*Jonathan Azose is a PhD student at the University of Washington, United States. His* *research group focuses on probabilistic population projections, which looks to be an upcoming **area in demography.*

In July 2014, the UN published probabilistic population projections for the first time. Probabilistic projections are nice in that they provide not only point projections, but also uncertainty bounds. This particular set of population projections incorporates uncertainty about fertility and mortality trends, but doesn’t extend this probabilistic modeling methodology to migration, the third necessary component of population projections.

Migration is the black sheep of the population projection family. In many cases, migration is difficult to measure accurately, let alone predict. Rwanda provides a good example. Imagine it’s 1990 and you’re trying to project future migration rates for Rwanda. Here are the data you’d have to work from:

Any reasonable person (or algorithm) would look at Rwanda’s history and predict continued stability in migration rates. From those data, it looks like Rwanda’s migration rate fort he next time period is unlikely to be more than 5 (the units here are net annual migrants per thousand). And here’s what actually happened:

Again, the scale here is “net annual migrants per thousand”, so the value of nearly -50 in 1990-1995 means that roughly 5% of Rwanda’s population emigrated annually over that period, a net loss of nearly 25% of Rwanda’s population through emigration. The following time period saw a similarly high in-migration rate. Of course, these mass migration events were a result of refugee movements related to the Rwandan genocide. This was a particularly high-volume migration, but Rwanda is not unique. Large migration movements in the 20th century have resulted from political changes, wars and economic booms–factors which are nearly unpredictable in the long run. All of which is to say that projecting migration is tricky, because migration is known to depend on a host of unpredictable factors.

But an optimistic statistician might look at all those unpredictable factors and say they’re good news, because we can compress them into a random error term in a statistical model. We’ve developed a statistical model for migration projection that does exactly that. We model migration rates *rc,t * in country *c *and time *t* as being related to migration rates in the previous time period rc,t-1 using the following formula:

where εc,t follows a normal distribution with variance δ^{2}. Each country has a δ^{2} parameter that controls variance, a μc parameter that controls long-term mean behavior, a φc that controls rapidity of decay towards that mean. The country-specific mean parameters account for the fact that countries may have different long term average tendencies—some may tend to be net receivers of migrants and others may be net senders, for example. A requirement that the φc parameters all be between 0 and 1 forces each country’s migration rates to trend on average back towards their long-term mean. (The model is really a Bayesian hierarchical model, and full specification of prior distributions can be found here)

In practice, to make projections, we first estimate parameters from historical data. We then apply our model to generate migration trajectories for all countries. There’s a little wrinkle because we know that net migration must sum to zero across all countries. (That is, there’s no migration into our out of the planet.) So we fix our simulated trajectories by redistributing overflow migrants in proportion to population size. Finally, summaries of those trajectories give us the predictive medians and intervals that make up our probabilistic migration projections. One example of such probabilistic projections with a few sample trajectories is shown below.

Denmark is a typical European country from a migration perspective. In recent history it has been a net receiver of migrants, but we only have to go back to the 1950s to find a time when it was a net sender. If we only reported the median projections (the red “x”s on the plot), it would look like Denmark was projected to have continue in-migration indefinitely into the future. With the full probabilistic projections, though, we get a more detailed picture. The two sets of intervals shown for the future are 80% and 95% predictive bounds. From these bounds, our predictions give at least a 10% probability of Denmark becoming a net out-migration country again at all time points from 2015 onwards, with increasing probability as time goes on.

Our methodology does a lot of things well. The easiest way to evaluate performance is to hold out some data for evaluation. We imagine, for example, that we only had data from 1950 to 1980 and then use our model to “predict” migration rates for 1980-2010. From this sort of out-of-sample comparisons, we find that our technique outperforms existing migration projection methodology. It provides probabilistic projections which are suitable for incorporation into a probabilistic population projection framework.

One final important aspect is that we accurately reproduce global trends in migration. One underreported trend in the data is the fact that nearly all of the world’s largest countries have higher absolute migration rates now than they did 60 years ago. In the plot below, absolute migration rates for the 25 largest countries in 2005-2010 are plotted in orange next to the 1950-1955 corresponding figures in blue. Out of 25 countries, 23 had higher absolute migration rates in 2005-2010. Our model on average predicts continued growth in absolute migration rates among these very large countries.

Why does this trend matter? Absolute migration rates serve as a measure of the contribution of migration to population change. This plot shows us that the impact of migration as a source of population change is growing in the largest countries, and growing substantially at that. Migration projections have often been overlooked in population projections, since they’ve historically been a small source in population variation. The plot above suggests that now is an excellent time to think about the future of migration.

*Interested in more methodological details, comparisons to other models, and a more detailed discussion? See the full paper here. Migration projections for all countries are available on request from jonazose@uw.edu.*

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