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We investigate the relationship between migration and productivity in the UK, using an instrumental variable along the lines suggested by Bianchi, Buonanno and Pinotti (2012). Our results suggest that immigration has a positive and significant impact (in both the statistical sense and more broadly) on productivity, as measured at a geographical level; this appears to be driven by higher-skilled workers. The results for training are less clear, but suggest that higher-skilled immigration may have a positive impact on the training of native workers. We discuss the implications for post-Brexit immigration policy.