Two methods of using collateral information from similar institutions to predict college freshman grade average were investigated. One central prediction model, referred to as pooled least squares with adjusted intercepts, assumes that slopes and residual variances are homogeneous across selected colleges. The second model, referred to as Bayesian m-group regression, allows estimates of slopes and variances to vary across colleges without ignoring the available collateral information. These models were compared with the more usual procedure of deriving regression equations within each college considered in isolation from other colleges. Data were obtained from colleges that participated in the American College Testing predictive research services program during the 1983 and 1984 years, and that had fewer than 100 records in 1983. Two groups of colleges were used: (1) 9 four-year colleges with "liberal", or "open," enrollment; and (2) 10 two-year colleges with more than 20 freshmen over the age of 25 years. It was found that both models using collateral information resulted in more accurate predictions, on cross validation, than did the within-college model, and that the Bayesian approach slightly outperformed the pooled least squares approach. It is noted that the Bayesian simultaneous regression model is highly adaptive to different regression structures and therefore can be expected to perform as well as the other two models across most situations. Seven tables present study data. (Author/SLD)