# 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.

# A conceptual framework for empirical analysis of migration (part 1: direct empirical measurement)

This post is part 1 of a series outlining a conceptual framework for the empirical analysis of migration. Read the introductory post to the series here. This post focuses on a particular form of comparison that can be carried out through direct empirical measurement. The questions directly answered this way aren’t the ones we are usually most interested in. But at least these are questions for which we can obtain precise answers in principle. That’s a start.

Questions about how different groups of people compare for a given regime at a given point in time (or over an interval of time) can be answered by direct empirical measurement, at least for existing regimes. They cannot be directly answered for hypothetical regimes. But the fact that they can be answered at all differentiates them from other, more speculative, questions.

#### (Source country, target country) pairs as the basis of aggregation

The conceptual model we use identifies two attributes of a person: the person’s source country (also known as the sending country, and defined as the country that person was born in) and the person’s target country (also known as the receiving country or recipient country, and defined as the country the person now lives in). For non-migrants, the source and target country coincide. For migrants, the source and target country differ. For every individual, therefore, we can write down a (source country, target country) pair. For instance, somebody born in Mexico who stays in Mexico gets the pair (Mexico,Mexico). Somebody born in Nepal who moves to India gets the pair (Nepal,India). (This is obviously a very crude simplified model, because some people migrate temporarily, some migrate to one country and then to another, etc. But it’s good enough to get us started).

We’re interested in the performance on indicator X both for people who stay put in their countries, and for people with particular (source country, target country) combinations. For instance, we may be interested in asking: how does the (Nepal, India) combination fare on indicator X? Explicitly, that’s asking: how do people who are from Nepal and living in India perform on indicator X?

Mathematical digression: using a matrix representation to store the information

We can use a matrix representation where the rows correspond to source countries and the columns correspond to target countries (both rows and columns should be the same list of countries in the same order for the observations below to hold). The entry in a given cell provides information on indicator X about the collection of people whose source country is the row country and whose target country is the column country.

Let’s explicitly consider the case of three countries. Let’s say country 1 is France, country 2 is Germany, and country 3 is the United Kingdom. The indicator X values for these source and target countries can be codified via a matrix:

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

The nine entries are interpreted as follows:

• $latex x_{11}$ is the performance on indicator $latex X$ of the people in country 1 (France) who stay in France.
• $latex x_{12}$ is the performance on indicator $latex X$ of the people who migrate from country 1 (France) to country 2 (Germany).
• $latex x_{13}$ is the performance on indicator $latex X$ of the people who migrate from country 1 (France) to country 3 (the UK).
• $latex x_{21}$ is the performance on indicator $latex X$ of the people who migrate from country 2 (Germany) to country 1 (France).
• $latex x_{22}$ is the performance on indicator $latex X$ of the people in country 2 (Germany) who stay in Germany.
• $latex x_{23}$ is the performance on indicator $latex X$ of the people who migrate from country 2 (Germany) to country 3 (the UK).
• $latex x_{31}$ is the performance on indicator $latex X$ of the people who migrate from country 3 (the UK) to country 1 (France).
• $latex x_{32}$ is the performance on indicator $latex X$ of the people who migrate from country 3 (the UK) to country 2 (Germany).
• $latex x_{33}$ is the performance on indicator $latex X$ of the people in country 3 (the UK) who stay in the UK.

Note that the entries on the main diagonal (the one from top left to bottom left), namely $latex x_{11}$, $latex x_{22}$, and $latex x_{33}$, correspond to the non-migrants, i.e., the people who stay put in their country. The off-diagonal entries, i.e., the entries $latex x_{ij}, i \ne j$, correspond to migrants. In this case, there are six such entries: $latex x_{12}, x_{13}, x_{21}, x_{23}, x_{31}, x_{32}$.

The three countries in the example above weren’t ordered in any particular way, so there is no significance of an entry being above or below the diagonal. If the countries had been ordered based on some criterion (such as GDP (PPP) per capita), then the entries above and below the diagonal would reflect different types of migration based on whether the sending or receiving country had higher GDP (PPP) per capita.

The simplified example here considers migration between three countries. However, if we want to study migration worldwide, we’d need to include all countries. If there are 200 countries, then we’d have a $latex 200 \times 200$ matrix, with a total of 40,000 entries. In general, if there are $latex n$ countries, the matrix is a $latex n \times n$ matrix with a total of $latex n^2$ entries, of which there are $latex n$ diagonal entries (corresponding to the people who stay put in their respective countries) and $latex n^2 – n = n(n-1)$ off-diagonal entries (corresponding to people who migrate from one country to another). Half of them ($latex n(n – 1)/2$) are above the diagonal. and the other half are below the diagonal, but the above/below distinction is of importance only if the countries are ordered according to some criterion.

Now, there may be cases where migration between two countries is so quantitatively small, or even actually zero, that it’s not meaningful to compute that particular matrix entry. For instance, I think there is zero migration from North Korea to Somalia. So, some entries of the matrix are not defined. This means that we need to be careful if we intend to subject the matrix to techniques of linear algebra. However, we’re using the matrix only to store information, and we don’t perform matrix operations.

End mathematical digression

#### Totals versus averages

In some cases, we care about the per capita level of an indicator. This is usually the case for indicators such as GDP per capita, crime, or unemployment. In cases where fixed resources are being used up, however, we may care more about the total use. An example may be water use in a country that has a fairly limited water supply. If we’re concerned about total use, then in addition to knowing the per capita value on indicator X for (source country, target country) pairs, we also need to know the size of the population.

The relative size of different populations may matter even if we are concerned only about averages, because we need relative sizes to compute weighted averages.

#### Weighted averages for residents, naturals, immigrants, and emigrants

In some cases, we are interested not in a particular (source country, target country) combination, but in combining information for all people in a particular source or target country. The following are four typical weighted averages we are interested in. If we are looking at a total of $latex n$ countries, then there are $latex n$ weighted averages of each type (one for each country) and therefore a total of $latex 4n$ weighted averages to consider.

• The weighted average for all residents of a country, including natives of the country who stay put and migrants from other countries to that country.
• The weighted average for all naturals of a country, including natives of that country who stay put and people from that country who migrate to other countries.
• The weighted average for all immigrants to a country, i.e., people who have that as their target country but are from other source countries.
• The weighted average for all emigrants from a country, i.e., people who have that as their source country but now live in other countries.

#### Typical forms of comparison

After figuring out how various (source country, target country) combinations, or weighted averages thereof, fare, we can then ask how they compare with one another. Here are some typical questions that can be asked. We’ll use the letter A to denote a hypothetical source country and the letter B to denote a hypothetical target country, but you can replace these with concrete instances (such as France and the United Kingdom):

1. How do migrants from country A to country B compare with natives of country B (the target country) on indicator X?
2. How do migrants from country A to country B compare with natives of country A (the source country) on indicator X?
3. How do migrants to country B compare with resident natives of that country on X?
4. How do migrants from country A compare with resident natives of that country on X?
5. How do migrants from country A compare with the natives of the countries they go to on X (combined analysis for all countries they go to)?
6. How do migrants to country B compare with the natives of their source countries on X (combined analysis for all source countries)?
7. How do migrants in general compare with non-migrants in general on X?
8. How do natives of a country receiving migrants compare with natives of a country sending migrants on X? One advantage of this question is that it can be asked without collecting separate statistics on migrants, and can also be asked prior to implementation of migration policies, although the answer might change after implementation of the migration policies.

Mathematical digression: interpretation of the questions in matrix terms

Here is how each of the questions would look like in terms of the matrix representation. For illustrative purposes, we will continue to draw on the three-country setup above with country 1 as France, country 2 as Germany, and country 3 as the United Kingdom.

1. Compare a matrix entry with the diagonal entry in its column. If we’re interested in studying migration from the UK to France, we compare the entry $latex x_{31}$ (migrants from the UK to France) with the entries $latex x_{11}$ (French natives who stay put).
2. Compare a matrix entry with the diagonal entry in its row. If we’re interested in studying migration from the UK to France, we compare the entry $latex x_{31}$ (migrants from the UK to France) with the entries $latex x_{33}$ (UK natives who stay put).
3. Compare the (weighted) average of the off-diagonal entries in a column with the diagonal entry of that column. If we are interested in understanding migration to Germany, we need to compare the entries $latex x_{12}$ and $latex x_{32}$ (migrants from France and the UK to Germany) with $latex x_{22}$ (Germans who stay put). We would usually compute the average of $latex x_{12}$ and $latex x_{32}$ weighted by the respective population sizes.
4. Compare the (weighted) average of the off-diagonal entries in a row with the diagonal entry of that row. If we are interested in understanding migration from France, we need to compare the entries $latex x_{12}$ and $latex x_{13}$ (migrants from France to Germany and to the UK) with the entry $latex x_{11}$ (French who stay put).
5. A bunch of pairwise comparisons of the type seen in Question 1 (with pairs in the same column). If we’re interested in figuring out how migrants from France compare with the natives wherever they go. Then, we will compare $latex x_{12}$ with $latex x_{22}$ (comparing French migrants to Germany with Germans who stay put), and separately compare $latex x_{13}$ with $latex x_{33}$ (comparing French migrants to the UK with UK natives who stay put).
6. A bunch of pairwise comparisons of the type seen in Question 2 (with pairs in the same row). If we’re interested in figuring out how migrants to the UK fare relative to the natives of their source country. Then, we will compare $latex x_{13}$ with $latex x_{11}$ (French who move to the UK versus French who stay put), and separately compare $latex x_{23}$ with $latex x_{22}$ (Germans who move to the UK versus Germans who stay put).
7. The off-diagonal entries represent migrants, and the diagonal entries represents people who do not migrate. This question therefore involves a comparison of the off-diagonal entries and the diagonal entries.
8. This compares two diagonal entries. If we’re interested in comparing Germany and the UK, we’ll compare $latex x_{22}$ and $latex x_{33}$.

End mathematical digression

#### Remarks on selection and treatment effects

We’ll return to this in more depth in part 2, but here are a few preliminary remarks.

The significance of the migration policy regime and other aspects of the scenario (economic policies, economic performance, linguistic differences, etc.) on the indicator matrix is two-fold:

• A compositional selection effect (for short, we’ll call this a selection effect or a compositional effect) for the groupings, i.e., the choice of the migration policy scenario determines who migrates and who doesn’t, and therefore affects what set of people get included in various (source country, target country) pairs.
• A treatment effect for the groupings, i.e., some people being able to migrate affects their own performance on indicator X, and also affect the performance on the indicator of others who stay behind in their own countries.

In Part 2, we will look more closely at how to isolate selection and treatment effects when comparing different policy regimes.

#### Remarks on measurability

For existing policy regimes, the performance on particular indicators of particular (source country, target country) pairs can be computed in principle. Some methods involve complete measurement: for instance, census data that asks people to identify their country of origin, or computerized records of all residents along with their source country. Other methods involve the use of partial data along with sampling techniques to extrapolate to the general population.

Some challenges:

• In some cases, there is ambiguity, both conceptual and empirical, on the source country of individuals, or on what it means to be a resident (for instance, do we count crimes by tourists?)
• In some cases, people deliberately conceal or misrepresent information about themselves where the stakes are high. For instance, a foreign-born person may claim to be a native-born when arrested for a misdemeanor, in order to avoid deportation. On the other hand, those who prefer deportation to another country to spending time in prison may misrepresent themselves as foreign-born. People may lie to get access to welfare benefits. False identity documentation may be produced in order to be eligible to work.
• In some cases, the population involved is so small that the indicator cannot be measured from small samples of the overall population. For instance, there are about 100 people in the US who were born in North Korea. A random sample would probably not pick any of them. Even if it did, statistical averages for the population would not be robust.
• There are challenges when considering the comparability of indicators across different target countries (and in some cases even within a particular country), because different countries (and different jurisdictions within a country) use different protocols for measurement and have different sources of bias. For instance, the rate of crime reporting may differ considerably between countries, particularly for rape and minor theft. Similarly, when comparing income values, purchasing power parity estimates are not necessarily reliable.

#### Normative significance of comparisons

The measurements and comparisons here offer only a starting point for investigating the effects of migration: we’d need comparative statics between different regimes in order to tease out the effects of migration. We’ll talk about this more in part 2 and in part 3. But in many cases, our only reliable empirical measurements are the direct ones discussed here, and people often draw conclusions based on this evidence. The following are three typical styles of crude conclusion people draw.

• Immigrants to country B do better (respectively, worse) on the indicator than natives of country B who stay in their country $latex \implies$ immigration “good” (respectively, “bad”) for country B.
• Emigrants from country A do better (respectively, worse) on the indicator than natives of country A $latex \implies$ emigration “bad” (cf. brain drain) (respectively, “good”) for country A.
• Natives of country A worse on the indicator than natives of country B $latex \implies$ Migration from country A to country B good for country A and bad for country B.

Of course, put so bluntly, the claims seem obviously ill-substantiated, and they often break down in practice.

But apart from the need to do more sophisticated counterfactual analysis to actually talk about the effects of migration, there’s another important point: the overall levels of an indicator might matter more than how different groups compare on it. The relative crime rates of natives and migrants are not as important as knowing whether either group has a high crime rate. The relative fertility rates are similarly less important than the overall fertility level. Too much focus on the question of “are immigrants better than natives?” can lead us to ignore other questions of greater moral and practical relevance.