How much do migrants stand to gain in income from moving across borders? Answering this question is complicated by non-random selection of migrants from the general population, which makes it hard to obtain an appropriate comparison group of non-migrants. New Zealand allows a quota of Tongans to immigrate each year with a random ballot used to choose among the excess number of applicants. A unique survey conducted by the authors allows experimental estimates of the income gains from migration to be obtained by comparing the incomes of migrants to those who applied to migrate, but whose names were not drawn in the ballot, after allowing for the effect of non-compliance among some of those whose names were drawn. We also conducted a survey of individuals who did not apply for the ballot. Comparing this non-applicant group to the migrants enables assessment of the degree to which non-experimental methods can provide an unbiased estimate of the income gains from migration. We find evidence of migrants being positively selected in terms of both observed and unobserved skills. As a result, non-experimental methods other than instrumental variables are found to overstate the gains from migration by 20–82%, with difference-in-differences and bias-adjusted matching estimators performing best among the alternatives to instrumental variables. (JEL: J61, F22, C21)