Methods that accurately predict the grade of a student at a given activity or course can identify students that are at risk in failing a course and allow their educational institution to take corrective actions. Though a number of prediction models have been developed, they either estimate a single model for all students based on their past course performance and interactions with learning management systems (LMS), or estimate student-specific models that do not take into account LMS interactions; thus, failing to exploit fine-grain information related to a student's engagement. In this work we present a class of collaborative multi-regression models that are personalized to each student and also take into account features related to student's past performance, engagement and course characteristics. These models use all historical information to estimate a small number of regression models shared by all students along with student-specific combination weights. This allows for information sharing and also generating personalized predictions. Our experimental evaluation on a large set of students, courses, and activities shows that these models are capable of improving the performance prediction accuracy by over 20%. In addition, we show that by analyzing the estimated models and the student-specific combination functions we can gain insights on the effectiveness of the educational material that is made available at the courses of different departments.