One can apply selection methods such as the Heckman selection model to the OLS framework to try and correct for possible selection bias. However, such methods either require the identification of an instrumental variable (see below), or more typically, rely on functional form assumptions that can be a weak basis for policy recommendations.

A comment on the choice of the specification and variables is in order. First, the model should entail the proper causality, otherwise coefficient β would be meaningless. Second, for the reasons stated above, the migration variable, if possible, should be indicating migration status rather than related phenomena, such as the amount of remittances. Third, the model should include all relevant variables – all of them exogenous. Specifically, it should _not_ include variables that are impacted by migration. Recall that the interpretation of OLS results would be “the impact of migration on the outcome (say, children’s education), holding all control variables constant”. So, if income is included among control variables, the interpretation would imply “impact of migration on children’s education holding income constant”, which is not necessarily what one would like to measure. However, including pre-migration income could improve the model, controlling for household wealth status.

# _3.5.4. Reconstruct the counterfactual_

The fundamental problem in identifying the impact of migration is a lack of a counterfactual, so one way forward is to try to (re-)construct it.

One approach is to “manually” reconstruct what a household’s situation would have been had a household member not migrated. For example, when we are interested in income, we would deduct remittances, add (impute) a hypothetical income had a migrant stayed, and correct (impute) the labor participation decisions (and, hence, incomes) of other members. The imputation part is problematic. Can we use the non-migrant earning function to predict home earning of a migrant, even if correcting for selection on observable characteristics? Should we use average or marginal earnings?<sup>3</sup> When interested in the impact on inequality, in addition to the procedure outlined above, one would ideally add to the imputed income a non-insignificant random error component with the same variance as in the estimated income equation and average it over, say, a thousand of repetitions to get the mean impact (and standard errors as a byproduct, see e.g Acosta, Calderon Fajnzylber and Lopez, 2006b).

A second approach is to compare a migrant household to an “identical” household that has not migrated. Obviously, the “identical” household does not exist, but by using a propensity score matching technique we can equally well compare to a non-migrant household with the same propensity to migrate. This means running a migration decision regression. The typical migration determinants are age, sex, education, existing migrant networks, household composition, and wealth. The last one is most likely to be itself

> 3 In the agricultural setting a marginal productivity of a migrant at home can be close to zero. In this case we should control for household composition when estimating the earning function. Similarly, in the high unemployment settings a migrant at home might not be able to get any job at all.

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