An improved Bayesian method, due to Lindley, for the simultaneous estimation of multiple regressions inmgroups is studied by applying the method to test data. Evidence is found to support the belief that in many testing applications the collateral information obtained from each subset ofm− 1 colleges will be useful for estimating the regression in themth college, especially when the sample sizes are small. Using a 25 per cent sample, the Bayesian prediction equations achieved an average 9.7 per cent reduction in mean square error, as compared with the within‐group least squares equations, when cross‐validated with a later sample. More importantly, the mean square error for the Bayesian equations based on the 25 per cent sample was only barely greater than that for the least squares equations based on the full sample data. Thus the main virtue of the method is that it permits predictions to be made separately for relevant subpopulations (e.g. male‐female) where sample sizes would otherwise be too small to achieve an acceptable degree of a