example, Clemens et al. (2019) compare the wages of immigrants from 42 countries in the United States with those of individuals of the same gender, age, and education level in the origin countries to get an estimate of the wage gain from migrating. Methods such as propensity score matching attempt to match migrants to similar non-migrants on the basis of much larger sets of covariates, whilst ensuring that these comparisons lie in the common support and do not involve the linear extrapolation that regression can entail. For example, Gibson and McKenzie (2014) match Tongan and ni-Vanuatu households that participate in a seasonal worker program to similar households matched on household demographics, characteristics of the males most likely to migrate such as their health status and work experience, previous migration histories and networks, household assets and infrastructure, geography, and past wage and salary history. Another example is Dustmann and Fabbri (2003), who use a matching approach to estimate the impact of English language proficiency among UK immigrants.

The challenge for these methods is that migration theory is full of examples of how individuals may self-select into migrating on the basis of variables that may be difficult to observe or measure, such as unobserved ability (Borjas, 1987), risk preferences, language skills, exposure to shocks, entrepreneurial aptitude, etc. This raises two issues for practical use. The first is that since many surveys will not measure all of these features, it will typically be difficult to convince a skeptical reader that one has controlled for all of the important characteristics of individuals that both influence their migration decisions and that may also be correlated with the outcomes of interest. The second is that because migration is typically a rare event, most individuals in the population will not be good matches for those who have migrated, dramatically reducing the effective sample size. For example, McKenzie et al. (2010) match migrants from Tonga who applied for a migration lottery program in New Zealand to Tongans in a nationally representative labor force survey. They find that their sample drops from over 4,000 observations to just 354 when restricting to individuals with propensity scores between 0.05 and 0.95 – reflecting that most individuals in the labor force survey are too dissimilar to be used as controls for the migrants.

The result is that in many cases we do not believe matching (or regression) will yield reliable and precise point estimates of the impacts of migration. How much this matters depends a bit on the question of interest. In some cases, the impacts of migration may be so large that the bias from selection is second-order (McKenzie, 2020). For example, McKenzie et al. (2010) use a randomized lottery to estimate the income gain from migrating to be NZ$274 per week, a 263% increase on the control mean. Using regression or matching overstates this gain, with estimates of around NZ$330-350 per week, which are 20-28% higher than the experimental estimate. But in both cases one would conclude that migrating resulted in a massive gain in income, and this bias from matching is an order of magnitude smaller than the treatment effect. Bounding methods can also be used to ask how sensitive 15