# A conceptual framework for empirical analysis of migration (part 2: comparative statics, multiple matrices)

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

The questions discussed in this post are often difficult or impossible to resolve empirically, because one or more of the scenarios being compared is counterfactual. Techniques used include comparison of different time periods or different regimes. Regression analysis may be used to isolate the relevant factors. Conclusions drawn here are suspect even if the data collected is impeccable, because the theoretical model used for analysis may be invalid.

The simplest form of comparison is to consider the indicator values for various (source country, target country) pairs under the different possible migration policy regimes, and compare corresponding indicator values between the two regimes. For instance, how do French natives who stay in France under the pre-EU migration policy regime compare with French natives who stay in France under the EU migration policy regime?

Mathematical digression: multiple matrices

The earlier static framework considered a single matrix that encapsulated information on the performance of migrants as well as people who stay put for various source and target countries. Now, we’re trying to compare different scenarios. Now, each scenario has its own matrix. Our goal then is to compare the entry in one matrix with the corresponding entry in another matrix. In some cases, what we’re interested in is not a single entry, but a weighted average, or ratio, or difference, of entries. We then compute and compare that expression for the different countries.

For instance, consider the three-country scenario with France, Germany and the UK again (from part 1). Now, consider two policy regimes: the pre-EU regime and the EU regime. These are qualitatively different regimes: in the former, migration between the countries is not completely free, so there are stronger selection effects for migrants. Therefore, the matrices for the two regimes are probably different.

Suppose the matrix with the pre-EU regime is as follows (the superscript $latex {}^o$ is not an exponent, but indicates that the matrix refers to indicator values under the old policy regime):

$latex \begin{pmatrix} x^o_{11} & x^o_{12} & x^o_{13} \\ x^o_{21} & x^o_{22} & x^o_{23} \\ x^o_{31} & x^o_{32} & x^o_{33} \\\end{pmatrix}$

and the matrix with the EU regime is as follows (the superscript $latex {}^n$ is not an exponent, but indicates that the matrix refers to indicator values under the new policy regime):

$latex \begin{pmatrix} x^n_{11} & x^n_{12} & x^n_{13} \\ x^n_{21} & x^n_{22} & x^n_{23} \\ x^n_{31} & x^n_{32} & x^n_{33} \\\end{pmatrix}$

We can then compare the entries. For instance:

• The comparison of $latex x^o_{11}$ and $latex x^n_{11}$ reveals how the French who stay in France under the pre-EU regime compare with the French who stay in France under the EU regime.
• The comparison of $latex x^o_{12}$ and $latex x^n_{12}$ reveals how the people from France and in Germany under the pre-EU regime compare with the people from France and in Germany under the EU regime.
• The comparison of $latex x^o_{13}$ and $latex x^n_{13}$ reveals how the people from France and in the UK under the pre-EU regime compare with the people from France and and in Germany under the EU regime.

End mathematical digression

Note that any such comparison between different policy regimes has two components:

• Selection effect: The set of people in each of the categories is different under the two regimes. In particular, people who might not have been able to migrate under the pre-EU regime can migrate under the EU regime. Thus, even if the indicator value is the same between the two regimes for every individual (i.e., the changes to migration patterns don’t actually affect how any individual performs on the indicator), the difference in the labels means a different matrix for the two regimes.
• Treatment effect: The marginal migrants under the new policy experience changes relative to what they would have if they had stayed put, and they may also influence the indicator values for the people who stay put, or the others who would have migrated under the old regime as well.

Separating the selection and treatment effects requires us to consider separate matrices of indicator values using groupings from one regime, but measurements from the other regime. For instance, we ask: how do the people who would have stayed in France under the EU migration policy regime fare under the non-EU migration policy regime? We then compare these matrices to the matrices where the grouping and performance are measured for the same regime.

Mathematical digression: the matrices that use grouping and indicator values from different regimes

We continue with our three-country representation: country 1 (France), country 2 (Germany) and country 3 (the UK). Recall that the superscript $latex {}^o$ was used for the old policy regime (the pre-EU regime) and the superscript $latex {}^n$ was used for the new policy regime. We now consider some new matrices that can be constructed in principle but are hard to measure because they require a mix of information about two policy regimes.

Consider the matrix that uses grouping from the EU regime but indicator values from the pre-EU regime, denoted with superscript $latex {}^{n,o}$.

$latex \begin{pmatrix} x^{n,o}_{11} & x^{n,o}_{12} & x^{n,o}_{13} \\ x^{n,o}_{21} & x^{n,o}_{22} & x^{n,o}_{23} \\ x^{n,o}_{31} & x^{n,o}_{32} & x^{n,o}_{33} \\\end{pmatrix}$

The matrix is interpreted as follows: it represents the average values of the indicators under the pre-EU regime but using the groupings under the EU regime. For instance, the entry $latex x^{n,o}_{12}$ measures how the people who would migrate from France to Germany under the EU regime fare under the pre-EU regime. We can similarly consider another matrix with entries denoted $latex x^{o,n}$ that uses the groupings from the pre-EU regime but the indicator values from the EU regime. Entry comparisons between the four matrices reveal different types of information. The various combinations are discussed below:

• A direct comparison of $latex x^o$ and $latex x^n$ is comparing different regimes, using the grouping for each regime when considering it. This incorporates both a compositional selection effect arising from the difference in grouping and the treatment effect arising from a different set of people being able to migrate, affecting themselves and others.
• The comparison of $latex x^n$ and $latex x^{n,o}$ isolates for the treatment effect using the grouping of the new regime.
• The comparison of $latex x^n$ and $latex x^{o,n}$ isolates for the selection effect using the grouping of the new regime.
• The comparison of $latex x^o$ and $latex x^{o,n}$ isolates for the treatment effect using the grouping of the old regime.
• The comparison of $latex x^o$ and $latex x^{n,o}$ isolates for the selection effect using the grouping of the old regime.

End mathematical digression

#### Changes in weights

The number of migrants, as well as the number of non-migrants, differs under the various policy regimes. Therefore, the weights needed to take a weighted average (when computing average indicators — “per natural” for people born in a country or “per resident” for people living in a country) differ between the policy regimes.

Mathematical digression

The choice of weights depends on the grouping, so $latex x^n$ and $latex x^{n,o}$ use the same weights as each other, whereas $latex x^o$ and $latex x^{o,n}$ use the same weights as each other, but different from the other two.

End mathematical digression

#### Same set of people in the two regimes?

One of the points we’ve elided somewhat in our framing above is that we’re assuming that the set of people is the same in both regimes, and in fact, that the set of naturals for each country (i.e., the set of people with that source country) is the same in both regimes. What differs between the regimes is what country they land up in (the compositional selection effect) and how this affects the value of the indicator for them (the treatment effect).

But the assumption that the set of people itself is the same doesn’t actually hold water. People have children, and their decision of whether or not to migrate affects the identity and affiliation of the children. It might also affect how many children they have. Similarly, people may die, and migration policies may affect how long people live. We’re abstracting away from these issues for now, but will return to them in parts 5 and 6, before we start applying the framework in earnest to real-world migration questions.

#### Different normative perspectives

The individualist utilitarian universalist perspective is concerned with the weighted average of the indicator over the whole matrix for the two different policy regimes.

Once we leave the utilitarian universalist perspective, however, we have a bewildering array of normative choices. There are three big dimensions to the normative choices:

1. The dimension of what particular indicator or weighted combination of indicators we care about. One may care about:
• A particular (source country, target country) combination.
• All naturals of a country (all people with that source country, including those who stay and those who leave).
• All residents of a country (all people with that target country, including natives and immigrants).
• All immigrants to a country.
• All emigrants from a country.
2. The method used for grouping:
• We could use, for each regime, the grouping of that regime. For instance, we could compare the performance on indicator X of the French who stay in France under the EU regime, with the performance on indicator X of the French who stay in France under the pre-EU regime. This is problematic because selection effects can lead to the compositional effects paradoxes where all individuals are better off but some indicators still get worse due to the change in grouping. Territorialism has this flavor in practice, though it could in principle be of the other type below.
• We could privilege a particular regime to determine the grouping. For instance, we could say “I’m interested in maximizing the welfare of the set of people who would be French natives staying in France under the pre-EU regime, regardless of where they go under the EU regime.” Citizenism, though it isn’t exactly in this framework (since it favors citizenship and not necessarily birthplace) has this flavor: citizenists explicitly reject changing the idea of “who are we” in the face of new migration policy when deciding ex ante what policy regime is favorable.
3. Whether one looks at only a single instance, or at all. For instance, we could imagine somebody who cares about French natives only, or German natives only, versus somebody who cares about “natives” as a reference class, or “whoever gets to be resident in a country” as what we’re trying to improve, for each country. This could well be universalist (if the set of things we care about encompass everybody) and yet be different from individualistic utilitarian universalism, because we care about averages for particular groupings rather than about individuals qua individuals. While these different forms of universalism often agree, they don’t always do, thanks to compositional effects paradoxes.

#### First-order and second-order effects

The most direct treatment effect of migration is on migrants: they move to a new place, and experience a new environment. Assuming that migrants are a relatively small share relative to both their source and target countries, this effect will dominate at a per capita level, though possibly not at the aggregate (total) level.

An indirect, second-order, treatment effect is on the natives of the sending countries and receiving countries. Migrants leave the sending countries, thereby changing the nature of the society in these countries. They enter the receiving countries. and similarly change the societies there. Effects here are likely to be small on a per capita basis, but comparable in the aggregate to the effects on migrants themselves.

Note also that individual migrants affect other migrants, because a lot of migrants interact with fellow migrants to a greater extent than would be predicted by their proportion in the population. There is some terminological ambiguity on whether to consider this a first-order or a second-order effect. On the one hand, it’s an effect directly experienced by “migrants” as a class. On the other hand, it is an effect that people’s migration has on other migrants. This idea is closely related to diaspora dynamics, and we’ll get to it somewhere in parts 5 and 6.

#### Crossed dependencies: how the migration policy regime of one country affects migration between other pairs of countries

When we talk of a particular policy regime or scenario, we’re talking of a particular combination of immigration and emigration policy regimes for all countries. For any given country, its own migration policy is the most relevant when considering migration flows to and from that country. But the migration policies of other countries matter too:

• The immigration policies of countries that may receive migrants from the country, and the emigration policies of the countries that may send migrants to the country, matter.
• The immigration policies of countries that may “compete” with the given country for migrants also matter. Similarly, the emigration policies of countries that may compete with the country for sending migrants to a third country also matter.

To complicate matters even further, migration policies of countries are often linked with each other based on reciprocity and multilateral agreements (the EU is one example; temporary visa programs around the world are another).

#### Policies not directly related to migration affect migration

In a sense, all policies are relevant to migration, because they affect the economic, social, and cultural indicators of the country, and these in turn affect how attractive a destination it is for potential migrants. Some policies more directly affect migrants. For instance, high minimum wage laws might deter migration from places where workers are unlikely to have sufficient skills to get jobs that command the high minimum wage.

Vipul Naik founded the Open Borders website in March 2012. See also: