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I have tried to look for an answer online but so far I have not succeeded. Is there a Bayesian equivalence of the Breusch-Godfrey test to no autocorrelation or of the Ljung-Box test? I have found some references on the Durbin-Watson test but as is well known this test is not of much use in applied work.

In a classical framework one would estimate the model of interest and then run an auxiliary regression to test for autocorrelation up to lag $p$. How is this done in a Bayesian setting?

I could think of obtaining the distribution of residuals and then do an auxiliary regression of these. I could then test the significance of the coefficients on lags using a Bayesian F-test or using the marginal likelihood to compare models. However, running the auxiliary regression would require a prior. Something which I do not have. Could the prior mean of the residuals be set to zero as that is what I expect them to be?

Further, would it be enough to just select a model based on the marginal likelihood and just disregard a test for autocorrelation?

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