Admixture regression methodology exploits the natural experiment of random mating between individuals with different ancestral backgrounds to infer the environmental and genetic components to trait variation across culturally-defined racial and ethnic groups. This paper provides a new statistical framework for admixture regression based on the linear polygenic index model widely used in behavioural genetics. Using this framework we develop a new test of the differential impact of multi-racial identities on trait variation, an orthogonalization procedure for added explanatory variables, and a partially linear semiparametric functional form. The methodology is illustrated with data from the Adolescent Brain Cognitive Development Study.