I present two flexible models of multivariate, count data regression that make use of the Sarmanov family of distributions. This approach overcomes several existing difficulties to extend Poisson regressions to the multivariate case, namely: i) it is able to account for both over and underdispersion, ii) it allows for correlations of any sign among the counts, iii) correlation and dispersion depend on different parameters, and iv) constrained maximum likelihood estimation is computationally feasible. Models can be extended beyond the bivariate case. I estimate the bivariate versions of two of these models to address whether the pricing strategies of competing duopolists in the early U.S. cellular telephone industry can be considered strategic complements or substitutes. I show that a Sarmanov model with double Poisson marginals outperforms the alternative count data model based on a multivariate renewal process with gamma distributed arrival times because the latter imposes very restrictive constraints on the valid range of the correlation coefficients.
