Many model selection criteria proposed over the years have become common procedures in applied research. However, these procedures were designed for complete data. Complete data is rare in applied statistics, in particular in medical, public health and health policy settings. Incomplete data, another common problem in applied statistics, introduces its own set of complications in light of which the task of model selection can get quite complicated. Recently, few have suggested model selection procedures for incomplete data with varying degrees of success. In this paper we explore model selection by the Akaike Information Criterion (AIC) in the multivariate regression setting with ignorable missing data accounted for via multiple imputation.