Dynamic Count Models with Flexible Innovation Processes for Irregular Maritime Migration

Abstract

Motivated by the challenge of analyzing the dynamics of weekly sea border crossings in the Mediterranean (2015-2025) and the English Channel (2018-2025), we develop a Bayesian dynamic framework for modeling heteroskedastic count time series. Building on theoretical considerations and empirical stylized facts, our approach utilizes a Poisson random walk model that allows for heavy-tailed innovations or stochastic volatility dynamics, while incorporating an explicit mechanism to separate structural from sampling zeros. Posterior inference is carried out via a straightforward Markov chain Monte Carlo algorithm. Applying this methodology to the Mediterranean and English Channel data, we compare alternative model specifications through a comprehensive out-of-sample density forecasting exercise. Evaluating each model using log predictive scores and empirical coverage up to the 99th percentile, we find strong evidence for stochastic volatility in the migration innovations, with these models producing well-calibrated forecasts even at extreme quantiles. Our framework can be used to develop risk indicators with direct policy implications for improving governance and preparedness for migration surges. More broadly, the methodology extends to other zero-inflated non-stationary count time series applications, including epidemiological surveillance and public safety incident monitoring.

Figures (8)

• Figure 1: Irregular maritime border crossing case study datasets. Left: Mediterranean Sea border crossings to Italy by sample week from 2015W40 to 2025W11. Right: English Channel crossings by sample week from 2018W01 to 2025W11. Top panels display log-transformed counts, with zero counts highlighted separately as dots on the zero line. Bottom panels show log differences of counts for observations where both $y_t>0$ and $y_{t-1}>0$.
• Figure 2: Top row: One-step ahead predictive densities and realized observations (black line) over the 250 holdout periods. Zero observations are highlighted separately on the $x$-axis. Grey shaded areas are 90% credible intervals. Bottom row: Cumulative log predictive scores relative to the stochastic volatility model. The Poisson model without zero inflation is not shown due to non-competitive performance. Left column: Mediterranean crossing data. Right column: English Channel crossing data. Holdout period runs from 2020W23 to 2025W11. Results are computed across out-of-sample periods where $y_{T+1}>0$ and conditional on $s_{T+1}=1$.
• Figure 3: Top row: Crossing counts and density fit by sample week on a logarithmic scale. Zero counts are highlighted separately on the horizontal axis. Middle row: Estimated posterior density of $h_t$. Bottom row: Estimated posterior mean probability of $y_t$ being a structural zero. Left column: Mediterranean crossings by sample week from 2015W40 to 2025W11. Right column: English Channel crossings by sample week from 2018W01 to 2025W11. Solid lines are posterior means. Grey shaded areas indicate 90% credible intervals.
• Figure S1: Simulation results. In each panel, the top row shows counts and fitted predictive distributions by time period on a logarithmic scale (grey shaded areas denote 90% credible intervals; zero counts are highlighted separately on the zero line). The bottom row shows estimated 90% credible intervals for $h_t$ (grey shaded) and true values (black line). Note that the $y$-axes differ across panels.
• Figure S2: Sea border crossing case study datasets. Left: Mediterranean Sea border crossings to Italy by sample week from 2015W40 to 2025W11. Right: English Channel sea border crossings by sample week from 2018W01 to 2025W11. Panels display counts, with zero counts highlighted as dots on the zero line.