This paper discusses likelihood-type inference in parameter-driven models for regression analysis of non-normal data in presence of serial correlation. Since the ordinary likelihood function involves an intractable high-dimensional integral, we consider a pairwise likelihood approach that requires to approximate a limited set of two-dimensional integrals. Maximization of the pairwise likelihood is carried out with a pairwise version of the expectation-maximization algorithm. The methodology is illustrated with surveillance data to evaluate the relationship between influenza and menigoccocal infections. Results are in close agreement with Bayesian inference based on the integrated nested Laplace approximation.