# Comparing and Proposing Methods for Estimating Bilateral Migration Flows

Thibault Laurent, Gautier Debaud, Paula Margaretic, Zaineb Smida,
Christine Thomas-Agnan

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Thibault Laurent, Gautier Debaud, Paula Margaretic, Zaineb Smida, Christine Thomas-Agnan. Comparing and Proposing Methods for Estimating Bilateral Migration Flows. Journées de Statistique, Société Française de Statistique, Jun 2025, Marseille, France. hal-05028780

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Distributed under a Creative Commons Attribution 4.0 International License
# COMPARING AND PROPOSING METHODS FOR ESTIMATING BILATERAL MIGRATION FLOWS

Thibault Laurent,¹ Gautier Debaud,² Paula Margaretic,³ & Zaineb Smida,⁴ & Christine Thomas-Agnan⁵

¹ Toulouse School of Economics, CNRS, France, Thibault.Laurent@tse-fr.eu

² GAEL, Université Grenoble Alpes, France, Gautier.Debaud@grenoble-inp.fr

³ University of Chile, Chili, paumargaretic@gmail.com

⁴ Univ Lyon, INSA Lyon, UJM, UCBL, ECL, CNRS UMR 5208, ICJ, F-69621, France, zaineb.smida@insa-lyon.fr

⁵ Toulouse School of Economics, France, Christine.Thomas@tse-fr.eu

**Résumé.** Les données sur les flux migratoires entre paires de pays sont peu accessibles, malgré leur importance capitale pour les chercheurs — notamment en économie, sociologie et démographie — ainsi que pour l’élaboration de politiques publiques efficaces. Face à cette problématique, la seule solution disponible consiste à utiliser des estimations basées sur les données de stocks bilatéraux de migrants fournies par la Banque mondiale et les Nations Unies. Abel and Cohen (2019) ont recensé au moins cinq méthodes d’estimation possibles, mais aucune étude n’a encore effectué une comparaison exhaustive de ces méthodes. Dans cet article, nous évaluons et comparons ces approches à l’aide de simulations, tout en montrant que certains résultats peuvent être établis de manière théorique. Après avoir mis en évidence l’importance des ajustements démographiques pour chaque méthode, nous proposons une nouvelle approche d’estimation qui intègre le taux de retour dans la méthode de différence des stocks. Nos résultats montrent que cette méthode est comparable à la méthode Pseudo-Bayes, la plus couramment utilisée, et pourrait même offrir de meilleurs résultats si des données auxiliaires sur les pourcentages de retours migratoires entre paires de pays étaient disponibles.

**Mots-clés.** Données de migration, Méthodes d’estimation, Simulation, Statistique publique.

**Abstract.** Reliable data on international migration flows remain limited, despite their crucial importance for researchers—spanning fields such as economics, sociology, demography, and epidemiology—and for the formulation of effective public policies. Understanding the drivers of migration, predicting flow patterns, and assessing their consequences are therefore significant challenges. To tackle this issue, at least five synthetic estimators have been developed using bilateral migrant stock data from the World Bank and the United Nations (Abel and Cohen, 2019). However, comprehensive comparisons of these methods are scarce. In this paper, we evaluate and compare these approaches through simulations, while also demonstrating that some results can be theoretically established. After highlighting the importance of demographic adjustments across all methods and underscoring the need for a standardized framework, we introduce a new estimation method that combines stock differencing with migration rates, offering a balance of flexibility, simplicity, and accuracy. Our results suggest that this approach performs as well as the Pseudo-Bayes method and may even
produce superior results if auxiliary data on return migration percentages between country
pairs were available.

**Keywords.** Migration data, Estimation methods, Statistical simulation, Public statistics.

# 1 Introduction

Migration flows between pairs of countries present significant challenges due to various factors, including economic disparities, political instability, and social networks that influence migration patterns. Countries with large income gaps often experience high levels of outward migration, while those with better economic opportunities attract migrants. Political conflicts or environmental factors like climate change also play a crucial role in shaping migration trends. Climate shifts and the resulting disruptions are likely to have severe consequences for human populations, forcing some, particularly the most vulnerable, to migrate. Moreover, migration policies, border control mechanisms, and international agreements further complicate the dynamics between countries. Understanding these flows is essential for designing effective public policies that address the economic, social, and humanitarian needs of both sending and receiving nations.

Although some statistical offices and international organizations provide data on mi-
gration trends, these remain incomplete. For example, the UNHCR (United Nations High
Commissioner for Refugees) publishes annual statistics on new asylum seekers and refugees.
Meanwhile, the Organization for Economic Co-operation and Development (OECD) compiles
annual data on labor migration, but only for its member countries, significantly limiting the
scope of possible analyses.

Therefore, these sources cover only specific segments of global mobility, making it difficult
to study migration trends as a whole, particularly in relation to climate change. Understand-
ing how environmental factors influence migration decisions requires a more comprehensive
dataset that captures global population movements. In the absence of direct data on mi-
gration flows, researchers must rely on estimates derived from bilateral migrant stock tables
published by the World Bank and the United Nations.

Abel and Cohen (2019, 2022) introduced a series of methods for estimating bilateral mi-
gration flows that cover more than 200 countries and are detailed by sex. Although these
estimators are widely used in migration studies, choosing the most appropriate method re-
mains a challenge. The literature compares different estimation techniques, but there is no
clear consensus on the best approach depending on the research question. Moreover, data
selection can significantly impact study results, particularly when analyzing climate-induced
migration. For example, Helbling et al. (2023) show that the use of one estimation method
over another can lead to divergent conclusions regarding the relationship between climate
change and migration flows.

In this study, we critically assess the methods proposed by Abel and Cohen (2019) and an-
analyze their estimation errors using a systematic simulation approach. Our preliminary results show that errors vary depending on the type of migration, with some methods performing better in specific contexts. We also identify certain limitations in existing approaches, particularly the selective application of demographic adjustments, which are used in minimization and pseudo-Bayesian estimators but not incorporated into other methods. In response, we propose a new estimation approach combining stock differencing and migration rates, aiming to balance flexibility, simplicity, and accuracy.

# 2 Notations

We aim to estimate a flow matrix using stock data available through the United Nations. In this section, we first establish the framework of the study and the main notations used.

## 2.1 Migration flow matrices

Consider individuals who may choose to migrate between four countries A, B, C and D over a given period $[t_0, t_1]$. Since direct migration flow data are unavailable, our goal is to estimate the corresponding migration flow matrix $F$:

$$F = \begin{pmatrix} A & B & C & D \\ A & F_{AA} & F_{AB} & F_{AC} & F_{AD} \\ B & F_{BA} & F_{BB} & F_{BC} & F_{BD} \\ C & F_{CA} & F_{CB} & F_{CC} & F_{CD} \\ D & F_{DA} & F_{DB} & F_{DC} & F_{DD} \end{pmatrix}, \quad (1)$$

where $F_{ij}$ is the flow from the country of origin $i$, to the country of destination $j$, over the chosen period.

Every individual involved in these migration flows is necessarily born in one of the four countries A, B, C or D. Consequently, the total migration flow matrix $F$ can be expressed as the sum of flow matrices by country of birth $F = F^A + F^B + F^C + F^D$, where $F^k$ represents the migration flow matrix for individuals born in country $F^k$, $k \in \{A, B, C, D\}$. Moreover, their elements belong to one of the four categories shown in the diagram below.

<figure><img src="image_2.png" /><figcaption>Figure 1: Decomposition of Flows by Country of Birth Into Four Categories</figcaption></figure>

More precisely, for $(i, j, k) \in \{A, B, C, D\}^3$:
## 2.2 Matrices of migrant stocks

* The stayers, denoted $Z_{ii}^k$, represent individuals born in $k$ who remained in country $i$ throughout the period considered.

* The outwards migrants, denoted $O_{kj}^k$, represent individuals born in $k$ who migrated to another country $j$ during the period considered.

* The returnees, denoted $R_{ik}^k$, represent individuals born in $k$ who previously lived in country $i$ and returned to $k$ over the period considered.

* The transit migrants, denoted $T_{ij}^k$, represent individuals born in $k$ who moved from country $i \neq k$ to country $j \neq k$ during the period under consideration.

Using this decomposition, we can express the general form of the flow matrices by country of birth as follows:

$$F^A = \begin{pmatrix} A & B & C & D \\ Z_{AA}^A & O_{AB}^A & O_{AC}^A & O_{AD}^A \\ R_{BA}^A & Z_{BB}^A & T_{BC}^A & T_{BD}^A \\ R_{CA}^A & T_{CB}^A & Z_{CC}^A & T_{CD}^A \\ R_{DA}^A & T_{DB}^A & T_{DC}^A & Z_{DD}^A \end{pmatrix}$$

$$F^C = \begin{pmatrix} A & B & C & D \\ Z_{AA}^C & T_{AB}^C & R_{AC}^C & T_{AD}^C \\ T_{BA}^C & Z_{BB}^C & R_{BC}^C & T_{BD}^C \\ O_{CA}^C & O_{CB}^C & Z_{CC}^C & O_{CD}^C \\ T_{DA}^C & T_{DB}^C & R_{DC}^C & Z_{DD}^C \end{pmatrix}$$

$$F^B = \begin{pmatrix} A & B & C & D \\ Z_{AA}^B & O_{BA}^B & T_{CA}^B & T_{AD}^B \\ R_{AB}^B & Z_{BB}^B & R_{CB}^B & Z_{CC}^B & T_{CD}^B \\ T_{DA}^B & R_{DB}^B & T_{DC}^B & Z_{DD}^B \end{pmatrix}$$

$$F^D = \begin{pmatrix} A & B & C & D \\ Z_{AA}^D & T_{AB}^D & R_{AC}^D & R_{AD}^D \\ T_{BA}^D & Z_{BB}^D & T_{BC}^D & R_{BD}^D \\ T_{CA}^D & T_{CB}^D & Z_{CC}^D & R_{CD}^D \\ O_{DA}^D & O_{DB}^D & O_{DC}^D & Z_{DD}^D \end{pmatrix}$$

As with the global flow matrix, terms in the same row have the same country of origin, those in the same column the same country of destination. The country of birth is added as a superscript. Naturally, departures (O), returns (R) and transits (T) will not be placed in the same way depending on the country of birth of the individuals observed.

All these flows can be linked with the following system of equations:

$$F_{ij} = \begin{cases} F_{ij}^i + F_{ij}^j + \sum_{k \neq \{i,j\}} F_{ij}^k = O_{ij}^i + R_{ij}^j + \sum_{k \neq \{i,j\}} T_{ij}^k & \text{if } i \neq j \\ \sum_k F_{ii}^k = \sum_k Z_{ii}^k & \text{if } i = j \end{cases}$$

## 2.2 Matrices of migrant stocks

To estimate flows, all methods rely on stock matrices. For four countries, these matrices take the following form:

$$S(t) = \begin{pmatrix} A & B & C & D \\ S_A^A(t) & S_B^A(t) & S_C^A(t) & S_D^A(t) \\ B & S_A^B(t) & S_B^B(t) & S_C^B(t) \\ C & S_A^C(t) & S_B^C(t) & S_C^C(t) \\ D & S_A^D(t) & S_B^D(t) & S_C^D(t) \\ TR_A(t) & TR_B(t) & TR_C(t) & TR_D(t) \end{pmatrix} \begin{pmatrix} TB^A(t) \\ TB^B(t) \\ TB^C(t) \\ TB^D(t) \\ N(t) \end{pmatrix} \quad (2)$$
where $S_j^i(t)$ is the number of people born in $i$ and living in $j$ at time $t$, $TB^i(t)$ is the number of people born in $i$ who are still alive at time $t$, $TR_j(t)$ is the number of people living in $j$ at time $t$, and $N(t)$ is the total number of people in the system at time $t$.

It is possible to express stocks using stayers, outwards migrants, returnees, and transit migrants. We apply this approach at two points in time, $t_0$ and $t_1$. The stocks of individuals born in $i$ and living in $j \neq i$ at time $t_0$ can be obtained by summing the individuals who: remain in $j$ at $t_1$, return to $i$ at $t_1$, or transit to another country $k \notin i, j$ at $t_1$. The stocks of individuals born in $i$ and living in $i$ at time $t_0$ are obtained by summing the individuals who: remain in $i$ at $t_1$ or leave for another country $k \neq i$ at $t_1$. Thus, we can write:

$$S_j^i(t_0) = \begin{cases} Z_{jj}^i + R_{ji}^i + \sum_{k \neq i, j} T_{jk}^i & \text{if } i \neq j \\ Z_{ii}^i + \sum_{k \neq i} O_{ik}^i & \text{if } i = j \end{cases}$$

The stocks of people born in $i$ and living in $j \neq i$ at time $t_1$ are obtained by summing the people who: are in $j$ at $t_0$ and remain there, are in $i$ at $t_0$ and leave for $j$, or are in another country $k \notin i, j$ at $t_0$ and transit to $j$. The stocks of people born in $i$ and living in $i$ at time $t_1$ are obtained by summing the people who: are in $i$ at $t_0$ and remain there, or are in another country $k \neq i$ at $t_0$ and return to $i$. Thus, we can write:

$$S_j^i(t_1) = \begin{cases} Z_{jj}^i + O_{ij}^i + \sum_{k \neq i, j} T_{kj}^i & \text{if } i \neq j \\ Z_{ii}^i + \sum_{k \neq i} R_{ki}^i & \text{if } i = j \end{cases}$$

Using the previous decompositions of flows and stocks, we can show that, on the one hand, the elements of the $k^{th}$ row of $S(t_0)$ are the row margins of the matrix $F^k$, and on the other hand, the elements of the $k^{th}$ column of $S(t_1)$ are the column margins of the matrix $F^k$. For example, if we add the margins of $F^k$:

$$F^A = \begin{array}{c} \begin{array}{c} A \\ B \\ C \\ D \end{array} \\ \begin{array}{c} Z_{AA}^A \\ R_{BA}^A \\ R_{CA}^A \\ R_{DA}^A \\ S_A^A(t_1) \end{array} \end{array} \begin{array}{c} \begin{array}{c} B \\ C \\ D \end{array} \\ \begin{array}{c} O_{AB}^A \\ Z_{BB}^A \\ T_{CB}^A \\ T_{DB}^A \\ S_B^A(t_1) \end{array} \end{array} \begin{array}{c} \begin{array}{c} C \\ D \end{array} \\ \begin{array}{c} O_{AC}^A \\ T_{BC}^A \\ Z_{CC}^A \\ T_{DC}^A \\ S_C^A(t_1) \end{array} \end{array} \begin{array}{c} \begin{array}{c} D \\ E \end{array} \\ \begin{array}{c} O_{AD}^A \\ T_{BD}^A \\ T_{CD}^A \\ Z_{DD}^A \\ S_D^A(t_1) \end{array} \end{array}$$

# 3 Estimation method

Building on the relationships between flows and stocks outlined above, several methods can be employed to estimate migration flows. To the best of our knowledge, five primary estimation methods are commonly used in migration studies, categorized into three groups: Stock differencing methods, migration rate-based methods and Model-based methods. We implement a new method that combines stock differencing with migration rates. Due to space constraints, we illustrate only the method based on the difference of stock matrices. It relies
on the difference $(\Delta S)_j^i = S_j^i(t_1) - S_j^i(t_0)$ in the size of migrant stocks at the start and end of a given period to estimate the corresponding migration flow. This difference can also be viewed as the net result of inflows and outflows in $j$ for individuals born in $i$.

$$ (\Delta S)_j^i = \begin{cases} (O_{ij}^i + \sum_{k \neq \{i,j\}} T_{kj}^i) - (R_{ji}^i + \sum_{k \neq \{i,j\}} T_{jk}^i) = \text{Inflows}_j^i - \text{Outflows}_j^i, & \text{if } i \neq j \\ \sum_{k \neq i} (R_{ki}^i - O_{ik}^i), & \text{if } i = j \end{cases} \quad (3) $$

where $\text{Inflows}_j^i$ refers to all individuals born in $i$ who move to $j$, whether from $i$ itself or from other countries, and $\text{Outflows}_j^i$ refers to all individuals born in $i$ who leave $j$, whether to return to $i$ or to transit through other locations. In the *drop negative* method, the flow from $i$ to $j$ ($i \neq j$) is estimated by calculating the difference in stocks over the period of interest. Any negative differences are treated as zero flows and are set to zero:

$$ \hat{F}_{ij}^{\text{drop}} = (\Delta S)_j^i \times \mathbb{1}_{(\Delta S)_j^i > 0}. \quad (4) $$

The residuals $F_{ij} - \hat{F}_{ij}$ can be expressed in terms of stayers, outflows, returns, and transits. After simplification, the following cases emerge:

$$ \epsilon_{ij}^{\text{drop}} = \begin{cases} R_{ij}^j + R_{ji}^i + \sum_{k \neq \{i,j\}} T_{ij}^k - \sum_{k \neq \{i,j\}} (T_{kj}^i - T_{jk}^i) & \text{pour } (\Delta S)_j^i > 0 \\ O_{ij}^i + R_{ij}^j + \sum_{k \neq \{i,j\}} T_{ij}^k & \text{pour } (\Delta S)_j^i \le 0 \end{cases} $$

# 4 Simulation results

We run 100 simulations using parallel programming and graphically compare results:

<figure><img src="image_3.png" /><figcaption>Figure 2: Boxplot MSE and Overestimation/Underestimation</figcaption></figure>

Looking at the MSE boxplot, it is clear that the *pseudo-Bayes* method performs the best, with the *return coefficients* method trailing closely behind. The other methods appear fairly
similar to one another, but with much larger errors compared to these top two. Methods
like *Drop*, *reverse*, *migration rates*, and *minimization* tend to underestimate almost all flows,
while *return coefficients* and *pseudo-Bayes* may either overestimate or underestimate them.
Interestingly, the flows where *return coefficients* and *pseudo-Bayes* perform less well tend
to be those they overestimate—this is not particularly surprising, given that these flows are
relatively small in this initial example.

References

Abel, G. J. and Cohen, J. E. (2019). Bilateral international migration flow estimates for 200 countries. *Scientific data*, 6(1):82.

Abel, G. J. and Cohen, J. E. (2022). Bilateral international migration flow estimates updated and refined by sex. *Scientific Data*, 9(1):1–11.

Helbling, M., Rybski, D., Schewe, J., Siedentop, S., Glockmann, M., Heider, B., Jones, B., Meierrieks, D., Rikani, A., and Stroms, P. (2023). Measuring the effect of climate change on migration flows: Limitations of existing data and analytical frameworks. *PLOS Climate*, 2(1):e0000078.