  In this paper we propose a new time-varying econometric model, called
Time-Varying Poisson AutoRegressive with eXogenous covariates (TV-PARX), suited
to model and forecast time series of counts. {We show that the score-driven
framework is particularly suitable to recover the evolution of time-varying
parameters and provides the required flexibility to model and forecast time
series of counts characterized by convoluted nonlinear dynamics and structural
breaks.} We study the asymptotic properties of the TV-PARX model and prove
that, under mild conditions, maximum likelihood estimation (MLE) yields
strongly consistent and asymptotically normal parameter estimates.
Finite-sample performance and forecasting accuracy are evaluated through Monte
Carlo simulations. The empirical usefulness of the time-varying specification
of the proposed TV-PARX model is shown by analyzing the number of new daily
COVID-19 infections in Italy and the number of corporate defaults in the US.
