $\beta^i$ is a $(k\times1)$ vector of the regression coefficients and $\epsilon^i$ is the vector of errors for the $T$ observations of the $i^{th}$ regression.

If the above is an SUR model (seemingly unrelated regressions), does it make sense to compute the BIC (Bayesian Information Criterion) for the model?

If so, how would that be computed? Should the dependent variables be stacked into one vector, in order to get only one regression equation?

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    $\begingroup$ You should stack them if you would like to use them for ease of computation. For example, check out how you can stack the values into vectors and matrices to estimate $\hat\beta$ en.wikipedia.org/wiki/… $\endgroup$ – user25658 Sep 11 '13 at 19:50
  • $\begingroup$ Thanks for the link. So the case I am exposing here is a special case of the one in Wikipedia, where X1 = X2 = ...= Xm = X. Would it still make sense to stack them the dependent variables the same way, even if they all have the same matrix X of independent variables? $\endgroup$ – Mayou Sep 11 '13 at 19:53
  • $\begingroup$ The other question is: is it appropriate to compute BIC for such model? $\endgroup$ – Mayou Sep 11 '13 at 19:54
  • $\begingroup$ Related: stats.stackexchange.com/questions/395695/… $\endgroup$ – Christoph Hanck Mar 18 '19 at 13:52
  • $\begingroup$ Are you sure there are really $i=1,\ldots,T$ equations and $T$ observations? $\endgroup$ – Christoph Hanck Mar 18 '19 at 14:05
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In fact, R will help you do most of that. Here is an example from the systemfit package, which works because there exists a logLik method for the fitted model object from systemfit:

library(systemfit)
data("Kmenta")
system <- list(consump ~ price + income, consump ~ price + farmPrice + trend)
fitsur <- systemfit(system, "SUR", data = Kmenta)    
BIC(fitsur)
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