IV regressions. The OLS estimation of model (19) rests on the assumption that all covariates are independent of "t. Nevertheless, it might be argued that some variables might depend on fertility, invalidating this assumption. In particular, higher fertility rates should increase labor supply and depress wages in domestic countries, a¤ecting international migration. In other words, ln(p0) that captures the incentive channel might depend on the level of the home country fertility rate ln (nt) : In this case, reverse causality might a¤ect directly the quality of the estimation of a3 but also the ones of other coe¢ cients such as a1<sup>9</sup> . If such an e¤ect is quantitatively important, it might be desirable to carry out instrumental variable estimation. Note that since we are using migration shares across destination countries (�d) rather than stocks of migrants to build the norm variable; we can rule out any reverse causality running from ln (nt) to ln �n<sup>~~d~~</sup> t �.

Table 3 reports the IV estimates of model (19) for the whole sample of countries. We consider the following instruments of the (log of) emigration rate: a dummy variable for islands, the (log of the) size of the country measured by its surface (in squared kilometers) and (the log of the) distance to main destination of the migrants.<sup>10</sup> It is worth emphasizing that the two necessary conditions for instrumentation are ful...lled in our regressions. First, the ...rst stage estimation results indicate that we have strong instruments. The F statistics of the ...rst stage regressions are most of the time above 10. The partial correlations of the ...rst stage regression show that the 3 instruments explain a signi...cant part of the variability of emigration rates. Second, as suggested by the p-value of the Hansen overidenti...cation test, those instruments are found to be independent of the fertility rates. The test fails to reject the null hypothesis of independence between the instrument set and the error term.

The main ...ndings of the IV estimations are broadly similar to those of the OLS estimations. We ...nd evidence of a shifting norm e¤ect (a1) and reject the existence of an interaction e¤ect with the emigration intensity (a2). The elasticity of the transfer of norm also remains quite in line with the one estimated by the OLS regressions. It is however slightly lower for the sample of countries for which data of remittances are available. We ...nd moderate evidence of an incentive e¤ect of migration on fertility, as suggested by the estimate of a3 in the parsimonious speci...cation including the remittances. In this set up, the slightly positive impact of remittances on fertility rates is also con...rmed. On the whole, the impact of the other variables are in general in line with the one found in OLS regressions.

> 9Theoretically speaking, the existence of a reverse causality between migration and fertility implies that the interaction term (emigration rate times the fertility norm) should also be instrumented. Nevertheless, since this term is insigni...cant in OLS regressions, we focus on the instrumentation of the emigration rate only.

> 10The ...rst stage regressions of the IV estimations yield estimates that are in line with intuition. In particular, size is negatively associated to emigration rates. Islands are found to display higher migration rates and higher distance to main destination negatively a¤ects emigration rates. The estimation results are not reported here to save space but can be obtained upon request to the ...rst author.

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