The amount of error that synthetic respondents are projected to use when answering the questionnaire has a huge effect on the scale factor for the estimated utilities. When I create synthetic respondents and add Gumbel error to the total utility of each alternative before projecting the first choices, I can exactly recover the right scale factor through aggregate logit. But, I have found that when I do the same with CBC/HB, I'm off by a scale factor. I've heard of others experiencing this same issue, but I cannot recall if there is a solution for choosing the right scaling of the Gumbel error together with the settings in the CBC/HB setup (prior variance, prior D.F.) that leads to exact recovery of the original scale factor (as baked into the original "true" synthetic utilities. Perhaps one of the HB experts out there has the answer?