Using microdata from the 1960-2000 decennial censuses, this paper explores how large initial differences in immigrant earnings by country of origin change with duration in the United States. One analysis reveals that country of origin adds less to the explanation of earnings, among working-age adult male immigrants, the longer they reside in the United States. Another discovers that the earnings dispersion of demographically comparable immigrants across countries of origin diminishes with time in the United States. Both indicate convergence in immigrant earnings by country of origin. To probe the sensitivity of these results to immigrant emigration, we pursue a theoretical analysis, which gauges how hypothetical patterns of selective emigration affect the convergence results, and an empirical analysis, which could be more broadly applied as a test for emigration bias. Both suggest that immigrant earnings convergence by country of origin is not an artifact of emigration. The convergence has methodological ramifications for the measurement of immigrant economic assimilation – in studies that follow cohorts and in studies that follow individuals with longitudinal data – and more generally for the study of any process in which unmeasured variables jointly affect initial conditions and subsequent growth.