  The United Nations is the major organization producing and regularly updating
probabilistic population projections for all countries. International migration
is a critical component of such projections, and between-country correlations
are important for forecasts of regional aggregates. However, there are 200
countries and only 12 data points, each one corresponding to a five-year time
period. Thus a $200 \times 200$ correlation matrix must be estimated on the
basis of 12 data points. Using Pearson correlations produces many spurious
correlations. We propose a maximum a posteriori estimator for the correlation
matrix with an interpretable informative prior distribution. The prior serves
to regularize the correlation matrix, shrinking a priori untrustworthy elements
towards zero. Our estimated correlation structure improves projections of net
migration for regional aggregates, producing narrower projections of migration
for Africa as a whole and wider projections for Europe. A simulation study
confirms that our estimator outperforms both the Pearson correlation matrix and
a simple shrinkage estimator when estimating a sparse correlation matrix.
