This post is part 4 of a series outlining a conceptual framework for the empirical analysis of migration. Read the introductory post to the series here, part 1 here, part 2 here, and part 3 here.

#### Migrant performance as a combination of source and target country performance?

A simple model against which we could compare reality is that migrant performance is a function of the native performance in their source and target countries. In other words, if we knew the performance of source country natives and we knew the performance of target country natives, we would be able to predict how migrants perform.

Qualitatively, here are some possibilities:

- Migrant performance falls somewhere in between the performance of their source and target countries. For instance, perhaps the performance of migrants falls midway between the source and target countries. Note in particular that if the source and target countries have identical values for natives, then migrants are also identical to them, suggesting that there is no effect coming from
*being a migrant per se*. - Migrant performance is nearly identical to that of natives in the target country, and is independent of the source country.
- Migrant performance is nearly identical to that of natives in the source country, and is independent of the target country.
- Migrant performance is
*determined*by performance in the target country, but is not equal to it. For instance, perhaps migrants have incarceration rates that are 0.7 times the incarceration rates of natives in the target country, regardless of their source and target countries. - Migrant performance is
*determined*by performance in the source country, but is not equal to it. For instance, perhaps migrants have fertility rates that are 1.2 times those of their source countries, regardless of where they come from and where they go.

**Mathematical digression: a linear combination model**

As in part 1, denote by $latex x_{ij}$ the performance of migrants from country $latex i$ to country $latex j$ on indicator *X*, and denote by $latex x_{ii}$ and $latex x_{jj}$ respectively the performance of natives of the countries who stay put. We claim that, to a reasonable approximation, there is a (nice enough) function $latex F$, independent of $latex i$ and $latex j$, such that:

The simplest possible example of such a function is a linear combination. In this model, we have the following, where $latex \alpha$ and $latex \beta$ are nonnegative reals:

We now revisit the five cases above in terms of the linear combination model:

- $latex \alpha + \beta = 1$, i.e., the performance of migrants is a
*convex*combination of that of natives from the source and target countries, and therefore in particular lies somewhere in between those two values. In that case, we can write $latex x_{ij} = \alpha x_{ii} + (1 – \alpha) x_{jj}$. The special case $latex \alpha = 0.5$ is the one where migrant performance is midway between the natives of the source and target countries. - $latex \alpha$ is close to 0 and $latex \beta$ is close to 1.
- $latex \alpha$ is close to 1 and $latex \beta$ is close to 0.
- $latex \alpha$ is close to 0 and $latex \beta$ is positive but not close to 1.
- $latex \alpha$ is positive but not close to 1, and $latex \beta$ is close to 0.

Linear models are not the only ones possible: one can imagine more complicated functional relationships, including power relationships (which would be linear once you take the logarithm). Linear models are the ones people generally look for when predicting performance, and that’s what linear regressions are generally used for. Anyway, the best type of model to use depends on the type of indicator we have and what we understand about how it’s determined, i.e., we need a phenomenological story first (more on this later in the post).

**End mathematical digression**

#### Separating selection and treatment: potential migrant performance and actual migrant performance in terms of source and target country performance

The above discusses the performance of people *who actually migrate* in terms of their source and target countries. But, building on the discussion in part 2 and (more directly relevant) part 3, we’re also interested in how potential migrants would perform if they weren’t allowed to migrate. This allows us to separate out the selection and treatment effects.

Unlike the case of people who do migrate, it’s not *a priori* clear why the indicator value in the target country should be a predictor for people who don’t migrate. One argument that it should: the very fact that they are considering migration to the target country, or that a potential migration policy is considering them, suggests potential affinity with the target country. It may happen in some cases that the function doesn’t depend on $latex x_{jj}$ at all.

**Mathematical digression: two linear combinations**

To stay similar to the earlier notation (from parts 2 and 3 of the series), we denote the “how migrants would do if they were’t allowed to migrate” quantity as $latex x_{ij}^{n,o}$. We are thus interested in understanding the function $latex G$ such that:

$latex x_{ij}^{n,o} = G(x_{ii},x_{jj})$

The simple case is a linear function, i.e., we have:

$latex x_{ij}^{n,o} = \alpha^{n,o}x_{ii} + \beta^{n,o}x_{jj}$

We can now make cases based on the values of these numbers. We list some possibilities:

- Suppose $latex \alpha/\beta < \alpha^{n,o}/\beta^{n,o}$. This means that for people who do migrate, their performance is predicted more by the target country than if they were not allowed to migrate.
- $latex \beta^{n,o} = 0$ suggests that the performance of potential migrants, if they stay in their source country, is determined completely by their source country. In the case $latex \alpha^{n,o} = 1$, the potential migrants are indistinguishable on the indicator from others in their source country. In other cases, migrants differ from others in their source country, but by a constant factor.
- $latex \alpha = 0$ suggests that the performance of people who actually migrate is determined completely by their target country. In the case $latex \beta = 1$, the migrants become indistinguishable from natives of the target country. In other cases, they differ by a constant factor.
- If $latex \alpha < \alpha^{n,o}$ and $latex \beta < \beta^{n,o}$, that implies that migrants score lower on the indicator if they're allowed to migrate than if they're not, regardless of how the source and target country compare on the indicator. The opposite conclusion holds if $latex \alpha > \alpha^{n,o}$ and $latex \beta > \beta^{n,o}$.

**End mathematical digression**

#### Phenomenological stories

The above were purely mathematical models of migrant performance, and didn’t provide a story as to why a particular functional expression works, of why particular parameter values are right. But what’s going on? Why might we expect a functional relationship, linear or otherwise, between migrant performance and the performance of natives in the source and target countries?

Some possible stories:

- Migration policy explicitly selects for people based on how they fare relative to the native population of the recipient country, so that the similarity across countries between the relative performance between natives and migrants is largely because most countries’ migration policies revolve around similar explicit objectives in terms of how the migrants should compare with the natives.
- Immigrants self-select for countries where their performance will be at a particular level relative to natives.
- People self-select to emigrate if their performance relative to their source country is at a particular level relative to the natives of that source country.
- People’s intrinsic characteristics (that they transport with themselves when they migrate) only determine their performance
*relative*to where they live, rather than in absolute terms. For instance, a person’s inclination to criminality may determine how much crime the person commits*relative*to natives of the region. Similarly, a person’s skill level may determine how much money the person can earn*relative*to natives of whatever country he or she is in, rather than in absolute terms.

**Mathematical digression: translating the phenomenology to the linear combination model**

(1) and (2) explain the parameters $latex \beta^{n,o}$ and $latex \beta^n$. (3) explains the parameters $latex \alpha^{n,o}$ and $latex \alpha^n$. In the extreme case that (4) holds completely, $latex \alpha^{n,o} = \beta^n$ and $latex \beta^{n,o} = \alpha^n = 0$.

**End mathematical digression**