# Bringing Age Back In: Accounting for Population Age Distribution in
Forecasting Migration

Nathan G. Welch  Department of Statistics, University of Washington,Box
354322, Seattle, WA 98195-4322, USAEmail<EMAIL_ADDRESS>Hana Ševčíková  Center
for Statistics and the Social Sciences, University of Washington,Box 354322,
Seattle, WA 98195-4320, USAEmail<EMAIL_ADDRESS>Adrian E. Raftery  Departments
of Statistics and Sociology, University of Washington,Box 354322, Seattle, WA
98195-4322, USAEmail<EMAIL_ADDRESS>
## Abstract

The link between age and migration propensity is long established, but
existing models of country-level net migration ignore the effect of population
age distribution on past and projected migration rates. We propose a method to
estimate and forecast international net migration rates for the 200 most
populous countries, taking account of changes in population age structure. We
use age-standardized estimates of country-level net migration rates and in-
migration rates over quinquennial periods from 1990 through 2020 to decompose
past net migration rates into in-migration rates and out-migration rates. We
then recalculate historic migration rates on a scale that removes the
influence of the population age distribution. This is done by scaling past and
projected migration rates in terms of a reference population and period. We
show that this can be done very simply, using a quantity we call the migration
age structure index (MASI). We use a Bayesian hierarchical model to generate
joint probabilistic forecasts of total and age- and sex- specific net
migration rates over five-year periods for all countries from 2020 through
2100. We find that accounting for population age structure in historic and
forecast net migration rates leads to narrower prediction intervals by the end
of the century for most countries. Also, applying a Rogers & Castro-like
migration age schedule to migration outflows reduces uncertainty in population
pyramid forecasts. Finally, accounting for population age structure leads to
less out-migration among countries with rapidly aging populations that are
forecast to contract most rapidly by the end of the century. This leads to
less drastic population declines than are forecast without accounting for
population age structure.

keywords: migration, probabilistic, population, projection, forecast

##  1 Introduction

Preparations for the next great demographic era are underway around the world
(CJ. Kim, 2019; Director of National Intelligence, 2021). Decades of declining
and then sub-replacement fertility created a demographic dividend, where the
share of the working-age population far outweighed the share of children and
older adults no longer participating in the workforce (Bongaarts, 2009). Now
the average ages of populations around the world are increasing at the fastest
rate in history (Bloom  Zucker, 2023). International migration may blunt the
impact of this dynamic, but migration alone will not fully mitigate the
realities of the next demographic era (RAND, 2005; Coleman, 2008). Accounting
for past and forecast population age structure reveals that migration may be
an even less effective mitigation than currently understood (Münz, 2013; Lee,
2011). However, existing forecasting methods fail to account for this
eventuality, leaving the role of population age structure in migration
dynamics to imprecise qualitative conjecture.

In the period from 2000 through 2015, growth in the total number of migrants
in developed countries averaged 3.0% annually, outpacing the 0.6% annual
population growth in these nations (Woetzel ., 2016). More robust forecasts of
migration dynamics are critical to the long-term policymaking process as
countries compete for the same migrants to mitigate the impacts of sustained
sub-replacement birth rates. In the coming decades, the portfolio of migrant
supplying countries will have to evolve as birth rates fall, origin
populations age, and the share of migration-age people in historic migrant
supplying countries falls.

We propose a probabilistic net migration forecasting approach that accounts
for the impacts that population age structure had on historic migration rate
estimates. We use these to generate new probabilistic population forecasts for
all countries. The resulting forecasts quantify how different the global
population distribution could be after accounting for past and projected
shifts in the population age structure.

The article is organized as follows. Section 2 describes the background to our
study. Section 3 summarizes the data and methods used to generate age
standardized migration forecasts. It details each step of the population
forecasting method. This is followed in Section 4 by an evaluation of the age-
standardized migration method compared to forecasts that do not account for
the role of population age structure in population forecasting. Then Section 5
summarizes differences between population forecasts with and without
accounting for population age structure in the migration component of the
forecasts. We conclude in Section 6 with a discussion of the contributions and
limitations of our approach.

##  2 Background

The association between migration and age is long established. Rogers  Castro
(1981) identified a consistent pattern in the age profile of out-migrants and
proposed a model migration age schedule. Even though modern migration
estimation and forecasting methods use migration age schedules to account for
the impact of migration on the sending and receiving populations, population
age structure is rarely an explicit consideration in estimation or
forecasting. This is reasonable over short-term forecast horizons given the
relatively slow pace of population age structure changes, but less so over the
long term.

Kupiszewski (2002) discussed differences in migration forecasts when the
sending population age-structure is ignored. Specifically, it was argued that
ignoring the limited number of migration-age individuals in Poland and
ignoring natural population aging contributed to unrealistically large out-
migration forecasts from Poland to other European Union (EU) countries before
accession to the EU in 2004.

Fertig  Schmidt (2005) argued that neglecting population age in calculating
migration rates obscures the underlying population dynamics. They proposed an
adjusted net migration rate that is calculated as the number of migrants aged
0-39 divided by the sending country population aged 0-39. This approach
reduces distortions created by the population of older people who tend to make
up a relatively small proportion of migrants, but effectively sets older age
migration to zero and ignores the age structure of the critical 0-39 age
group.

Kolk (2019) reviewed age-specific migration and the analogy to age-specific
and total fertility rate definitions. It was argued that population-level
measures should be adopted for both subnational and international migration
estimation. The authors outlined key challenges to extending such methods to
an international context, specifically properly specifying the population at
risk of in-migration and differences in the definition of an international
migrant from country to country.

Fully probabilistic population forecasting is an active area of demographic
research (e.g. Hyndman  Booth, 2008; Raftery ., 2012, 2014; Wiśniowski .,
2015; Yu ., 2023). Azose . (2016) developed a probabilistic net migration
model and used it to produce probabilistic population forecasts for all
countries through 2100. Welch  Raftery (2022) proposed a fully probabilistic
population forecasting method that uses a probabilistic model of bilateral
migration flows for all countries through 2045. However, none of these
population forecasting methods systematically account for population age
structure changes in both the historic data used to fit these models and in
forecasting.

Raftery  Ševčíková (2023) proposed a method for accounting for population age
structure in probabilistic migration and population forecasts, and applied it
to very long term population forecasts, to 2300, motivated by the problem of
estimating the social cost of carbon. Like us, they used net migration data,
which has advantages of data availability and analytical simplicity over
methods based on more complete migration data, such as between-country flows
or in- and out-migration flows for countries. However, a problem with this is
that it does not take account of the different age distributions of in- and
out-migrants (Rogers, 1990). They avoided this problem by assuming that when a
country experienced net in-migration in a period, it was all in-migration and
there was no out-migration, and conversely for out-migration. Here we develop
a new method that avoids this unrealistic assumption while still using net
migration data.

##  3 Data & Methods

### Data

Recent methodological innovations make it possible to estimate globally
consistent country-level total migration inflows and outflows from Census
counts of the number of people by country of birth in all countries (Abel,
2013; Azose  Raftery, 2019). Abel  Cohen (2019) found that the pseudo-Bayes
method of Azose  Raftery (2019) to estimate bilateral migration flows led to
the most accurate migration estimates among several extant methods.
Unfortunately, pseudo-Bayes migrant flow estimates over five-year periods are
available for only six five-year periods starting in 1990 and ending in 2020
(Abel  Cohen, 2019). Estimating and forecasting bilateral migration flows has
a number of advantages (Welch  Raftery, 2022), but population age structure
changes too slowly to solely rely on these estimates for long-term migration
forecasting.

The United Nation’s World Population Prospects (WPP) publishes globally
consistent net migration estimates back to the 1950-1955 period (United
Nations, 2022). The net migration rate is defined as the difference between
migration inflows and outflows divided by population size. As such, the
population age structure at the given time is not accounted for.

The pseudo-Bayes flow estimates and net migration rate estimates have
independent value. The pseudo-Bayes flow estimates contain more detailed
information in the form of bilateral flows, but the length of such time series
is limited (Abel  Cohen, 2019). On the other hand, the availability of net
migration rate estimates going back to 1950 provides a measure of migration
over many more historic periods (United Nations, 2022). We use strengths of
both data sets to develop a new combined data series. Migration flow estimates
are calculated with respect to specific revisions of the WPP. The most recent
flow estimates were estimated using WPP 2019 (Abel  Cohen, 2019). We use WPP
2019 (United Nations, 2019) and the associated flow estimates for all
analyses.

We fit a model, described in Section 3, to the 1990-2020 period from which
both in-migration and net migration rates can be estimated for all countries
with populations of 100,000 or more. This model is then used to approximate
the contribution that in-migration made to the net migration rate prior to
1990. Our model-based approach is an efficient solution that allows for the
estimation of globally consistent age-standardized in-migration and out-
migration rates for five-year periods starting in 1950 and running through
2020. The successful decomposition of net migration rates into in-migration
and out-migration rates is critical to calculating migration rates adjusted
for the population age structure that gave rise to those rates.

### Age Standardization of Net Migration Rates

For reasons of data availability across all countries, and analytic
simplicity, the UN uses all-age net migration as part of the basis for its
population projections for all countries (United Nations, 2019). They project
this forwards deterministically, combining it with probabilistic projections
of fertility and mortality, to obtain probabilistic population projections
conditional on the projected levels of future migration. As an alternative,
Azose  Raftery (2015) proposed a probabilistic approach to forecasting future
net migration rates (i.e. net migration divided by population). Neither the
UN’s deterministic approach nor the probabilistic approach of Azose  Raftery
(2015) takes account of historic or future changes in population age
structure.

We propose to modify the approach of Azose  Raftery (2015) to account for
population age structure. We do this by creating an age-standardized version
of the net migration rate. This is tricky, because historic net migration data
are not disaggregated by age. Also, net migration is equal to in-migration
minus out-migration, and the age profiles of in- and out-migration can differ.
As pointed out by Rogers (1990), this makes it hard to standardize net
migration rates directly, and indeed this was one of the arguments of Rogers
(1990) for not using net migration at all. Here we propose a method for doing
so which hopefully reduces these concerns.

We start by developing a method for decomposing historic net migration into
in- and out-migration. This uses a subset of the data for which estimates of
both in- and out-migration are available (i.e. 1990-2020), and uses this to
estimate in- and out-migration for the entire data period (i.e. 1950-2020). We
then develop a method for age-standardizing out-migration. This turns out to
be very simple, relying on a quantity we call the migration age structure
index (MASI). We extend this to in-migration, and hence to net migration.
Finally we apply the probabilistic forecasting method of Azose  Raftery (2015)
to the age-standardized net migration rates, and input the resulting forecasts
to the overall probabilistic population projection method. The notation used
in describing our method is shown in Table 1.

Table 1: Notation for migration age-standardized migration method $O_{i,t}$ |  Integer-valued outflow from origin $i$ in period starting with year $t$
---|---
$O_{i,t,a}$ |  Integer-valued outflow from origin $i$ in period starting with year $t$
|  of age group $a$-$(a+4)$
$I_{i,t}$ |  Integer-valued inflow to destination $i$ in period starting with year $t$
$N_{i,t}$ |  Integer-valued net migrant flow in country $i$ in period starting with year $t$
$P_{i,t,a,s}$ |  Integer-valued population of origin $i$ at the start of period $t$ for age group
|  $a$-$(a+4)$ and sex $s\in\\{\text{male},\text{female}\\}$
$P_{i,t,+,+}$ |  Integer-valued population of origin $i$ at the start of period $t$
$\tilde{P}_{i,t,+,+}$ |  Integer-valued population of origin $i$ at risk of migration over period $t$ to $t+5$,
|  defined as $P_{i,t+5,+,+}-N_{i,t}$
$\mbox{OMR}_{i,t}$ |  $=O_{i,t}/\tilde{P}_{i,t,+,+}$. Out-migration rate for country $i$ in the period starting in year $t$
$\mbox{OMR}_{i,t,a}$ |  $=O_{i,t,a}/\tilde{P}_{i,t,a,+}$. Out-migration rate for country $i$, period starting in
|  year $t$ and age group $a$
$G_{i,t}$ |  $=\sum_{a}\mbox{OMR}_{i,t,a}$. Gross Migraproduction Rate (GMR) for country $i$ and period
|  starting in year $t$
$\mbox{IMR}_{i,t}$ |  $=I_{i,t}/\tilde{P}_{i,t,+,+}$. In-migration rate for period starting in year $t$ & country $i$
$\mbox{NMR}_{i,t}$ |  $=\mbox{IMR}_{i,t}-\mbox{OMR}_{i,t}$. Net migration rate for country $i$ & period $t$
$\pi_{i,t,a}$ |  $=\tilde{P}_{i,t,a,+}/\sum_{a}\tilde{P}_{i,t,a,+}$. Share of population from age group $a$ in period $t$
$\pi^{*}_{i,t,a}$ |  Reference population age distribution for country $i$ in period $t$
$R_{a}$ |  Reference migration age schedule normalized so that $\sum_{a}R_{a}=1$
$C_{i,t}$ |  $=\sum_{a}R_{a}\pi_{i,t,a}$. The migration age structure index (MASI), a scalar accounting
| for the population and migrant age structure in the migration rate at
|  time $t$ for country $i$
$\check{C}_{t}$ |  $=\sum_{i,a}R_{a}\pi_{i,t,a}$. MASI for the world at time $t$
$\mbox{OMR}_{i,t}^{\star}$ |  $=\mbox{OMR}_{i,t}\times C_{i,\text{baseline}}/C_{i,t}$. Age-standardized out-migration rate
$\mbox{IMR}_{i,t}^{\star}$ |  $=\mbox{IMR}_{i,t}\times\check{C}_{\text{baseline}}/\check{C}_{t}$. Age-standardized in-migration rate
$\mbox{NMR}_{i,t}^{\star}$ |  $=\mbox{IMR}_{i,t}^{\star}-\mbox{OMR}_{i,t}^{\star}$. Age-standardized net migration rate

#### Decomposing Historic Net Migration into In- and Out-migration

Net migration age-standardization takes place on the inflow and outflow
components of the net migration rate. Pseudo-Bayes estimates (Azose  Raftery,
2019; Abel  Cohen, 2019) of migration flows are available over five-year
periods starting in 1990 and ending in 2020 (6 periods). Net migration rate
estimates are available for five-year periods back to 1950 (United Nations,
2019). We use a mixed-effects model to relate the 1990-2020 in-migration rate,
$\mbox{IMR}_{i,t}$, to the net migration rate, $\mbox{NMR}_{i,t}$, for each
country $i$ over these six five-year periods, $t$, from 1990 through 2020. Our
mixed-effects model is defined as follows:

$\displaystyle\mbox{IMR}_{i,t}$
$\displaystyle=\beta_{0,i}+\beta_{1}\max\left(\mbox{NMR}_{i,t},0\right)+\varepsilon_{i,t},$
$\displaystyle\beta_{0,i}$
$\displaystyle\sim\text{Normal}\left(\beta_{0},\sigma^{2}_{between}\right),$
$\displaystyle\varepsilon_{i,t}$
$\displaystyle\sim\text{Normal}\left(0,\sigma^{2}_{within}\right).$ (1)

This model uses a random intercept term to account for the average country-
specific in-migration rate associated with the net migration rate for each
period from 1990-2020. Country intercepts, $\beta_{0,i}$, are concentrated
around a global mean intercept, $\beta_{0}$. The difference in a country’s
random intercept from the global mean, $\beta_{0}$, is determined by the
average difference from the global mean for that country in all periods.
Outflow rates are implicitly defined by this model. Table 1 defines each term
in the model.

The random intercept model was fit using the linear mixed-effects model
implementation in the lme4 R package implementation (Bates ., 2015). Starting
with $\mbox{NMR}_{i,t}$, the estimated in-migration rate is
$\mbox{IMR}_{i,t}=\beta_{0,i}+\beta_{1}\max\left(\mbox{NMR}_{i,t},0\right)$.
The net migration rate and estimated in-migration rate are then used to
calculate the out-migration rate,
$\mbox{OMR}_{i,t}=\mbox{IMR}_{i,t}-\mbox{NMR}_{i,t}$.

Figure 1 summarizes the 1990-2020 data, model fit, and association between
estimated and actual country-level migration flow rates. Points in Figure 1(a)
show the observed $\max\left(\mbox{NMR}_{i,t},0\right)$ on the horizontal axis
and $\mbox{IMR}_{i,t}$ on the vertical axis. The discontinuity at
$\mbox{NMR}_{i,t}=0$ reflects the fact that the in-migration rate must be non-
negative. In the $\mbox{NMR}_{i,t}\leq 0$ portion of the domain, the country-
specific intercept, $\beta_{0,i}$, of model 1 establishes the minimum average
magnitude of in-migration corresponding to a negative net migration rate for
that country. When $\mbox{NMR}_{i,t}>0$, the model in-migration rate increases
from this minimum average inflow at a rate of $\beta_{1}$ per unit increase in
$\mbox{NMR}_{i,t}$. The mean model in-migration rate is shown by the blue line
in Figure 1(a).

Figure 1: (a) Observed in-migration rates versus net migration rates for all
countries (points) with mean model (blue), LOESS line (black dashed), and
country-specific models for China (CHN, red) and Qatar (QAT, maroon); (b)
Observed in-migration rates versus fitted in-migration rates compared to $y=x$
(dashed line); (c) Observed out-migration rates versus fitted out-migration
rates compared to $y=x$ (dashed line) for five-year-periods from 1990-2020.

Many observations in Figure 1(a) are highly concentrated around some values.
The LOESS curve helps clarify the location of points that fall too close to
each other to be seen separately. In the $\mbox{NMR}_{i,t}<0$ portion of the
domain, the similarity between the LOESS line and the $\beta_{0,i}$ values
shows that the mean intercept term provides an accurate summary of the
observed in-migration and net migration rates. Near $\mbox{NMR}_{i,t}=0$,
however, the disagreement between the model mean and LOESS line shows that
many $\mbox{IMR}_{i,t}$ observations near $\mbox{NMR}_{i,t}=0$ could be
overstated with the model mean alone. The LOESS line departs from the mean
model most for large positive values of $\mbox{NMR}_{i,t}$. This is primarily
due to the influence of a few large positive $\mbox{NMR}_{i,t}$ observations
for a small number of countries.

Finally, Figure 1(a) shows two examples of country-specific average
association between $\mbox{IMR}_{i,t}$ and $\mbox{NMR}_{i,t}$. The top line
shows the model fit for Qatar (maroon), which experienced some of the highest
migration rates in the world since 1990. The bottom line shows the model fit
for China, a country with stable net out-migration over the same period. The
differences between these two examples highlight the need for a model flexible
enough to account for such wide variation between countries.

Figure 1(b) shows the fitted and observed $\mbox{IMR}_{i,t}$ for all countries
from 1990-2020. The high level of agreement between fitted and observed in-
migration rates with an $R^{2}=0.98$ is notable considering the simplicity of
model (1). The agreement between fitted and observed out-migration rates in
Figure 1(c) is lower with $R^{2}=0.91$, but is still strong. Taken together,
Figures 1(b)-(c) show that our model-based decomposition of the net migration
rate is reasonable for the periods where net migration, in-migration, and out-
migration rates are all available.

Figure 2 shows the observed net migration rate for five-year periods starting
in 1950 and running through 2020 on the original scales. The inflow and
outflow rates from 1990 through 2020 used for the estimation of our model are
shown in the second and third columns along with the mixed-effects model
decomposition. Our model-based estimates of inflow and outflow rates applied
to years prior to 1990 are also shown in the plots. Model-based estimates
relating net migration rates to in-migration and out-migration rates are
similar to the observed rates for 1990-2020. Estimated inflow and outflow
rates prior to the 1990-1995 period appear plausible for the periods where
pseudo-Bayes estimates are available.

Figure 2: Observed net migration (left column), decomposed into in-migration
(middle column), and out-migration rates (right column), on the scale of
annual migrants per thousand people, compared to mixed-effects model estimates
for the United States, Germany, Turkey, and Saudi Arabia. Solid blue lines
show the model-based estimates. Dashed black lines show the observed migration
rates used for the estimation.

While the model-based decomposition is not perfect, Figure 2 shows that the
mixed-effects net rate decomposition approach leads to plausible estimates of
in-migration and out-migration rates. The correlation between the observed and
predicted rates was over 0.97. Broad agreement between observed inflow and
outflow rates for 1990-2020 periods suggests that net migration rates prior to
1990 can be similarly decomposed into inflow and outflow rates.

Raymer . (2023) proposed an alternative approach to decomposing net migration
into in-migration and out-migration. This method uses a fixed constant to
approximate the magnitude of net migration attributed to total inflows and
outflows in proportion to a country’s population. This method could also be
used to decompose the net flow into total inflow and outflow in place of model
(1) for a subset of countries or all countries if desired. However, we found
the proportion of the in-migration to net migration rates using model (1) too
variable to justify using a single constant for all countries. The model (1)
offers a systematic approach to specifying a quantity similar to the fixed
constant used in Raymer . (2023), but one that is country-specific, which our
analyses suggest is needed.

#### Age Standardization of Out- and In-migration Rates

The net migration rate, NMRi,t, for country $i$ over period $t$ to $t+5$ is
defined as the difference between the in-migration rate, $\mbox{IMR}_{i,t}$,
and out-migration rate, $\mbox{OMR}_{i,t}$:

$\mbox{NMR}_{i,t}=\mbox{IMR}_{i,t}-\mbox{OMR}_{i,t}=\frac{N_{i,t}}{\tilde{P}_{i,t,+,+}}=\frac{I_{i,t}-O_{i,t}}{\tilde{P}_{i,t,+,+}}.$
(2)

Here, $I_{i,t}$ denotes the total inflow count, $O_{i,t}$ the total outflow
count, and $N_{i,t}=I_{i,t}-O_{i,t}$ the total net flow count over the period
starting in year $t$. We define the denominator of the net migration rate,
$\tilde{P}_{i,t,+,+}=P_{i,t+5,+,+}-N_{i,t}$, as the population of country $i$
at the end of the period starting in year $t$ before factoring in the change
due to net migration. This denominator specifies the population at risk of
migration over the period.

Historic net migration rate estimates are not disaggregated by age, but we aim
to standardize migration rates to remove the effects of population age
structure differences among countries in the same period and within countries
across periods. The age pattern of age-specific migration rates is known to be
relatively consistent over time and country of origin (Rogers  Castro, 1981),
but the rate defined in equation (2) obscures the influence of the sending and
receiving country population age structures, as discussed by Rogers (1990). A
population age-standardized net migration rate should account for the age
structure of the sending populations in both components of the net rate as the
migrant age distribution and totals are primarily linked to the origin
population age structure.

Let $\pi_{i,t,a}$ denote the population in age group $a$ as a proportion of
the total population in country $i$ at time $t$, namely

$\pi_{i,t,a}=\frac{\tilde{P}_{i,t,a}}{\sum_{a}\tilde{P}_{i,t,a}}\,.$ (3)

Since the origin population of inflows to country $i$ consists of every
country other than $i$ in period starting in year $t$, we approximate the
proportion of the global population in age-group $a$ by

$\check{\pi}_{t,a}\approx\frac{\tilde{P}_{+,t,a,+}}{\tilde{P}_{+,t,+,+}}\quad\text{where
}\quad\tilde{P}_{+,t,a,+}=\sum_{i,s}\tilde{P}_{i,t,a,s}\quad\text{and}\quad\tilde{P}_{+,t,+,+}=\sum_{i,s,a}\tilde{P}_{i,t,a,s}\,.$
(4)

The out-migration rate for country $i$, period $t$ and age group $a$ is
OMRi,t,a, and $G_{i,t}=\sum_{a}\mbox{OMR}_{i,t,a}$ is the Gross
Migraproduction Rate (GMR) for country $i$ and period $t$, as defined by
Rogers  Castro (1981). This is a measure of overall migration that is not
affected by population age distribution, rather as the Total Fertility Rate
(TFR) is a measure of overall fertility that is independent of the age
distribution of women.

The values of OMR${}_{i,t,a}/G_{i,t}$, which give the age pattern of age-
specific out-migration rates, tend to be stable over time and place,
reflecting a tendency for international migration to be largely concentrated
among people aged 15–35 and their dependent children, peaking in the twenties,
as pointed out by Rogers  Castro (1981). They also proposed a parametric model
for this pattern, the famous Rogers-Castro curve. As an approximation, we thus
consider the situation where this ratio is constant over time and space, so
that OMR${}_{i,t,a}/G_{i,t}=R_{a}$ for all $i,t$, where $\sum_{a}R_{a}=1$.
Under this assumption, the quantity $R_{a}$ is the same for all countries and
time periods, and could be modeled by a Rogers-Castro curve, or estimated
empirically.

We then have an exact result for the out-migration rate for a reference
population with a given age distribution:

######  Theorem 1.

Consider a population that has the same age-specific out-migration rates as
country $i$ in period $t$, but a different population age distribution given
by $\pi^{*}_{i,t,a}$. Then this population has out-migration rate

$\mbox{OMR}^{*}_{i,t}=\mbox{OMR}_{i,t}\frac{\sum_{a}\pi^{*}_{i,t,a}\mbox{OMR}_{i,t,a}}{\sum_{a}\pi_{i,t,a}\mbox{OMR}_{i,t,a}}.$
(5)

If both populations have the same age-specific pattern of migration rates,
$R_{a}$, then

$\mbox{OMR}^{*}_{i,t}=\mbox{OMR}_{i,t}\frac{C^{*}_{i,t}}{C_{i,t}},$ (6)

where $C_{i,t}=\sum_{a}\pi_{i,t,a}R_{a}$ is the migration age structure index
(MASI).

The proof of this theorem can be found in the Appendix.

This indicates that one can age-standardize the out-migration rate to a given
reference population using equation (6). This is a remarkably simple method,
since it involves only a simple multiplication by the MASI, $C_{i,t}$. The
MASI involves only the population age distribution and the invariant migration
age-pattern $R_{a}$, but not the age-specific migration rates themselves,
which cancel.

We can age-standardize in-migration by viewing it as out-migration to country
$i$ from the rest of the world. We approximate the age distribution of the
rest of the world by the age distribution of the world as a whole. This yields
an age-standardized in-migration rate, IMR${}^{*}_{i,t}$. Finally, we obtain
the age-standardized net migration rate as
NMR${}^{*}_{i,t}=\mbox{IMR}^{*}_{i,t}-\mbox{OMR}^{*}_{i,t}$.

Typically the reference population age distribution will be the distribution
in a particular year. In this article, we standardize to the age pattern of
2020. To calculate the age-standardized in-migration and out-migration rate of
period $t$ in terms of the 2020 population age structure, remove the
population age structure effects in period $t$ and scale the migration rate in
terms of the 2020 reference population:

$\displaystyle\mbox{IMR}_{i,t}^{\star}$
$\displaystyle=\mbox{IMR}_{i,t}\frac{\check{C}_{2020}}{\check{C}_{t}}$
$\displaystyle\mbox{OMR}_{i,t}^{\star}$
$\displaystyle=\mbox{OMR}_{i,t}\frac{C_{i,2020}}{C_{i,t}}$ (7)

where $\check{C}_{t}$ denotes the MASI for the world at time $t$.

This implies that the age-standardized net migration rate is given by
$\mbox{NMR}_{i,t}^{\star}=\mbox{IMR}_{i,t}^{\star}-\mbox{OMR}_{i,t}^{\star}$.
This measure of migration puts historic and forecast rates of migration into
the context of the 2020 population age structure, removing variation in
migration rates attributable to population age structure differences from
period to period. This specification also shows how to convert between the
age-standardized rates and the historic or future rates in terms of the
reference population.

After estimating historic inflow and outflow rates calculated from model (1),
age-standardized inflow, outflow, and net flow rates can be calculated for
forecasting.

#### Net Migration Rate Model

We use the resulting rates to develop a probabilistic model for age-adjusted
net migration rates. Using the above estimates of inflows and outflows, we
compute the age-standardized net migration rates, $\mbox{NMR}_{i,t}^{\star}$,
for all countries $i$ and for $t$ from 1950 through 2020, using the 2020
population age structures as the baseline population, as shown in Section 3.
We then fit the Bayesian hierarchical model of Azose  Raftery (2015).
Hyperparameter values for this model required no adjustment since the default
specifications were broadly defined and our specification of the net rate is
on the same scale as that of Azose  Raftery (2015). This is done using a
Markov chain Monte Carlo (MCMC) algorithm. This yields a sample of model
parameters.

### Forecasting

The goal here is to forecast migration and population probabilistically by
generating jointly a set of future migration and population trajectories. We
use the same approach to forecasting fertility and mortality as the UN
projections for the UN’s 2019 World Population Prospects (WPP) (United
Nations, 2019). Our approach to forecasting migration builds on the methods of
Azose  Raftery (2015) and Azose . (2016), but modifies them to adjust for
differences in population age distribution over time and between countries.

For each future trajectory $j\in\\{1,\dots,J\\}$, we independently project the
population for each country $i$ by five-year age group $a$ through $a+5$ and
sex $s$ for year $t+5$ without migration. The resulting projection,
$\tilde{P}_{i,t+5,a,s}^{(j)}$, denotes a realization of the total population
that would be observed in year $t+5$ had no one in the population migrated in
or out of each country. This approximates the population at risk of migrating
for the period $t$ to $t+5$, by age and sex. We then independently generate an
age-standardized net migration rate for the period $t$ to $t+5$,
$\mbox{NMR}_{i,t}^{\star(j)}$, using the parameter sample from the Bayesian
hierarchical model of Azose  Raftery (2015) estimated in the previous
subsection. The sampled age-standardized net migration rate is then decomposed
into the age-standardized in-migration rate, $\mbox{IMR}_{i,t}^{\star(j)}$,
and age-standardized out-migration rate, $\mbox{OMR}_{i,t}^{\star(j)}$, using
the mixed-effects model (1) that has been fit to age-standardized historical
rates (i.e., $\mbox{IMR}^{\star}$ and $\mbox{OMR}^{\star}$). The resulting
age-standardized coefficients are denoted by $\beta_{0}^{\star}$ and
$\beta_{1}^{\star}$.

The mean age-standardized in-migration rate for trajectory $j$ and country $i$
is calculated as

$\mbox{IMR}^{\star(j)}_{i,t}=\beta_{0,i}^{\star}+\beta_{1}^{\star}\max\left(\mbox{NMR}^{\star(j)}_{i,t},0\right).$
(8)

The corresponding out-migration rate for the period starting in year $t$ is
then given by

$\mbox{OMR}^{\star(j)}_{i,t}=\mbox{IMR}^{\star(j)}_{i,t}-\mbox{NMR}^{\star(j)}_{i,t}.$
(9)

Age-standardized inflow and outflow rates then need to be converted back to
period-specific rates to calculate inflow and outflow counts corresponding to
the projected population age structure:

$\displaystyle\mbox{IMR}_{i,t}^{(j)}$
$\displaystyle=\mbox{IMR}_{i,t}^{\star(j)}\times\check{C}_{t}^{(j)}/\check{C}_{2020},$
$\displaystyle\mbox{OMR}_{i,t}^{(j)}$
$\displaystyle=\mbox{OMR}_{i,t}^{\star(j)}\times C_{i,t}^{(j)}/C_{i,2020}.$
(10)

In-migration rates then are converted to inflow counts by age and sex,
$I_{i,t,a,s}^{(j)}$, by multiplying the inflow rate by the total country
population for trajectory $j$ at time $t$ and then applying a Rogers & Castro-
like migration age schedule. These counts are further disaggregated by sex in
proportion to the males and females in the population. Out-migration counts by
age and sex, $O_{i,t,a,s}^{(j)}$, are similarly calculated.

While a Rogers & Castro-like age schedule is appropriate for out-migration in
most countries, the migration schedules in Gulf Cooperation Council (GCC)
member states (Bahrain, Kuwait, Oman, Qatar, Saudi Arabia, United Arab
Emirates) call for a different approach. In recent decades, GCC country
migration has been dominated by large inflows of primarily male migrant
workers. This workforce supplies the labor required to power some of the
globe’s largest construction and infrastructure projects. These temporary
workers typically enter GCC countries on visas valid for two years or less.
The path from temporary worker to long-term resident or citizen is nearly
nonexistent in GCC countries. As a result, the population age structure of GCC
countries should be influenced by changes in the volume of migrant workers,
but the age distribution of the population should remain relatively constant
as worker visas expire, those workers leave the country, and new workers
arrive to replace them or add to their ranks. To maintain the population age
structure of the population due to temporary workers, we take a different
approach to the migration age schedule.

Instead of applying the standardized model migration age schedule in GCC
countries, we used a different schedule to maintain the foreign-worker-
dominated age structure. This modification ensures that the population age
distribution in GCC countries remain consistent with the distribution of
migrant worker stocks replaced by departing foreign workers. The out-migration
schedule was modified so that outflows are dominated by older working age
migrants, instead of the Rogers & Castro-like schedule used for all other
countries. After accounting for the age distribution of the population due to
in-migration, the outflow migration age schedule was determined by the
difference between the population age distribution and the normalized Roger &
Castro-like migration age schedule. This formulation effectively re-weights
the outflow schedule so that outflows are composed of older workers aging out
of prime working age groups (i.e., 15–65). Outflow rates in GCC countries have
been much smaller than inflow rates over recent decades, but the population
age distribution has changed little over this time. Hence, these inflow and
outflow migration schedule modifications ensure that the migrant worker
population remains in the prime working age range rather than having large
groups of migrant workers aging in place with the resident population. We
found that these adjustments led to much more plausible forecasts of GCC
migration and population dynamics through 2100 compared to the approach used
for the rest of the world.

It was necessary to rebalance global inflows and outflows each period to
maintain global net-zero migration. Azose  Raftery (2015) and Azose . (2016)
proposed methods for doing so, but their methods are not directly applicable
to a forecasting method that produces inflow and outflow counts for each
country. Adjusting net migration counts without specifying the proportion
allocated to inflows or outflows leads to inconsistencies in the projected
inflow and outflow totals compared to the net flow totals. We adjusted inflow
and outflow totals by the global net overflow/underflow by adjusting inflow
and outflow totals at the country-level in proportion to the country
population as a share of the globe.

The global rebalancing procedure is as follows. Let
$\tilde{I}_{i,t,a,s}^{(j)}$ denote the $j^{\text{th}}$ inflow count trajectory
for country $i$ for period starting in year $t$ for age group $a$ to $a+5$ and
sex $s$. Similarly, let $\tilde{N}_{i,t,a,s}^{(j)}$ denote the corresponding
net migration count before global rebalancing. Then the adjusted inflow and
outflow counts were calculated as

$\displaystyle I_{i,t,a,s}^{(j)}$
$\displaystyle=\tilde{I}_{i,t,a,s}^{(j)}-w\left(\sum_{k\in
K}\tilde{N}_{k,t,a,s}^{(j)}\right)\frac{\tilde{P}_{i,t,a,s}^{(j)}}{\sum_{k\in
K}\tilde{P}_{k,t,a,s}^{(j)}},$ $\displaystyle O_{i,t,a,s}^{(j)}$
$\displaystyle=\tilde{O}_{i,t,a,s}^{(j)}+(1-w)\left(\sum_{k\in
K}\tilde{N}_{k,t,a,s}^{(j)}\right)\frac{\tilde{P}_{i,t,a,s}^{(j)}}{\sum_{k\in
K}\tilde{P}_{k,t,a,s}^{(j)}},$ (11)

with $w=0.5$, splitting the global net overflow/under-flow equally among
inflows and outflows. The index $K$ defines two groups of countries used for
normalization. Rebalancing was done among GCC countries (Bahrain, Kuwait,
Oman, Qatar, Saudi Arabia, United Arab Emirates) and the labor supplying
origin countries (Bangladesh, India, Indonesia, Philippines, Pakistan) in one
group. The second group was composed of all other countries outside the GCC
labor corridor. After this adjustment, the global net migration for each age
group and sex equals zero and the net flow of migrants for trajectory $j$ in
country $i$ at time $t$ by age and sex becomes
$N_{i,t,a,s}^{(j)}=I_{i,t,a,s}^{(j)}-O_{i,t,a,s}^{(j)}$. As in Azose  Raftery
(2015), these adjustments led to relatively small changes to forecasts.

We then recalculated the balanced age-standardized net migration rate,
$\mbox{NMR}_{i,t}^{\star(j)}$, as

$\mbox{NMR}_{i,t}^{\star(j)}=\left(\frac{I_{i,t,+,+}^{(j)}}{\tilde{P}_{i,t+5,+,+}^{(j)}}\right)\times\check{C}_{2020}/\check{C}_{t}^{(j)}-\left(\frac{O_{i,t,+,+}^{(j)}}{\tilde{P}_{i,t+5,+,+}^{(j)}}\right)\times
C_{i,2020}/C_{i,t}^{(j)},$ (12)

which will be the rate jump-off for sampling in the next time period. Finally,
we account for net migration in the population projection generated at the
start of the forecasting routine,

$P_{i,t+5,a,s}^{(j)}=\tilde{P}_{i,t+5,a,s}^{(j)}+N_{i,t,a,s}^{(j)},$ (13)

before moving to the next forecast period.

##  4 Validation

Age-standardized forecast efficacy was evaluated in terms of out-of-sample
performance. Figure 3 shows the observed and four-period migration and
population forecasts for four countries starting in the 2000-2005 period
through the 2015-2020 period. Columns show the United States, El Salvador,
South Africa, and Saudi Arabia forecasts using 1995-2000 migration rate
persistence (effectively the current UN method) for every period, the age-
agnostic method of Azose . (2016), and the age-standardized method described
above. Rows show the migration age structure index (MASI) ratio, i.e. the
ratio of the MASI for the period to the MASI for the baseline year 2000, the
net migration rate, and population for each country. Prediction intervals for
both the age-agnostic and age-standardized method included the observed
population and net migration rates in all cases. Population and migration
forecasts using the age-standardized approach were as good or better than the
age-agnostic forecasts by the end of the period for all cases except in El
Salvador in terms of agreement with the observed quantities. Persistence of
1995-2000 migration rates performed especially poorly for El Salvador and
Saudi Arabia.

Figure 3:  Migration age structure index (MASI) ratio for each country ( ) and
the globe ( ) with base-year 2000, out-of-sample validation forecast of
population (millions of people), and age-standardized and age-agnostic net
migration rate (net annual migrants per thousand), for four countries.
Forecasts use probabilistic age-standardized net migration ( ), probabilistic
age-agnostic net migration ( ), observed fertility, and observed mortality.
Dashed lines in each plot indicate the observed values. Solid lines indicate
the median forecast. Shaded regions show the 80% prediction interval.
Migration models were fit to 1950-2000 data. Forecasts are for the 2000-2005,
2005-2010, 2010-2015 and 2015-2020 periods.

Age-standardized and age-agnostic forecasts largely agree with one another
since population age structure differences make little impact over such a
short period of time. Note that the proportion of the migration age population
increased relative to the baseline population in El Salvador. The 0–15 year
olds were the largest proportion of the population in 2000; however, the
proportion of the population aged 0–15 fell during the validation period and
the large 0–15 population cohort aged into the prime migration age cohorts by
the end of the 2015-2020 period. As a result, El Salvador’s age structure
ratio increased from the 2000 baseline period. The age structure ratio
forecast outpaced the observed age structure ratio for the first two periods
(2000-2010), but the prediction interval contained the observed age structure
ratio for the last two periods (2010-2020).

Global age ratio forecasts were well aligned with the true global age
structure. Country-level population age ratio forecasts showed mixed results,
however. Prediction intervals for the MASI ratio captured all or most of the
observed values for the United States and South Africa. The forecast MASI
ratio for Saudi Arabia was lower than the true value for the first two
periods, but included the observed value by the end of the forecast period.
The forecast MASI ratio for El Salvador was consistently higher than the
observed value from 2000 to 2010. These departures from the forecast are
explained by the higher net migration observations compared to the forecasts
for those countries.

Global and country-level MASI ratio values equal one for the baseline
population structure (2000 in this case) by definition. When the global MASI
is higher than the country-level MASI, then we expect higher in-migration
using the age-standardized method compared to the age-agnostic method.
Conversely, when the country-level MASI ratio is below 1, we expect lower net
out-migration compared to the baseline period using the age-standardized
method and a model that does not account for population age structure. The
MASI for the United States was below the global MASI and the baseline MASI
value for the whole forecast period. The United States’ MASI ratio also
indicated an aging population compared to the baseline population age
structure. Both of these factors should contribute to less out-migration from
the United States as the global population had a net positive supply of
migration-age people and the United States had a shrinking net positive supply
of migration age people compared to the baseline population. Indeed, the age-
standardized net migration forecast for the United States was shifted towards
higher net in-migration compared to the age-agnostic forecast.

The forecast MASI ratios for El Salvador and for South Africa were higher than
the global MASI ratios and above the baseline period, indicating that the age-
standardized migration forecast should be shifted towards more net out-
migration compared to the age-agnostic forecast. The age-standardized net
migration rate was indeed more negative than the age-agnostic migration rate
for El Salvador. However, age-standardized and age-agnostic net migration
forecasts were indistinguishable for South Africa given the relatively small
share of population change attributable to migration.

The forecast MASI ratio for Saudi Arabia was higher than the global MASI ratio
and above the baseline period, indicating that the age-standardized migration
forecast should be shifted towards more net out-migration compared to the age-
agnostic forecast if the forecasting methods were otherwise identical.
However, the age-standardized approach to forecasting Gulf Cooperation Council
(GCC) uses a different approach to GCC net migration forecasting that makes
the age-standardized and age-agnostic comparisons less relevant in this case.
Still, net migration forecasts generated from the age-standardized method were
more similar to the observed rates. Since migration is the main contributor to
population change in GCC countries, the small improvement in the age-
standardized forecast led to a substantial improvement in the population
forecast compared to the age-agnostic approach.

Multiple other forecast horizons were also evaluated. Predictions were
generated one to four periods ahead of the last observed data. Out-of-sample
forecasts use the last observed population age structure as the baseline, fit
each model to data available prior to the first forecast period, and generate
forecasts for each period through the 2015-2020 period. We generated one-
period-ahead forecasts for periods starting in 2000, 2005, 2010, and 2015.
Two-period-ahead forecasts were generated for periods starting in 2000, 2005,
and 2010. Three-period-ahead forecasts were generated for periods starting in
2000 and 2005. The four-period-ahead forecast was generated for the period
starting in 2000 only. We did not evaluate the five-period-ahead forecast as
it is not possible to fit the mixed-effects model (1) with only one period of
inflow and outflow data.

We evaluated the point forecasts in terms of the Mean Absolute Error (MAE),
the Log Mean Absolute Error (LMAE), and the Mean Absolute Scaled Error (MASE).
The LMAE is defined as follows:

$\displaystyle l(y)$
$\displaystyle=\text{sign}(y)\left[\log\left(\left|y\right|+c\right)-\log
c\right]\text{ with }c>0$ $\displaystyle\text{sign}(y)$
$\displaystyle=\begin{cases}1&y>0\\\ 0&y=0\\\ -1&y<0\end{cases}$
$\displaystyle LMAE_{t}$
$\displaystyle=\frac{1}{200}\sum_{i=1}^{200}\left|l(f_{i,t})-l(r_{i,t})\right|.$
(14)

We used $c=1$. The LMAE formulation prevents large errors in a few forecasts
from dominating the error metric.

The MASE, recommended by Hyndman  Koehler (2006), summarizes forecast
performance in terms of the mean ratio of errors from a proposed method in the
numerator and errors from a naïve method such as persistence forecast in the
denominator. Error estimates in the denominator are estimated from mean errors
of the naïve method using the in-sample data, e.g. mean net migration rate
errors generated from the 1950-2000 data using the naïve method in the
denominator compared to mean forecast error in the 2000-2020 data. Let
$f_{i,t}$ denote the net migration rate forecast for country $i$ and period
starting in year $t$ when the true net migration rate is $r_{i,t}$. The MASE
of the forecast for $T_{k}$ forecast periods and $S_{k}$ in-sample periods
starting in forecast period $t_{0}$ to $t_{0}+5$ and in-sample period $s_{0}$
to $s_{0}+5$ with a forecast horizon $k$ periods ahead of the last observation
is defined as

$\text{MASE}_{k}=\frac{\frac{1}{200\times
T_{k}}\sum_{i=1}^{200}\sum_{\\{t_{0}\\}_{k}}\left|r_{i,t_{0}+5(k-1)}-f_{i,t_{0}+5(k-1)}\right|}{\frac{1}{200\times
S_{k}}\sum_{i=1}^{200}\sum_{\\{s_{0}\\}_{k}}\left|r_{i,s_{0}+5k}-r_{i,s_{0}}\right|}.$
(15)

This formula leads to the ratio of average $k$-period ahead forecast errors
for all possible horizons $k$ in the in-sample and out-of-sample periods. For
the $k=1$ one-period-ahead forecasts,
$\\{t_{0}\\}_{1}=\\{2000,2005,2010,2015\\}$, $T_{1}=4$,
$\\{s_{0}\\}_{1}=\\{1950,1955,\dots,1990\\}$, and $S_{1}=9$. For the $k=4$
four-period-ahead forecast, $\\{t_{0}\\}_{4}=\\{2000\\}$, $T_{4}=1$,
$\\{s_{0}\\}_{4}=\\{1950,1955,\dots,1975\\}$, and $S_{4}=6$.

Forecast calibration was evaluated in terms of 95% prediction interval
coverage and average prediction half-interval width. If a model is well-
calibrated, then 95% prediction intervals generated from the model should
contain about 95% of the true net migration rate observations. Prediction
intervals that contain less than 95% of the true values are too narrow,
indicating that the model underestimates forecast variation. Prediction
intervals that contain more than 95% of the true values are too wide,
indicating that the model overestimates forecast variation. Prediction half-
interval width is calculated as half the difference between the upper
prediction interval quantile and lower prediction interval quantile. Accurate,
well-calibrated models with smaller half-interval widths are preferred to
models with the same accuracy and calibration estimates.

Table 2 summarizes out-of-sample predictive performance of the age-
standardized, age-agnostic, and net rate persistence methods. Errors are
calculated as the difference between the observed value and the median of
2,000 posterior predictive distribution draws. Rate persistence uses the last
observed net rate in each country as the forecast. The best score for each
method is shown in bold font.

Table 2: Mean predictive performance of different methods for net migration rate (migrants per thousand period person years): mean absolute error (MAE), log mean absolute error (LMAE) with $c=1$, mean absolute scaled error (MASE), 95% prediction interval coverage, and mean prediction interval half-width (HW). | Method | MAE | LMAE | MASE | Cover | HW
---|---|---|---|---|---|---
5 Years | Persistence | 4.02 | 0.68 | 1.42 | — | —
| Agnostic | 3.44 | 0.65 | 1.25 | 93 | 10.47
| Standardized | 3.54 | 0.65 | 1.28 | 93 | 10.39
10 Years | Persistence | 5.33 | 0.88 | 1.63 | — | —
| Agnostic | 3.86 | 0.76 | 1.16 | 91 | 11.63
| Standardized | 3.93 | 0.77 | 1.17 | 90 | 11.44
15 Years | Persistence | 5.16 | 1.00 | 1.44 | — | —
| Agnostic | 3.49 | 0.83 | 1.10 | 92 | 12.32
| Standardized | 3.51 | 0.82 | 1.11 | 92 | 11.85
20 Years | Persistence | 4.77 | 1.02 | 1.27 | — | —
| Agnostic | 2.91 | 0.82 | 1.00 | 94 | 12.54
| Standardized | 2.86 | 0.79 | 0.98 | 94 | 12.04

The age-agnostic and net rate persistence models were fit using the bayesPop R
package (Ševčíková  Raftery, 2016). The age-standardized model was fit using a
custom implementation of the bayesPop R package. The age-standardized and the
age-agnostic model accuracy were similar across all horizons, but age-
standardized forecast accuracy overtakes age-agnostic forecast accuracy as the
number of periods between the last observations and the forecast period
increased for all metrics. The age-standardized model was better calibrated in
terms of 95% prediction interval coverage for all horizons except the one-
period-ahead forecast, but both models had slight under-coverage compared to
the 95% nominal rate. The age-agnostic and age-standardized models
outperformed the rate persistence model forecast in all horizons. Similarities
between the agnostic and standardized model accuracy are expected over short
forecast horizons since population age structure changes slowly.

##  5 Results

Figure 4 summarizes probabilistic population forecasts in four countries using
the age-agnostic migration model from Azose . (2016), the population forecasts
using the age-standardized migration model, and the projections from the 2019
World Population Prospects (WPP 2019) (United Nations, 2019). Columns of the
figure correspond to the same country. Rows of the figure show forecasts
through 2100 for the MASI ratio, the net migration rate, and total population.

Figure 4: 2020 MASI ratios (top row), net migration rate as net annual
migrants per thousand (middle row), and probabilistic forecast of population
(in millions) age-standardized and age-agnostic (bottom row) for the United
States, El Salvador, South Africa, and Saudi Arabia. Forecasts use
probabilistic net migration ( =age-standardized and  =age-agnostic), as well
as probabilistic fertility, and mortality. The MASI ratio plots show the
values for each country ( ) and the world ( ). Solid lines in each plot
indicate the observed and median forecast. Shaded regions show the 80%
prediction interval. Forecasts start in the 2020-2025 period.

The MASI ratios in the first row of Figure 4 summarize the differences in the
population age structure from the 2020 baseline for each country (green)
compared to the population age structure for the world (purple). Countries
with a higher migration-age population than the 2020 population baseline rise
above 1. Countries aging faster than the world population fall below the
global index. Forecast population age structures in the United States and
South Africa generally follow the global aging trend.

When the country-level MASI ratio is similar to the global one, then net
migration forecasts from the age-standardized model and age-agnostic model
should be more similar than in countries where the country-level and global
MASI ratios diverge. The population age structures of El Salvador and Saudi
Arabia are forecast to age much faster than the global average. In these
cases, the age adjustment shifts the migration forecast towards higher net
inflows, as we see for El Salvador and Saudi Arabia.

Median net migration forecasts from both probabilistic net migration models
are similar in many countries, but long-term age-standardized net migration
forecast intervals tend to be narrower than age-agnostic forecast intervals.
This is explained by the population age normalization step used in the age-
standardized method. Age-standardized net migration rate variation
attributable to variation in historic population age structure is removed
before fitting the Azose  Raftery (2015) net migration model. Removing
variation in past net migration estimates attributable to population age
structure reduces the variance in the net migration model parameter estimates
and hence the migration and population forecasts. Prediction intervals (80%)
for both probabilistic net migration models include the United Nations’ WPP
2019 projections through 2100 for all countries shown in Figure 4.

Median net migration forecasts from the age-agnostic and age-standardized
methods are most similar in countries that have population age indices similar
to the global population age index. The population age indices along with the
median migration forecasts for the United States and South Africa in Figure 4
demonstrate this trend. Age-standardized migration forecasts in countries
where the population age structure diverges from the global norm often leads
to much different median migration forecasts compared to the age-agnostic
migration model. For example, El Salvador’s population is forecast to age much
more rapidly from 2020 to 2100 compared to the global age structure. As a
result, the median age-standardized net migration forecast is shifted towards
lower net out-migration compared to the age-agnostic forecast. Since the age
adjustment leads to less negative net migration, the age-standardized
population forecast is higher than the forecast using the age-agnostic model.
Furthermore, the age-standardized 80% prediction interval upper bound shows
that there is less certainty that the population size will peak by 2100
compared to the age-agnostic model.

Figure 4 also shows that Saudi Arabia’s population age structure is forecast
to change about as fast as El Salvador’s compared to the 2020 baseline
population age structure. Saudi Arabia’s net migration forecast is also
shifted towards higher net in-migration, but this leads to a higher net
positive migration forecast compared to the age-agnostic model. The age-
standardized net positive migration forecast is better aligned to the historic
net rate data in Saudi Arabia and the UN’s WPP 2019 net migration forecast.
Differences in Saudi Arabia’s migration and population forecast using the age-
standardized approach arise from both population age structure effects and
adjustments to modeling unique features of migration in GCC countries. These
adjustments lead to a substantially higher median population forecast and
prediction interval. The UN WPP 2019 population forecast falls to the bottom
of the age-standardized prediction interval compared to the upper middle of
the interval using the age-agnostic model. See the supplemental index for
detailed forecasts for all countries.

Figure 5 summarizes past and median forecast MASI ratios for the globe and
several countries by global region. The MASI ratio provides a concise one-
number summary of historic and forecast population age structure dynamics.
While mean country and global age also summarize the degree of aging, the MASI
is more informative in terms of assessing changes to the inherent force of
migration attributable to the relevant populations at risk. The MASI
explicitly weights age groups by a Rogers & Castro-like model migration age
schedule. Taking the ratio of period MASI values puts the structural
demographic force of migration potential within a country into the context of
the reference population. The role of the MASI in removing the influence of
population age structure is a more relevant measure of population age dynamics
for migration than other measures that are less directly related to a
population’s structural migration potential, such as its average age.

Figure 5: Historic and median country-level forecasts of MASI ratios for 2020
baseline by region compared to the world.

Figure 5 shows that the median population age structure forecasts in Western
countries follows the global population aging trend. This indicates that the
underlying force of out-migration in Western countries is forecast to align
with the global average compared to the population age structure in 2020 and
the median net migration forecasts from the age-standardized and age-agnostic
methods will largely agree with one another. The underlying force of migration
from Latin America by contrast falls sharply in coming decades compared to the
2020 baseline and the world overall, indicating that the force of out-
migration will fall faster than the world on average and lead to higher net
migration forecasts using the age-standardized method. The population age
index forecast for some countries has yet to reach their peak, e.g. Rwanda in
Africa and Samoa in Oceania. This implies that forecast out-migration from
these countries is too small in the age-agnostic method.

Age-standardized migration forecasts led to less steep population declines in
countries facing the greatest demographic challenges. Figure 6 shows the age
index, net migration, and population forecasts among large countries that
anticipate some of the most drastic population contractions by 2100. The age-
standardized net migration forecast leads to less severe population forecast
declines than the age-agnostic forecast as there are fewer people of prime
migration age in the population. The age index shows that the force of out-
migration falls in these countries as they age compared to the rest of the
world, shifting the net migration forecast towards higher in-migration
compared to the age-agnostic forecast. Median age-standardized net migration
forecasts in these countries are more similar to the UN’s WPP 2019 migration
and population forecasts than the age-agnostic forecasts.

Figure 6: 2020 base-year MASI ratios (top row), age-standardized and age-
agnostic net migration rate as net annual migrants per thousand (middle row),
and probabilistic forecast of population (in millions) age-standardized and
age-agnostic (bottom row) for Singapore, South Korea, Greece, Poland.
Forecasts use probabilistic net migration ( =age-standardized and  =age-
agnostic), fertility, and mortality. Age-index ratio plots show the age
structure ratios for each country ( ) and world ( ). Solid lines in each plot
indicate the observed and median forecast. Shaded regions show the 80%
prediction interval. Forecasts start in the 2020-2025 period.

Figure 7 shows the world and regional forecasts using the age-agnostic and
age-standardized migration forecast models. World forecasts of population are
indistinguishable, which makes sense as both methods use the same fertility
and mortality models. Global forecasts could have differed due to differences
in fertility and mortality norms among migrants’ destination countries, but
these differences were not large enough to substantially alter the global
population forecasts or overall uncertainty.

Figure 7: Probabilistic forecast of population (millions of people) by region.
Forecasts use probabilistic net migration ( =age-standardized and  =age-
agnostic), fertility, and mortality. Solid lines indicate the median forecast.
Dashed lines indicate the observed population. Shaded regions show the 80%
prediction interval. Forecasts start in the 2020-2025 period.

Africa’s population forecast by the end of the century is lower under the age-
standardized model compared to the agnostic migration model. This makes sense
because the force of migration associated with population age structure in
many African countries is forecast to far exceed the global median for most of
the remainder of the century. As a result, long-term forecasts that ignore
population age structure for many countries in Africa are slightly overstated
compared to the age standardized migration model. End-of-century median
population forecasts for all other regions are higher using the age-
standardized migration model.

The most pronounced differences are for Europe and North America. Age-
standardized posterior prediction intervals for Europe and North America lie
within the age-agnostic prediction intervals. The most extreme population
declines in Europe and Latin America are less likely under the age-
standardized model. Migration forecasts from the age-standardized model are
lowered by aging populations in both regions. The most extreme population
growth and decline trajectories in North America are less likely under the
age-standardized model.

Table 3 details the differences in the population forecasts using the age-
standardized and age-agnostic migration forecasting methods shown in Figure 7.

Table 3: Median population and net migration forecasts at the end of the century by United Nations Area and the world using age-standardized (S) and age-agnostic (A) migration forecasting in millions of people along with the net difference (S-A) and percent difference 100$\times\frac{S-A}{A}$ %.  UN Area | Measure | S | A | S - A |  100$\times\frac{S-A}{A}$ %
---|---|---|---|---|---
Africa | population | 4142.6 | 4231.3 | -88.7 | -2.0
| net migration | -10.5 | -6.3 | -4.3 | 68.0
Asia | population | 4880.3 | 4806.0 | 74.4 | 2.0
| net migration | 0.1 | -1.2 | 1.3 | -109.0
Europe | population | 631.6 | 621.8 | 9.8 | 2.0
| net migration | 3.4 | 2.8 | 0.7 | 24.0
Latin America | population | 694.4 | 668.3 | 26.2 | 4.0
& the Caribbean | net migration | 0.7 | -1.4 | 2.1 | -153.0
Northern America | population | 439.3 | 434.6 | 4.6 | 1.0
| net migration | 4.4 | 3.9 | 0.5 | 12.0
Oceania | population | 65.1 | 62.6 | 2.5 | 4.0
| net migration | 0.4 | 0.2 | 0.2 | 132.0
World | population | 10863.3 | 10884.4 | -21.1 | 0.0

Figure 8 shows the population forecast differences at the end of this century
for each country for three different forecasts. Each point denotes the
departure of the median population forecast by 2100 using the age-agnostic or
age-standardized migration model from the median WPP 2019 forecast. These
ratios make it possible to evaluate differences in all three forecasts
together. Points falling near the origin indicate approximate agreement
between the age-agnostic, age-standardized, and WPP 2019 median population
forecasts. Points on along $y=0$ indicate agreement between median age-
agnostic and WPP 2019 forecasts that differ from the age-standardized
forecast. Points on $x=0$ indicate agreement between the age-standardized
forecast and the WPP 2019 forecast that differ from the age-agnostic forecast.
Points on $y=x$ far from the origin denote agreement between the age agnostic
and age-standardized forecasts that depart from the WPP 2019 forecast. All
other combinations are summarized by the plot regions bounded by the dashed
lines and labeled I-VI on the plot.

Figure 8: Three-way comparison of median 2100 population forecasts using age-
standardized migration model, age-agnostic migration model, and median WPP
2019 forecast. Point color indicates UN Area, point diameter indicates WPP
2019 median population forecast size (millions), and regions I-VI show
direction of forecast differences using the age-agnostic and age-standardized
migration models.

Figure 8 shows that all three median population forecasts differ for Eastern
Sahara (ESH). Both the age-agnostic and age-standardized methods lead to
higher population forecasts for the United Arab Emirates (ARE) than WPP 2019,
but the median age-standardized forecast is largest. The median age-
standardized population forecast was highest among the three for all GCC
countries. The age-standardized method generated fewer negative net migration
trajectories over the forecast period; however, this outcome is consistent
with the last three decades of sustained positive net migration that has been
dominated by foreign workers responding to labor opportunities in rapidly
expanding economies. Net migration rate forecasts using the age-standardized
approach continue the net positive trend in GCC countries with expanding
populations. While the age-standardized migration forecasts for GCC countries
is higher than the age-agnostic approach, the population age structure in most
GCC countries appears much more plausible and closer to the WPP 2019 forecast
than the age-agnostic method.

The Supplementary Information includes forecasts for the 200 largest
countries, associated quantiles, and additional information about the software
used to generate these results.

##  6 Discussion

The association between migration and age is long established. However, we are
unaware of any existing migration forecasting method that explicitly accounts
for the influence of overall population age structure, apart from that of
Raftery  Ševčíková (2023), which relied on an unrealistic assumption. The
potential support ratio (PSR), defined as the proportion of the working age
population to the total population, has been used as a proxy for population
age structure in other methods (K. Kim  Cohen, 2010). However this approach
does not distinguish a population age distribution concentrated at ages 20-39
from one more heavily concentrated at ages 40-65, although the implications of
these for migration are very different.

Our approach addresses multiple criticisms of using net migration as a unit of
analysis. Our model-based approach to decompose the net flow rate into inflow
and outflow rates alleviates the migration age schedule issues outlined in
Rogers (1990). He also pointed out that relatively small net migration counts
over a period can hide comparatively huge inflows and outflows of roughly
equal magnitude. While net migration is the unit of analysis here, we use the
historic data within a country to decompose net flow rate estimates into
inflow and outflow rates. Dividing inflow and outflow counts by the age-
standardized population size makes it possible to put flow rates from a
diverse population of countries around the globe on the same scale for
modeling and forecasting.

While the methods used in age-standardized net migration estimation and
forecasting partly address long-standing criticisms of net migration rate
analysis, a bilateral migration flow model among all countries as in Welch
Raftery (2022) eliminates theoretical challenges induced by the use of net
migration rate as the unit of analysis. However, the computational complexity
of migration flow forecasting and limited historic bilateral flow data reduce
the viability of this alternative for longer-term forecasting. International
bilateral migration flow forecasting among 200 countries involves estimating
and forecasting 39,800 bilateral flows as opposed to 200 net flow rates.
Furthermore, there are only six quinquennial periods of bilateral migration
flow estimates available at present (Abel  Cohen, 2019) compared to thirteen
periods for net migration (United Nations, 2019). Methods described in this
article offer a more computationally tractable alternative to bilateral
migration flow model fit to only half the historical net migration data.

Probabilistic forecasts offer a principled means to integrate uncertainty, but
historic shocks to long-term trends arise without warning. Multiple such
shocks occurred in the last ten years alone. The 2020 COVID-19 pandemic,
Syria’s decade old civil war, Russia’s invasion in Ukraine, and the economic
meltdown in Venezuela all upset long-term demographic norms—especially
migration. New norms could take hold as a result of these and other crises,
but the full effect of these historic events should be integrated into updated
forecasts once the data become available. The age-standardized probabilistic
migration forecasting method presented here includes well-defined mechanisms
to account for changes to variation around long-term trends. When the full
extent of recent crises emerges in the data, the forecasts should be revised
with these data.

Using age-standardized net migration aims to eliminate the influence of
population age structure on historic and forecast net migration rates,
improving on existing probabilistic net migration forecasting methods.
However, key drawbacks of the use of net migration are not completely
addressed by our method. Shifts in net migration forecast due to changes in
country population age structure are directly reflected in the forecast for
the country of origin; however, using the population age structure of the
globe to normalize inflow rates could be improved to more directly reflect the
changes in population age structures of sending countries. For example, Mexico
and Central American countries are currently the largest suppliers of
immigrants to the United States. Calculating a country-specific inflow age
ratio index to reflect that fact rather than a simple average of the global
population age distribution would more directly link changes in the population
age structure of top sending countries on the age-standardized net migration
rate. We chose to use the global average in our models because it is far
simpler analytically and to account for the possibility that migration
corridors that exist now could change in the future.

Our methods build on methodological advances made in the last decade (Azose
Raftery, 2015), address several known challenges with net migration as the
unit of analysis (Rogers, 1990), and offer a more computationally tractable
alternative to migration flow forecasting (Welch  Raftery, 2022). Our model-
based approach to decomposing net migration rates into inflow and outflow
rates offers an efficient alternative to more computationally challenging
analyses involving direct modeling of bilateral migration flows. We propose a
new method to account for unique differences in forecasting net flows in Gulf
Cooperation Council countries that more accurately reflects migration dynamics
in those countries in the future. Finally, we found the MASI ratio to be a
concise one-number summary of the population age structure that measures the
pace of aging expected to take place in the coming decades in the context of
migration. Reducing the influence of population age structure on historic and
forecast migration rates leads to more accurate long-term forecasts and
matches or narrows the migration and population prediction intervals compared
to a comparable age-agnostic method.

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##  Appendix A Proof of Theorem 1

###### Proof.

Starting from the definition of $\mbox{OMR}_{i,t}$,

$\displaystyle\mbox{OMR}_{i,t}$
$\displaystyle=\frac{O_{i,t}}{\tilde{P}_{i,t,+,+}}$
$\displaystyle\left(\mbox{definition of OMR}\right)$
$\displaystyle=\sum_{a}\frac{O_{i,t,a}}{\tilde{P}_{i,t,a,+}}\times\frac{\tilde{P}_{i,t,a,+}}{\tilde{P}_{i,t,+,+}}$
$\displaystyle\left(\mbox{OMR disaggregated by age}\right)$
$\displaystyle=G_{i,t}\sum_{a}\frac{\mbox{OMR}_{i,t,a}}{G_{i,t}}\times\pi_{i,t,a}$
$\displaystyle\left(\mbox{Multiplication by }\frac{G_{i,t}}{G_{i,t}}\right).$

The same derivation applies to $\mbox{OMR}_{i,t}^{*}$, implying that

$\mbox{OMR}_{i,t}^{*}=G_{i,t}\sum_{a}\frac{\mbox{OMR}_{i,t,a}}{G_{i,t}}\times\pi_{i,t,a}^{*}.$
(17)

Rearranging $\mbox{OMR}_{i,t}$ in terms of $G_{i,t}$ and substituting the
equation into (17) yields the desired result:

$\mbox{OMR}_{i,t}^{*}=\mbox{OMR}_{i,t}\frac{\sum_{a}\mbox{OMR}_{i,t,a}\pi_{i,t,a}^{*}}{\sum_{a}\mbox{OMR}_{i,t,a}\pi_{i,t,a}}.$
(18)

Multiplying (18) by $\frac{G_{i,t}}{G_{i,t}}$, assuming the same
$R_{a}=\mbox{OMR}_{i,t,a}/G_{i,t}$ in both populations, and applying the
definition of $R_{a}$, we obtain

$\mbox{OMR}_{i,t}^{*}=\mbox{OMR}_{i,t}\frac{\sum_{a}R_{a}\pi_{i,t,a}^{*}}{\sum_{a}R_{a}\pi_{i,t,a}}=\mbox{OMR}_{i,t}\frac{C^{*}_{i,t}}{C_{i,t}},$
(19)

as desired. This concludes the proof. ∎