Implications for Relationships in the Data

Panel C describes a scenario in which the return to migration experience at home rises steeply with baseline skill. In this scenario, more- skilled individuals have an incentive to make longer temporary trips abroad. Likewise, if the disutility of spending time abroad is negatively related to skill, then the more- skilled will be observed completing longer stays abroad. Such a scenario is depicted in Panel D.

IV. Implications for Relationships in the Data

Consider a cohort of individuals who behave according to the model presented here, and suppose that we have cross- sectional data on this cohort at age t back in the home country. In this cross- section, we observe nonmigrants with skill levels in the set: S nm = {s i | s i S or s i S} . We also observe the temporary migrants who have returned by age t. Let s ⋅ represent the inverse of the optimal duration func- tion. Then the return migrants that we observe in the data have skill levels that fall on the interval S = [ s ⋅, S], and optimal migration durations that fall on the interval T = [ τS , t]. Let M refer to the set of migrants returning by age t. Suppose that we observe the log of earnings, y i = logw h,i , where the true data generating process is given by the following log- linear specifi cation: 5 y i = γ h + γ 0s h s i + γ 1 h τ i + γ 2 h s i τ i For simplicity, we ignore covariates and suppose that all skill is unobserved. Then such a data generating process will leave us with observations of log- earnings, and migration durations for those individuals that migrate. A natural empirical strategy to learn about the relationship between migration experience and earnings is to regress log- earnings on a dummy for any migration experience and a measure of migration experience which is zero for nonmigrants: 6 y i = δ + δ 1 1 τ i 0 + δ 2 τ i + u i Where τ i is the optimal migration duration chosen by individual i we suppress the star notation here, and u i is an error term. What do the OLS estimates identify in this case? The point estimates of the parameters in Equation 6 can be decomposed into the fol- lowing summary statistics of observables and unobservables See the appendix for a derivation: 6 7 ˆ δ = y i | τ i =0 8 ˆ δ 1 = y i | i ∈M − y i | τ i =0 − τ i | i ∈M ˆ δ 2 9 ˆδ 2 = γ 1 h + γ 2 h s i | i ∈M Avg. effect for return migs. . + γ 0s h Covs i , τ i | i ∈ M Var τ i | i ∈ M Negative + γ 2 h τ i | i ∈M Covs i , τ i | i ∈ M Var τ i | i ∈ M Sign depends on γ 2 h 6. Here y i | i ∈M indicates the sample mean of y conditional on being a migrant. Similarly, Covs i , τ i | i ∈ M indicates the sample covariance of skill and the optimal migration duration. We can now give an interpretation to the OLS coeffi cients in this simple regression model. The coeffi cient on U.S. migration experience, ˆδ 2 , has three parts. The fi rst part refl ects the true average causal effect of U.S. labor market experience on wages in Mexico for return migrants, our main parameter of interest. The average effect on re- turn migrants could be higher or lower than the average effect for the whole population, depending on the sign of γ 2 h and the pattern of extensive margin selection into return migration. If there is negative to intermediate selection into return migration relative to the whole population which is consistent with our fi ndings and those of the previous literature, and if γ 2 h , then the true average return for the migrants in the sample will be less than the population average. If this pattern of selection holds and γ 2 h , then the average return for migrants will be greater than the population average. The second and third terms in Equation 9 introduce bias in the OLS estimate of the average return to migration experience. Rearranging, we see that the bias- inducing terms equal γ 0s h + γ 2 h τ i | i ∈M Covs i , τ i | i ∈ M Varτ i | i ∈ M. Given our assump- tions on the parameters, Covs i , τ i | i ∈ M Varτ i | i ∈ M 0 so the sign of the bias will depend on the sign of γ 0s h + γ 2 h τ i | i ∈M . This represents the average partial derivative of log- earnings with respect to skill among return migrants, holding migra- tion durations fi xed. Intuitively, the relationship between skill and wages should be positive even fi xing migration durations, making the OLS bias negative. However, if there is a suffi ciently large degree of negative complementarity between skill and mi- gration experience γ 2 h and large, then this average could be negative and the OLS bias could be positive. Suppose that skill in the model were perfectly measured by observed years of education. Then we could get positive bias only if the coeffi cient on the interaction between education and migration experience were greater in magnitude than 1 τ times the coeffi cient on schooling itself. In later empirical results, we will fi nd that the estimated coeffi cient on the interaction is far lower than this threshold value. Furthermore, if we make the assumption that there is no heterogeneity in the return to migration experience, we get: 10 ˆδ 2 = γ 1 h + γ 0s h Covs i , τ i | i ∈ M Var τ i | i ∈ M which indicates a negative bias. Our conclusion is that the endogeneity of τ is likely to bias the OLS estimate of δ 2 downward relative to γ 1 h . Under the assumptions made here, the model predicts that ˆδ 2 will provide a lower bound for the true average effect of migration experience on the earnings of return migrants. An important point related to the estimation of Equation 6 is that the return to mi- gration experience is identifi ed entirely from comparisons between return migrants. Indeed, the computation of ˆδ 2 does not involve any data on nonmigrants when other covariates are excluded from the regression. When both a dummy variable for any migration experience 1 τ 0, and the continuous experience measure τ are included, the coeffi cient on the dummy variable absorbs the difference in earnings due to the selectivity of migrants, since it is estimated by the difference in mean earnings be- tween return migrants and nonmigrants adjusting for estimated returns to migration. The coeffi cient on τ is still potentially biased, but only because migrants with different levels of skill choose different migration durations. Migrants could be selected from any portion of the skill distribution. They could be less- skilled or more- skilled, on average, than nonmigrants. However, as long as more- skilled migrants choose shorter trips, this generates a downward bias in the OLS estimate of γ 1 h . In this sense, the bias does not result from the selectivity of return migrants, but from the endogeneity of accumulated migration experience.

V. Data and Descriptive Statistics