The impacts of international emigration and remittances on incomes and poverty in sending areas are increasingly studied with household survey data. But comparing households with and without emigrants is complicated by a triple-selectivity problem: first, households self-select into emigration; second, in some emigrant households everyone moves while others leave members behind; and third, some emigrants choose to return to the origin country. Allowing for duration-dependent heterogeneity introduces a fourth form of selectivity—we must now worry not just about whether households migrate, but also when they do so. This paper sets out these selectivity issues and their implications for existing migration studies and then addresses them by using survey data designed specifically to take advantage of a randomized lottery that determines which applicants to the oversubscribed Samoan Quota (SQ) may immigrate to New Zealand. We compare incomes and poverty rates among left-behind members in households in Samoa that sent SQ emigrants with those for members of similar households that were unsuccessful in the lottery. Policy rules control who can accompany the principal migrant, providing an instrument to address the second selectivity problem, while differences among migrants in which year their ballot was selected allow us to estimate duration effects. We find that migration reduced poverty among former household members but also find suggestive evidence that this effect may be short-lived as both remittances and agricultural income are negatively related to the duration that the migrant has been abroad.