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I try to run the multivariate multilevel model with common coefficient to a predictor under MCMC estimation. However, when I try to run the model with MCMC estimation, the results with MCMC are substantially different from IGLS estimation and also the results look strange.

For example, in the equation attached data set, the coefficients of pre-score were 0.442, 0.409, and 0.233 for each outcome under IGLS and common coefficient of pre-score was 0.275 under IGLS. On the contrary, the coefficients of pre-score were 0.440, 0.409, and 0.231 for each outcome under MCMC and common coefficient of pre-score was 0.007 under MCMC.

I feel weird because, as I know, the fixed effect is not substantially different depends on estimation method. I know it might be my data problem but I would like to ask if there is technical issue of MLwiN program that can explain the reason of the results.

Have you checked the MCMC chains to ensure that they have converged to a stable distribution?

I checked this with Bill, who wrote the MCMC algorithm, and apparently it was not designed to allow common coefficients in multivariate models. As a workaround you can use Metropolis-Hastings for the fixed effects (via Model->MCMC->MCMC Methods), which does appear to work.

I will look into providing a warning for this in future versions of the software.