we interact the _Years since migration_ dummies with _Ri_ , a dummy variable for refugee status.<sup>23</sup> The coefficients _α_ 1 _j_ can then be interpreted as employment gaps between natives and refugees, for different categories _j_ of years since migration.

The vector _Xit_ contains individual characteristics (age, age squared, gender, marital status). We use a set of fixed effects that are introduced progressively in our estimations: fixed effects _φt_ control for current economic conditions in the host country, _φτ_ for heterogeneity among arrival cohorts, and _φa_ for heterogeneity in terms of the age structure of refugees upon arrival in Switzerland. Two remarks are in order regarding the identification of the integration trajectories. First, there is perfect collinearity for refugees between the year of arrival _τ_ , the number of years since migration ( _YSM_ ) and the current year _t_ . We identify the effects of _YSM_ and the year of arrival by assuming that the impact of the current year dummies ( _φt_ ) on employment rates is identical for refugees and natives. This seems a reasonable assumption in our case since the period 2010–2018 is characterized by small fluctuations of unemployment and per capita GDP in Switzerland, which is among the countries that have not been strongly affected by the global financial crisis. Second, it might seem that we assume in equation (1) that the impact of age on employment is identical for natives and refugees. This is, however, not the case due to the presence of fixed effects _φa_ , capturing the refugees’ age at arrival. There is perfect collinearity between the refugees’ age at arrival, their (current) age and the number of years since migration. Therefore, controlling for age at arrival is equivalent to controlling for the age of refugees separately from the age of natives. Third, there are no dummies for origin countries _c_ in equation (1). When comparing natives and refugees (or natives and migrants), this set of dummies adds up to the refugee dummy and is therefore perfectly collinear with the set of _YSM_ dummies. As a consequence, the vertical position of the employment gap profile cannot be identified if country of origin dummies are used. We address this question in the comparison between refugees and migrants, where we are able to account for heterogeneity in countries of origin.

The comparison between natives and refugees informs us if the employment rates of refugees catch up with similar natives in the long run, and how fast they converge to their long-run level. We then carry out a comparison between natives and migrants with a similar background as refugees, in order to check if the refugee status has a specific influence on employment trajectories.<sup>24</sup> We follow Fasani et al. (2020a) and select migrants from non EU-15 countries as the relevant comparison group. Although this choice excludes the largest migrant groups in Switzerland (Italy, Germany, France, Portugal, Spain etc.), the composition of non EU15 migrants and refugees still differs in terms of origin countries. To account for these differences, we finally proceed to a direct comparison between refugees and (non EU15) migrants. To do this, we estimate equation (1) for a sample including these two groups and add country of origin fixed effects to the estimating equation. One issue that arises in this estimation is the collinearity between the year of arrival, the number of years since migration and the current year. As all three variables are defined for migrants and refugees, we cannot address this problem in the same way as described above in the case of refugees and natives. Instead, we choose to drop the age at arrival dummies from the estimating equation.

A final issue concerns controls for education levels. In our main sample, which is drawn exclusively from administrative data, we do not observe individuals’ education levels, as mentioned above. However, for a subsample of natives and refugees, we are able to match the administrative dataset with data on education levels from the Swiss Structural Survey. Therefore we estimate all equations on the two samples: first, by using the main sample without education variables, second, by using the restricted sample, taking dummy variables for education levels into account. We also check if the restricted sample can be considered a random sub-sample of our main sample by estimating the equation using

> 23. To facilitate the interpretation of the estimation results, we do not drop a reference category for the _YSMitj_ dummies when we interact them with the refugee dummy _Ri_ . For this reason, the _Ri_ dummy itself does not appear in the equation since it is perfectly collinear with the interaction terms. 24. The comparison between natives and migrants is based on the same equation as (1), except that the refugee dummy is replaced by a dummy for migrant status.

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