We consider the role of unobservables, such as differences in search frictions, reservation wages, and productivities for the explanation of wage differentials between migrants and natives. We disentangle these by estimating an empirical general equilibrium search model with on-the-job search due to Bontemps et al. (1999) on segments of the labour market defined by occupation, age, and nationality using a large scale German administrative dataset. The native-migrant wage differential is then decomposed into several parts, and we focus especially on the component that we label migrant effect, being the difference in wage offers between natives and migrants in the same occupation-age segment in firms of the same productivity. Counterfactual decompositions of wage differentials allow us to identify and quantify their drivers, thus explaining within a common framework what is often labelled the unexplained wage gap.