Hi Xiaoxia,
At the very, very bottom line, stationarity requirement for VARs is needed to ensure the underlying theoretical distribution has finite variance, which in turn ensures that t-statistics can be computed. Since you don’t compute t-statistics in the Bayesian setup, you don’t necessarily require stationarity.
Now that said, there are other benefits to stationarity. First, if you put two non-stationary time series into a VAR, you will have spurious correlation between them, because the model will add the (potentially independent, but shared) trend into the estimated parameters. Basically, you will find stronger relationships than you should.
Second, non-stationary time-series can quickly become numerically unstable, especially for long time series. Overflows are possible after several hundred time steps, and even if your data doesn’t overflow, the MCMC sampler will take a much longer time to run if it has to consider scenarios where your data blow up to large numbers.
So while it’s not statistically required, stationarity is still desirable in a BVAR