Immigrant supply shocks are usually expected to lower wages for comparable workers. Indigenous people may respond to lower wages by moving to markets that were not directly targeted by immigrants and where wages are unlikely to have fallen. In this paper, we argue that the observed wage change in the target market depends not only on the magnitude of the indigenous response, but also on which indigenous residents choose to respond. A non-random response alters the composition of the sample of indigenous workers, mechanically changing the average wage of indigenous residents in the affected markets and distorting the estimated effect of immigration on wages. We document the importance of this selection bias in the French labor market, where women have accounted for a rapidly growing proportion of the foreign-born labor force since 1976. The raw correlations suggest that the immigration supply shock did not change French women's wages, but led to a significant decline in their employment rate. In contrast, immigration had little effect on men's employment rate, but led to a significant decline in their wages. We show that the near-zero correlation between immigration and women's wages is partly due to the fact that indigenous women who left the labor market had relatively low wages. After accounting for selection bias, wage elasticities are similar for both French women and French men (between -0.7 and -1.0).