Much of the socioeconomic mobility achieved by U.S. immigrant families takes place across rather than within generations. When assessing the long-term integration of immigrants, it is therefore important to analyze differences not just between the foreign-born and U.S-born, but also across generations of the U.S.-born. Because of data limitations, however, virtually all studies of the later-generation descendants of immigrants rely on subjective measures of ethnic self-identification rather than arguably more objective measures based on the countries of birth of the respondent and his ancestors. In this context, biases can arise from "ethnic attrition" (e.g., U.S.-born individuals who do not self-identify as Hispanic despite having ancestors who were immigrants from a Spanish-speaking country). Analyzing 1994-2010 data from the Current Population Survey (CPS), we present evidence that such ethnic attrition is sizeable and selective for the second- and third-generation populations of key Hispanic and Asian immigrant groups. In addition, our results suggest that ethnic attrition generates measurement biases that vary across national origin groups in direction as well as magnitude, and that correcting for these biases is likely to raise the socioeconomic standing of the U.S.-born descendants of most Hispanic immigrants relative to their Asian counterparts. Finally, although changes to the CPS Hispanic origin and race questions adopted in 2003 have substantially lowered attrition rates for second- and third-generation Hispanics and Asians, ethnic attrition remains a significant issue even with the improved questionnaire.